Tensor products of C -algebras and operator spacespisier/TPCOS.pdfTensor products of C-algebras and operator spaces The Connes-Kirchberg problem by Gilles Pisier November 9, 2019

Tensor products of C∗-algebras and operator spaces

The Connes-Kirchberg problem

byGilles Pisier

November 9, 2019

Contents

Introduction vi

1 Completely bounded and completely positive maps: basics 71.1 Completely bounded maps on operator spaces . . . . . . . . . . . . . . . . . . . . . . 81.2 Extension property of B(H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Completely positive maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4 Normal c.p. maps on von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . 221.5 Injective operator algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.6 Factorization of completely bounded (c.b.) maps . . . . . . . . . . . . . . . . . . . . 241.7 Normal c.b. maps on von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . 281.8 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 Completely bounded and completely positive maps: a tool kit 292.1 Rows and columns. Operator Cauchy-Schwarz inequality . . . . . . . . . . . . . . . . 292.2 Automatic complete boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3 Complex conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4 Operator space dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.5 Bi-infinite matrices with operator entries . . . . . . . . . . . . . . . . . . . . . . . . . 362.6 Free products of C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.7 Universal C∗-algebra of an operator space . . . . . . . . . . . . . . . . . . . . . . . . 412.8 Completely positive perturbations of completely bounded maps . . . . . . . . . . . . 422.9 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 C∗-algebras of discrete groups 453.1 Full (=Maximal) group C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Full C∗-algebras for free groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3 Reduced group C∗-algebras. Fell’s absorption principle . . . . . . . . . . . . . . . . . 513.4 Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.5 Group von Neumann algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.6 Amenable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.7 Operator space spanned by the free generators in C∗λ(Fn) . . . . . . . . . . . . . . . 603.8 Free products of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.9 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4 C∗-tensor products 624.1 C∗-norms on tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2 Nuclear C∗-algebras (a brief preliminary introduction) . . . . . . . . . . . . . . . . . 654.3 Tensor products of group C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4 A brief repertoire of examples from group C∗-algebras . . . . . . . . . . . . . . . . . 684.5 States on the maximal tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . 694.6 States on the minimal tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . 714.7 Tensor product with a quotient C∗-algebra . . . . . . . . . . . . . . . . . . . . . . . 744.8 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

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5 Multiplicative domains of c.p. maps 765.1 Multiplicative domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.2 Jordan multiplicative domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.3 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Decomposable maps 806.1 The dec-norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.2 The δ-norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.3 Decomposable extension property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.4 Examples of decomposable maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.5 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7 Tensorizing maps and functorial properties 977.1 (α→ β)-tensorizing linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.2 ‖ ‖max is projective (i.e. exact) but not injective . . . . . . . . . . . . . . . . . . . . 1017.3 max-injective inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.4 ‖ ‖min is injective but not projective (i.e. not exact) . . . . . . . . . . . . . . . . . . 1087.5 min-projective surjections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.6 Generating new C∗-norms from old ones . . . . . . . . . . . . . . . . . . . . . . . . . 1147.7 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8 Biduals, injective von Neumann algebras and C∗-norms 1168.1 Biduals of C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1168.2 The nor-norm and the bin-norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1168.3 Nuclearity and Injective von Neumann algebras . . . . . . . . . . . . . . . . . . . . . 1178.4 Local reflexivity of the maximal tensor product . . . . . . . . . . . . . . . . . . . . . 1228.5 Local reflexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1268.6 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

9 Nuclear pairs, WEP, LLP and QWEP 1309.1 The fundamental nuclear pair (C∗(F∞), B(`2)) . . . . . . . . . . . . . . . . . . . . . 1309.2 C∗(F) is residually finite dimensional . . . . . . . . . . . . . . . . . . . . . . . . . . . 1349.3 WEP (Weak Expectation Property) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1369.4 LLP (Local Lifting Property) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409.5 To lift or not to lift (global lifting) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1439.6 Linear maps with the WEP and the LLP . . . . . . . . . . . . . . . . . . . . . . . . 1469.7 QWEP C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1479.8 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

10 Exactness and nuclearity 15110.1 The importance of being Exact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15210.2 Nuclearity, exactness and approximation properties . . . . . . . . . . . . . . . . . . . 15610.3 More on nuclearity and approximation properties . . . . . . . . . . . . . . . . . . . . 16010.4 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

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11 Traces and ultraproducts 16211.1 Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16211.2 Tracial probability spaces and the space L1(τ) . . . . . . . . . . . . . . . . . . . . . 16411.3 The space L2(τ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16611.4 An example from free probability: semicircular and circular systems . . . . . . . . . 17011.5 Ultraproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17211.6 Factorization through B(H) and ultraproducts . . . . . . . . . . . . . . . . . . . . . 17811.7 Hypertraces and injectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18511.8 The factorization property for discrete groups . . . . . . . . . . . . . . . . . . . . . . 18711.9 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

12 The Connes embedding problem 18912.1 Connes’s question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18912.2 The approximately finite dimensional (i.e. “hyperfinite”) II1-factor . . . . . . . . . . 19512.3 Hyperlinear groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19612.4 Residually finite groups and Sofic groups . . . . . . . . . . . . . . . . . . . . . . . . . 19712.5 Random matrix models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19912.6 Characterization of nuclear von Neumann algebras . . . . . . . . . . . . . . . . . . . 20012.7 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

13 Kirchberg’s conjecture 20213.1 LLP ⇒ WEP ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20213.2 Connection with Grothendieck’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . 20413.3 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

14 Equivalence of the two main questions 20914.1 From Connes’s question to Kirchberg’s conjecture . . . . . . . . . . . . . . . . . . . . 20914.2 From Kirchberg’s conjecture to Connes’s question . . . . . . . . . . . . . . . . . . . . 21014.3 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

15 Equivalence with finite representability conjecture 21315.1 Finite representability conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21315.2 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

16 Equivalence with Tsirelson’s problem 21416.1 Unitary correlation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21416.2 Correlation matrices with projection valued measures . . . . . . . . . . . . . . . . . . 21616.3 Strong Kirchberg conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22116.4 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

17 Property (T) and residually finite groups. Thom’s example 22217.1 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

18 The WEP does not imply the LLP 22518.1 The constant C(n): WEP 6⇒ LLP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22718.2 Proof that C(n) = 2

√n− 1 using random unitary matrices . . . . . . . . . . . . . . 230

18.3 Exactness is not preserved by extensions . . . . . . . . . . . . . . . . . . . . . . . . . 23318.4 A continuum of C∗-norms on B⊗ B . . . . . . . . . . . . . . . . . . . . . . . . . . . 23518.5 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

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19 Other proofs that C(n) < n. Quantum expanders 23719.1 Quantum coding sequences. Expanders. Spectral gap . . . . . . . . . . . . . . . . . . 23719.2 Quantum expanders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23919.3 Property (T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24119.4 Quantum spherical codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24319.5 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

20 Local embeddability into C and non-separability of (OSn, dcb) 24520.1 Perturbations of operator spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24520.2 Finite dimensional subspaces of C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24720.3 Non-separability of the metric space OSn of n-dimensional operator spaces . . . . . 25020.4 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

21 WEP as an extension property 25521.1 WEP as a local extension property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25521.2 WEP versus approximate injectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 25721.3 The (global) lifting property LP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25921.4 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

22 Complex interpolation and maximal tensor product 26022.1 Complex interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26022.2 Complex interpolation, WEP and maximal tensor product . . . . . . . . . . . . . . . 26322.3 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

23 Haagerup’s Characterizations of the WEP 27223.1 Reduction to the σ-finite case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27223.2 A new characterization of generalized weak expectations and the WEP . . . . . . . . 27323.3 A second characterization of the WEP and its consequences . . . . . . . . . . . . . . 27523.4 Preliminaries on self-polar forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27723.5 max+-injective inclusions and the WEP . . . . . . . . . . . . . . . . . . . . . . . . . 28023.6 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28623.7 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

24 Full crossed products and failure of WEP for B ⊗min B 29024.1 Full crossed products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29124.2 Full crossed products with inner actions . . . . . . . . . . . . . . . . . . . . . . . . . 29424.3 B ⊗min B fails WEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29624.4 Proof that C0(3) < 3 (Selberg’s spectral bound) . . . . . . . . . . . . . . . . . . . . . 30424.5 Other proofs that C0(n) < n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30524.6 Random permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30624.7 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

25 Open problems 308

26 Appendix: Miscellaneous background 31026.1 Banach space tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31026.2 A criterion for an extension property . . . . . . . . . . . . . . . . . . . . . . . . . . 31226.3 Uniform convexity of Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . 31326.4 Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

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26.5 Ultraproducts of Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31526.6 Finite representability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31526.7 Weak and weak* topologies. Biduals of Banach spaces . . . . . . . . . . . . . . . . . 31626.8 The local reflexivity principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31726.9 A variant of Hahn-Banach Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 31826.10 The trace class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31926.11 C∗-algebras. Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32026.12 Commutative C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32126.13 States and the GNS construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32226.14 On ∗-homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32326.15 Approximate units, ideals and Quotient C∗-algebras . . . . . . . . . . . . . . . . . . 32526.16 von Neumann algebras and their preduals . . . . . . . . . . . . . . . . . . . . . . . . 32726.17 Bitransposition. Biduals of C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . 33126.18 Isomorphisms between von Neumann algebras . . . . . . . . . . . . . . . . . . . . . 33426.19 Tensor product of von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . . . 33526.20 On σ-finite (countably decomposable) von Neumann algebras . . . . . . . . . . . . . 33526.21 Schur’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

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Introduction

These lecture notes are centered around two open problems, one formulated by Alain Connes in hisfamous 1976 paper [61], the other one by Eberhard Kirchberg in his landmark 1993 paper [155]. Atfirst glance, these two problems seem quite different and the proof of their equivalence describedat the end of [155] is not so easy to follow. One of our main goals is to explain in detail the proofof this equivalence in an essentially self-contained way. The Connes problem asks roughly whethertraces on “abstract” von Neumann algebras can always be approximated (in a suitable way) byordinary matrix traces. The Kirchberg problem asks whether there is a unique C∗-norm on thealgebraic tensor product C ⊗C when C is the full C∗-algebra of the free group F∞ with countablymany generators.

In the remarkable paper where he proved the equivalence, Kirchberg studied more generally thepairs of C∗-algebras (A,B) for which there is only one C∗-norm on the algebraic tensor productA⊗B. We call such pairs “nuclear pairs”. A C∗-algebra A is traditionally called nuclear if this holdsfor any C∗-algebra B. Our exposition chooses as its cornerstone Kirchberg’s theorem asserting thenuclearity of what is for us the “fundamental pair”, namely the pair (B,C ) where B = B(`2)(see Theorem 9.6). Our presentation leads us to highlight two properties of C∗-algebras, the WeakExpectation Property (WEP) and the Local Lifting Property (LLP).

The first one is a weak sort of extension property (or injectivity) while the second one is aweak sort of lifting property. The connection with the fundamental pair is very clear: A has theWEP (resp. LLP) if and only if the pair (A,C ) (resp. (A,B)) is nuclear. With this terminology,the Kirchberg problem reduces to proving the implication LLP ⇒ WEP, but there are many moreinteresting reformulations that deserve mention and we will present them in detail. For instancethis problem is equivalent to the question whether every (unital) C∗-algebra is a quotient of onewith the WEP, or equivalently, in short, is QWEP. In passing, although the P stands for property,we will sometimes write for short that A is WEP (or A is LLP) instead of A has the WEP (resp.LLP).

Incidentally, since Kirchberg (unlike Connes) explicitly conjectured a positive answer to all theseequivalent questions in [155], we often refer to them as his conjectures.

One originality of our treatment (although already present in [155]) is that we try to underlinethe structural properties (or their failure), such as injectivity or projectivity, in parallel for theminimal and the maximal tensor product of C∗-algebras. This preoccupation can be traced backto the “fundamental pair” itself: Indeed, we may view B as “injectively universal” and C as“projectively universal”. The former because any separable C∗-algebra A is a subalgebra of B, thelatter because any such A is a quotient of C (see Proposition 3.39).

In particular, we will emphasize the fact that the minimal tensor product is injective but notprojective, while the maximal one is projective but not injective (see §7.4 and 7.2). This is analogousto the situation that prevails for the Banach space tensor products in Grothendieck’s classical work,but unlike Banach space morphisms (i.e. bounded linear maps) the C∗-algebraic morphisms areautomatically isometric if they are injective (see Proposition 26.24). The lack of injectivity of themax-norm is a rephrasing of the fact that if B1 ⊂ B2 is an isometric (or equivalently injective)∗-homomorphism between C∗-algebras and A is another C∗-algebra, it is in general not true thatthe resulting ∗-homomorphism

(0.1) A⊗max B1 → A⊗max B2

is isometric (or equivalently injective). This means that the norm induced by A ⊗max B2 on thealgebraic tensor product A⊗B1 is not equivalent to the max-norm on A⊗B1. In sharp contrast,

1

this does not happen for the min-norm: A⊗min B1 → A⊗min B2 is always injective (=isometric),and this is why one often says that the minimal tensor product is “injective”.

This “defect” of the max-tensor product leads us to single out the class of inclusions, B1 ⊂ B2,for which this defect disappears (i.e. (0.1) is injective for any A). We choose to call them “max-injective”. We will see that this holds if and only if there is a projection P : B∗∗2 → B∗∗1 with‖P‖ = 1. We will also show that if (0.1) is injective for A = C then it is injective for all A.

It turns out that a C∗-algebra A is WEP if and only if the embedding A ⊂ B(H) is max-injective or, equivalently, if and only if there is a projection P : B(H)∗∗ → A∗∗ with ‖P‖ = 1.All these facts have analogues for the min-tensor product, but now its “defect” is the failure of“projectivity”, meant in the following sense: Let q : B1 → B2 be a surjective ∗-homomorphismand let A be any C∗-algebra. Let I = ker(q). Then, although the associated ∗-homomorphismqA : A ⊗min B1 → A ⊗min B2 is clearly surjective (indeed, it suffices for that to have a denserange), its kernel may be strictly larger than A⊗min I. As a result, the min-norm on the algebraictensor product A ⊗ B2 (= A ⊗ (B1/I)) may be much smaller than the norm induced on it by(A ⊗min B1)/(A ⊗min I). In sharp contrast, this “defect” does not happen for the max-norm andwe always have an isometric identification

A⊗max (B1/I) = (A⊗max B1)/(A⊗max I).

Again this defect of the min-norm leads us to single out the quotient maps (i.e. the surjective∗-homomorphisms) q : B1 → B2 for which the defect does not appear, i.e. the maps such that forany A we have an isometry

(0.2) A⊗min B2 = (A⊗min B1)/(A⊗min I).

Here again, we can give a rather neat characterization of such maps, this time as a certain form oflifting property, see §7.5. It turns out that if (0.2) holds for A = B then it holds for all C∗-algebrasA. We call such a map q a “min-projective surjection”. The usual terminology to express that(0.2) holds for any A is that B1 viewed as an extension of B2 by I is a “locally split extension” (weprefer not to use this term). This notion is closely connected with the notion of exact C∗-algebra.

A C∗-algebra A is called exact if (0.2) holds for any surjective q : B1 → B2. This “exact”terminology is motivated by the fact that (0.2) holds if and only if the sequence

0→ A⊗min I → A⊗min B1 → A⊗min B2 → 0

is exact. But actually, for C∗-algebras, the exactness of that sequence boils down to the fact thatthe natural ∗-homomorphism

A⊗min B1

A⊗min I→ A⊗min B2

is isometric (=injective).Although our main interest is in C∗-algebras, it turns out that many results have better for-

mulations (and sometimes better proofs) when phrased using linear subspaces of C∗-algebras (theso-called operator spaces) or unital self-adjoint subspaces (the so-called operator systems). It isthus natural to try to describe as best as we can the class of linear transformations that preservethe C∗-tensor products. For the minimal norm, it is well known that the associated class is thatof completely bounded (c.b.) maps. More precisely, given a linear map u : A → B betweenC∗-algebras we have for any C∗-algebra C

(0.3) ‖IdC ⊗ u : C ⊗min A→ C ⊗min B‖ ≤ ‖u‖cb

2

where ‖u‖cb is the c.b. norm of u. Moreover, the sup over all C of the left-hand side of (0.3) isequal to ‖u‖cb, and it remains unchanged when restricted to C ∈ Mn | n ≥ 1. The space of suchmaps is denoted by CB(A,B).

The mapping u is called completely positive (in short c.p.) if IdC ⊗u : C⊗minA→ C⊗minB ispositive (=positivity preserving) for any C, and to verify this we may restrict to C = Mn for anyn ≥ 1. The cone formed of all such maps is denoted by CP (A,B).

For the max tensor product, there is an analogue of (0.3) but the corresponding class of map-pings is smaller than CB(A,B). These are the decomposable maps denoted by D(A,B), definedas linear combinations of maps in CP (A,B). More precisely, for any u as previously we have

(0.4) ‖IdC ⊗ u : C ⊗max A→ C ⊗max B‖ ≤ ‖u‖dec,

where ‖u‖dec is the norm in D(A,B). Moreover, the supremum over all C of the left-hand sideof (0.4) is equal to the dec-norm of u composed with the inclusion B ⊂ B∗∗. The dec-norm wasintroduced by Haagerup in [104]. We make crucial use of several of the properties established byhim in the latter paper. See chapter 6.

The third class of maps that we analyze are the maps u : A→ B such that for any C

‖IdC ⊗ u : C ⊗min A→ C ⊗max B‖ ≤ 1.

This holds if and only if u is the pointwise limit of a net of finite rank maps with ‖u‖dec ≤ 1 (seeProposition 6.13). When u is the identity on A this means that A has the c.p. approximationproperty (CPAP) which, as is by now well known, characterizes nuclear C∗-algebras (see Corollary7.12).

More generally, suppose given two C∗-norms α and β, defined on A ⊗ B for any pair (A,B).We denote by A⊗αB (resp. A⊗β B) the C∗-algebra obtained after completion of A⊗B equippedwith α (resp. β).Then we say that a linear map u : A → B between C∗-algebras is (α → β)-tensorizing if for anyC∗-algebra C

‖IdC ⊗ u : C ⊗α A→ C ⊗β B‖ ≤ 1.

In §7.1 we describe the factorizations characterizing such maps in all the cases when α and β areeither the minimal or the maximal C∗-norm. We also include the case when u is only defined ona subspace E ⊂ A using the norm induced on C ⊗ E by C ⊗α A. The main cases of interest aremin → max (nuclearity) and max → max (decomposability). For the former, we refer to chapter10, where we characterize nuclear C∗-algebras in parallel with exactness.

The bidual A∗∗ of a C∗-algebra A is isomorphic to a von Neumann algebra. In chapter 8 westudy the relations between C∗-norms on A and on A∗∗ and we describe the biduals of certain C∗-tensor products. The notion of local reflexivity plays an important role in that respect. We provein §8.3 the equivalence of the injectivity of A∗∗ and the nuclearity of A. In Corollary 7.12 (provedin §10.2) we show that for C∗-algebras nuclearity is equivalent to the completely positive approxi-mation property (CPAP). We also show in Theorem 8.12 that injective von Neumann algebras arecharacterized by a weak* analogue of the CPAP, which is sometimes called “semidiscreteness”.

But our main emphasis is on nuclear pairs: in §9.1 we prove the nuclearity of the fundamentalpair (B,C ) and in the rest of chapter 9 we give various equivalent characterizations of C∗-algebraswith the properties WEP, LLP and QWEP, that we choose to define using nuclear pairs. The mainones are formulated using the bidual A∗∗ of a C∗-algebra A (see §8.1). Let iA : A → A∗∗ be thenatural inclusion. For instance:(i) A is nuclear if and only if for some (or any) embedding A∗∗ ⊂ B(H) there is a projection

3

P : B(H)→ A∗∗ with ‖P‖cb = 1.(ii) A is WEP if and only if for some (or any) embedding A ⊂ B(H) there is a projection P :B(H)∗∗ → A∗∗ with ‖P‖cb = 1.(iii) A is QWEP if and only if for some embedding A∗∗ ⊂ B(H)∗∗ there is a projection P : B(H)∗∗ →A∗∗ with ‖P‖cb = 1.

We then come to the central part of these notes: the Connes embedding problem whether anytracial probability space embeds in an ultraproduct of matricial ones (chapter 12) and the Kirchbergconjecture (chapter 13) that C is WEP or that every C∗-algebra is QWEP. We show that theyare equivalent in chapter 14. We also show the equivalence with a well known conjecture fromBanach space theory (chapter 15). The latter essentially asserts that every von Neumann algebrais isometric (as a Banach space) to a quotient of B(H) for some H. In yet another direction weshow in chapter 16 that all these conjectures are equivalent to one formulated by Tsirelson in thecontext of quantum information theory.

In one of its many equivalent forms, Kirchberg’s conjecture reduces to LLP ⇒ WEP for C∗-algebras. Actually, he originally conjectured also the converse implication but in chapter 18 weshow that this fails, by producing tensors t ∈ B ⊗ B for which the min and max norms aredifferent; in other words the pair (B,B) is not nuclear. The proof combines ideas from finitedimensional operator space theory (indeed t ∈ E ⊗ F for some finite dimensional subspaces E,Fof B) together with estimates of spectral gaps, that allow us to show that a certain constantC(n) defined next is < n for some n. The latter constant involves a sequence of integers Nm

and a sequence (u1(m), · · · , un(m)) of n-tuples of unitary Nm × Nm-matrices and their complexconjugates (u1(m), · · · , un(m)). We then set

(0.5) C(n) = inf supm6=m′

‖∑n

1uj(m)⊗ uj(m′)‖,

where the last norm is meant in MNmNm′ and the infimum runs over all possible sizes (Nm) and allpossible sequences (u1(m), · · · , un(m)) of n-tuples of unitary Nm ×Nm-matrices.

Using unitary random matrices we will show that C(n) = 2√n− 1 (see §18.2). Nevertheless

other more explicit (deterministic) constructions of sequences (u1(m), · · · , un(m)) responsible forC(n) < n are of much interest such as property (T) groups, expanders, quantum expanders, orquantum analogues of spherical coding sequences. In each case we obtain a tensor t ∈ B⊗B suchthat ‖t‖min < ‖t‖max. We describe these delicate ingredients in chapter 19.

In chapter 20, we gather several applications of the preceding ideas to the structure of the metricspace of all finite dimensional operator spaces equipped with the cb-analogue of the Banach-Mazur“distance”, that is defined when dim(E) = dim(F ) by

dcb(E,F ) = inf‖u‖cb‖u−1‖cb | u : E → F invertible.

For instance, for any finite dimensional operator space E, the dual space E∗ admits a naturaloperator space structure (described in §2.4) so that we may view both E and E∗ as subspaces ofB. Thus the identity operator on E defines a tensor tE ∈ B ⊗B. We show that (see (20.6))

‖tE‖B⊗maxB = infdcb(E,F ) | F ⊂ C

where the infimum (which is actually attained) runs over all possible subspaces F ⊂ C withdim(F ) = dim(E).

The fact that (B,B) is not a nuclear pair actually implies that for arbitrary von Neumannalgebras (M,N) the pair (M,N) is nuclear only if either M or N is nuclear. This follows fromthe fact that a non-nuclear von Neumann algebra must contain as a subalgebra the direct sum in

4

the sense of `∞ of the family Mn | n ≥ 1 of all matrix algebras, and there is automatically aconditional expectation onto it. The latter is explained in §12.6.

In chapter 23 we present in detail two unpublished characterizations of the WEP due toHaagerup. The first one says that a C∗-algebra A has the WEP if and only if for any n andany linear map u : `n∞ → A the dec-norm of u coincides with its c.b. norm (see §23.2). Thisnaturally complements his earlier results from the 1980’s in [104]. Haagerup claimed this theoremat some point in the 1990’s but apparently did not circulate a detailed proof of it, as he did for thesecond (more delicate) one, that we give in §23.5.There, to put it very roughly `n∞ is replaced by `n2 . More precisely, the second characterization saysthat A has the WEP if and only if for any n and any (a1, · · · , an) ∈ An we have

‖∑

aj ⊗ aj‖1/2min = ‖∑

aj ⊗ aj‖1/2max.

An important ingredient for its proof is the identification, for any C∗-algebra A, of the norm

An 3 (aj) 7→ ‖∑

aj ⊗ aj‖1/2max

as the norm obtained on An (n ≥ 1) by the complex interpolation method of parameter θ = 1/2between the (“row and column”) norms

(aj) 7→ ‖∑

aj∗aj‖1/2 and (aj) 7→ ‖

∑ajaj

∗‖1/2.

In order to give a reasonably self-complete proof of the latter fact we give a brief basic descriptionof complex interpolation in chapter 22.

One important consequence of this particular characterization is the fact that the WEP isstable under complete isomorphisms. Explicitly, if two C∗-algebras A,B are completely isomorphicas operator spaces, then A WEP ⇒ B WEP. In other words, if we forget the algebraic structure ofa C∗-algebra, the WEP is “remembered” by its operator space structure.In a similar flavor (see chapter 23), let M ⊂M be von Neumann algebras, if there is a completelybounded projection P :M→M onto M (i.e. M is “completely complemented” in M) then thereis a projection Q : M → M that is completely positive with ‖Q‖cb = 1. Thus when M = B(H)we conclude that M is injective.

In chapter 24 we show that the tensor product M ⊗min N of two non-nuclear von Neumannalgebras M and N (for instance for M = N = B) fails the WEP (see Corollary 24.23). The proofis reminiscent of the earlier proof that (M,N) is not a nuclear pair. It makes crucial use of theconstant that we denote by C0(n), that is defined in the same way as C(n) in (0.5), but usingunitaries associated to permutations instead of plain unitary matrices and restricting them to theorthogonal of the constant vector. Again, the key point is that C0(n) < n. We review the recentresults that establish the latter. In analogy with the case of C(n) we can show that C0(n) = 2

√n− 1

using a very recent result on random permutation matrices, and also that C0(3) < 3 by delicatedeterministic arguments: we can use either Selberg’s famous spectral bound or known results onexpanders in permutation groups.

Lastly in chapter 25 we gather a collection of open questions related to our main topics.

Prerequisites These notes are written in a rather detailed style and should be accessible tograduate students and non-specialists. The prerequisite background is kept to a minimum. Ofcourse basic functional analysis is needed, but for operator algebras, the fundamental theorems weuse are the classical ones, such as the bicommutant theorem and Kaplansky’s Theorem, as well as

5

basic facts about states, ∗-homomorphisms and the GNS construction, and we review all those inour appendix §26.Sources The main source for these notes is Kirchberg’s fundamental paper [155]. However, we havemade extensive use of Haagerup’s treatment of decomposable maps in [104]. This allowed us toreformulate many results known for completely positive maps or ∗-homomorphisms for just linearmaps. In addition, Ozawa’s surveys [189, 191, 192] have been an invaluable help and inspiration,as well as the (highly recommended) book [39] by Brown and Ozawa.Many of Kirchberg’s results on exactness are already presented in detail in Simon Wassermann’sexcellent 1994 notes [259], the present volume can be viewed as a sequel and an updated complementto his.Almost all chapters are followed by a Notes and Remarks section where we try to complement thereferences given in the text, and sometimes add some pointers to the literature.About operator spaces Some results already appear in our 2003 book on operator spaces [208].When convenient, we used the presentation from [208]. We describe several applications of operatorspace theory when they are relevant for our topic, but our main focus being here on tensor productsof C∗-algebras, we will refrain from developing operator space theory for its own sake, and we referthe reader instead to [208], or to [80, 196].About operator systems Following Arveson’s pioneering papers [12], much work (notably byChoi, Effros and Lance) on operator systems appeared already in the 1970’s which marked a firstperiod when much progress on tensor products of C∗-algebras was achieved. In particular, Choiand Effros introduced in [47] a notion of duality for operator systems that prefigured the one foroperator spaces developed after Ruan’s 1987 PhD thesis. The emphasis then moved on to operatorspaces in the 1990’s, and C∗-tensor products were investigated (following Kirchberg’s impulse andHaagerup’s work) in the more general framework of operator space tensor products, by Effros, Ruan,Blecher, Paulsen and others. Curiously, operator systems made a reappearance more recently andtheir tensor products were investigated thoroughly in a series of papers, notably [150, 151]. Thisled to several characterizations of the WEP (see [90, 149, 153, 152, 91, 92]), connected to theConnes-Kirchberg problem, but for lack of space (and energy) we chose not to cover this.We also had to leave out the connections of the Connes-Kirchberg problem with noncommutativereal algebraic geometry, for which we refer the reader to [163, 40] and to Ozawa’s survey [191].Basic notation and conventions. The letter H ( or H) always stands for a Hilbert space. OurHilbert spaces all have an inner product

(y, x) 7→ 〈y, x〉

that is linear in x and antilinear in y.We denote by B(H) (resp. K(H)) the Banach algebra formed of all the bounded (resp. compact)linear operators on H equipped with the operator norm.Let K be another Hilbert space. We denote by K ⊗2H the Hilbert space tensor product, obtainedby completing K ⊗H equipped with the classical scalar product characterized by

〈k ⊗ h, k′ ⊗ h′〉 = 〈k, k′〉〈h, h′〉.

We denote by K the complex conjugate Hilbert space, which is classically identified with the dualK∗. Then K ⊗2 H can be identified with the space of all the Hilbert-Schmidt maps from K to H.The unitary group of a unital C∗-algebra A is denoted by U(A).The identity map on a linear space X is denoted by IdX .The unit ball of a normed space X is denoted by BX .Let 1 ≤ p ≤ ∞. Let I be an arbitrary index set. We denote by `p(I) the set of families of complex

6

scalars x = (xi)i∈I such that∑

i∈I |xi|p < ∞ (supi∈I |xi| < ∞ when p = ∞) equipped with the

norm ‖x‖p = (∑

i∈I |xi|p)1/p (supi∈I |xi| when p =∞).When I = N, we denote `p(I) simply by `p.Let (Xi)i∈I be a family of Banach spaces. We denote by(

⊕∑

i∈IXi

)p

their direct sum “in the sense of `p”, equipped with the norm (xi) 7→ (∑‖xi‖p)1/p.

When Xi = X for all i ∈ I, we denote(⊕∑

i∈I Xi

)p

by `p(I;X).

When Xi = C for all i ∈ I we recover `p(I).In the particular case when p = ∞ the space X =

(⊕∑

i∈I Xi

)∞ is the set of those x = (xi) with

xi ∈ Xi (∀i ∈ I) such that ‖x‖ = supi∈I ‖xi‖ <∞.The unit ball of this space X is just the product BX =

∏i∈I BXi .

Let n ≥ 1 be an integer. We denote by `np the space Cn equipped with the norm

x 7→ ‖x‖ =(∑n

1|xj |p

)1/p.

Thus `np = `p(I) for I = 1, · · · , n. When p = 2, the resulting space `n2 is the model for anyn-dimensional Hilbert space.When p = ∞, we set ‖x‖ = supj |xj |, the resulting space `n∞ is the model for any n-dimensionalcommutative C∗-algebra.

We denote by Mn (resp. Mn×m) the space of n×n (resp. n×m) matrices with complex entries.More generally, for any vector space E we will denote by Mn(E) (resp. Mn×m(E)) the space ofn× n (resp. n×m) matrices with entries in E. Thus Mn = Mn(C) (resp. Mn×m = Mn×m(C)).

A linear mapping u : X → Y between Banach spaces with ‖u‖ ≤ 1 is called “contractive” or“a contraction”. We say that u : X → Y is a metric surjection if u(X) = Y and the image of theopen unit ball of X coincides with the open unit ball of Y . Then passing to the quotient by ker(u)produces an isometric isomorphism from X/ ker(u) to Y .

A mapping u : A→ B between C∗-algebras is called a ∗-homomorphism if it is a homomorphismof algebras such that u(x∗) = u(x)∗ for all x ∈ A. When B = B(H) for some Hilbert space H theterm “representation” is often used instead of ∗-homomorphism.Some abbreviations frequently used: c.b. for completely bounded, c.p. for completely positive,u.c.p. for unital and completely positive, c.c. for completely contractive.Acknowledgment. These lecture notes are partially based on the author’s notes for topics coursesgiven at Texas A&M University (Fall 2014 and 2016, Spring 2018), and a minicourse at the WinterSchool on Operator Spaces, Noncommutative Probability and Quantum Groups, held in Metabiefin Dec. 2014, organized by the Laboratoire de Mathematiques de Besancon. I am indebted toMateusz Wasilewski whose careful reading led to a number of corrections and improvements of afirst draft. The author is very grateful to all the auditors and readers of the various drafts who bytheir questions and remarks helped improve several chapters, in particular to Roy Araiza, Li Gao,Kei Hasegawa, Guixiang Hong, Alexandre Nou, Mikael de la Salle, Andrew Swift, Simeng Wang,John Weeks, Hao Xing, and particularly Ignacio Vergara. Special thanks are due to Michiya Morifor his critical reading of a close to final version, which allowed me to correct numerous inadequacies.

1 Completely bounded and completely positive maps: basics

In this opening chapter, we present the fundamental extension and factorization properties of c.b.and c.p. maps.

7

1.1 Completely bounded maps on operator spaces

We start by a few basic facts about operator spaces.

Definition 1.1. (“Non-commutative Banach spaces”) An operator space E is a closed subspace ofB(H), i.e. we are given an inclusion mapping

(1.1) E ⊂ B(H).

Equipped with the norm induced by the one of B(H), the space E is then a Banach space.Thus if we wish, we may think of an operator space as a Banach space given with the additionalstructure of an isometric embedding as in (1.1). Then for each n ≥ 1 we have Mn(E) ⊂Mn(B(H)).Since Mn(B(H)) ' B(H⊕· · ·⊕H), (n-times) we can equip Mn(E) with the norm induced on it byMn(B(H)) or equivalently by B(H ⊕ · · · ⊕H). Thus the data of the embedding (1.1) immediatelyleads to the sequence of norms

‖.‖Mn(E) | n ≥ 1.

In operator space theory, the usual norm from Banach space theory is replaced by that sequenceof norms. The ordinary norm on E corresponds to n = 1.

Equivalently, we can think of an operator space E as a closed subspace of a C∗-algebra A itselfembedded in B(H) so we have

E ⊂ A ⊂ B(H).

For instance, we could use for A the C∗-algebra generated by E in B(H) (i.e. the smallest C∗-subalgebra containing E) or we could take A = B(H).

Actually, we can avoid reference to B(H): a more abstract but still equivalent viewpoint is todefine an operator space as a closed subspace of an (abstract) C∗-algebra A. Then the space Mn(A)of n× n matrices with entries in A is also a C∗-algebra for the usual matrix operations, and hence(since it is complete) it has a unique C∗-norm, which we can again induce on the subspace Mn(E).Thus the embedding E ⊂ A automatically yields a sequence of norms ‖.‖Mn(E) | n ≥ 1. It is easyto see (using Gelfand theory to embed A in some B(H), see §26.11) that these two definitions ofoperator spaces are equivalent.

Occasionally, we will consider a linear subspace E ⊂ B(H) that is not closed and treat it as anoperator space. This simply means that we are referring to the norm closure of E in B(H).

We will refer to the sequence of norms ‖.‖Mn(E) | n ≥ 1 as the “operator space structure”of E. We extract from the preceding discussion that we can equip any concrete operator spaceE ⊂ B(H) with a natural (somewhat “abstract”) operator space structure.

“Operator space Theory” (see the books [80, 208]) took off after Ruan’s 1987 thesis where heidentified the abstract sequences of norms on Mn(E) (E a vector space) that come from a concreterealization of E as a subspace of some B(H). Operator space theory was then developed in theworks of Effros-Ruan, Blecher-Paulsen, and others as a generalization of the Choi-Effros theory ofoperator systems developed in the 1970’s in the series [45, 46, 47, 48].

In this theory, the bounded linear maps (between Banach spaces) are replaced by the completelybounded linear ones (between operator spaces).

Let u : E → F be a linear map between operator spaces. For any given n ≥ 1, we denote byun : Mn(E)→Mn(F ) the linear map defined by

un([aij ]) = [u(aij)].

Definition 1.2. A map u : E → F is called completely bounded (in short c.b.) if

supn≥1 ‖un‖Mn(E)→Mn(F ) <∞.

8

We define ‖u‖cb = supn≥1 ‖un‖Mn(E)→Mn(F ) and we denote by CB(E,F ) the Banach space of allc.b. maps from E into F equipped with the c.b. norm.

In tensor product notation, using Mn(E) 'Mn⊗E we write un = IdMn⊗u : Mn⊗E →Mn⊗Fso that un(

∑ak ⊗ xk) =

∑ak ⊗ u(xk). We have then

(1.2) ‖u‖cb = supn≥1 supt∈BMn(E)‖(IdMn ⊗ u)(t)‖Mn(F ).

If G ⊂ B(H) is another operator space and if v : F → G is c.b., then the compositon vu : E → Gclearly remains c.b. and we have

‖vu‖cb ≤ ‖v‖cb‖u‖cb.

Of course, when n = 1, 1 × 1 matrices are just elements of E, so that u1 : M1(E) → M1(F ) isnothing but u itself. In particular we have ‖u‖ ≤ ‖u‖cb and CB(E,F ) ⊂ B(E,F ). In general thisis a strict inclusion, but for linear forms or if rk(u) = 1 we have ‖u‖ = ‖u‖cb. Indeed, assumingE ⊂ B(H), for any u : E → F of the form u(a) = ξ(a)b, ξ ∈ E∗, b ∈ F (note that ‖u‖ = ‖b‖‖ξ‖)we have for any a ∈Mn(E)

‖[u(aij)]‖ = ‖b‖‖[ξ(aij)]‖Mn = ‖b‖ supx,y∈B`n2

∣∣∣∑ ξ(aij)yixj

∣∣∣ ≤ ‖u‖ supx,y∈B`n2

∥∥∥∑ aijyixj

∥∥∥E

= ‖u‖ sup∣∣∣∑〈yik, aijxjh〉∣∣∣ | x, y ∈ B`n2 , h, k ∈ BH

and hence‖[u(aij)]‖ ≤ ‖u‖‖a‖Mn(E).

Thus, whenever u has rank 1, we have

(1.3) ‖u‖cb = ‖u‖.

Proposition 1.3 (c.b. with commutative range). Let F ⊂ B(H) be an operator space. Let AF bethe C∗-algebra generated by F .

(i) For any n ≥ 1 and any x in Mn(F ) we have

‖x‖Mn(F ) ≥ sup∥∥∥∑λiµjxij

∥∥∥F| λi ∈ C, µj ∈ C,

∑|λi|2 ≤ 1,

∑|µj |2 ≤ 1

.

(ii) If AF is commutative we have equality in (i). Then, if E is an arbitrary operator space, anybounded map u : E → F is c.b. and satisfies ‖u‖cb = ‖u‖.

Proof. (i) is an easy exercise. As for (ii), when AF is commutative (see §26.12), we can assumeAF = C0(Ω) and also Mn(AF ) = C0(Ω;Mn), for some locally compact space Ω. Then equality in(i) is very simple to check and (ii) is then immediate.

Remark 1.4 (c.b. with commutative domain). The preceding result is not valid if the domainis assumed commutative. For any n > 2 there is a map T : `n∞ → B(`2) with ‖T‖ ≤ 1 and‖T‖cb ≥

√n/2 > 1. Indeed, let (uj)1≤j≤n be a matricial spin system i.e. a system of unitary

self-adjoint N × N matrices that are anticommuting i.e. satisfying ∀i 6= j uiuj + ujui = 0. LetT : `n∞ → MN be defined by T (ej) = uj/(2n)1/2. Then T satisfies the announced bounds. Theproof that ‖T‖cb ≥

√n/2 uses the elementary identity ‖

∑n1 uj ⊗ uj‖min = n, valid for any n-tuple

of unitary matrices (see (18.5)). As for ‖T‖ ≤ 1 we refer to [104, p. 209] for a proof. Haagerupalso shows in [104] that ‖T‖ = ‖T‖cb when n = 2, which we will prove in Remark 3.13.

9

When ‖u‖cb ≤ 1, we say that u is “completely contractive” (or “a complete contraction”).The notion of isometry is replaced by that of “complete isometry”: a linear map u : E → F is

said to be a complete isometry (or completely isometric) if un : Mn(E) → Mn(F ) is an isometryfor all n ≥ 1.An invertible mapping u : E → F is said to be a complete isomorphism if both u and u−1 are c.b.Clearly, a completely isometric surjective map is a complete isomorphism.

Remark 1.5. For instance if S : H → H is an isometry, then the linear map uS : B(H) → B(H)defined by uS(x) = S∗xS is completely isometric. This is easily checked by observing that Sinduces an isometry Sn from `n2 (H) to `n2 (H), such that (uS)n(y) = S∗nySn for any y ∈Mn(B(H)) =B(`n2 (H)). Thus (uS)n is of the same form as uS . In particular if S : H → H is a surjective isometrythen uS is completely isometric isomorphism.

Remark 1.6. If E,F are C∗-algebras and u : E → F is a ∗-homomorphism then un : Mn(E) →Mn(F ) is also a ∗-homomorphism, and hence (see Proposition 26.24) we have ‖un‖ = 1 for all n(unless u = 0), which shows that u is automatically a complete contraction.Moreover, if u is injective then un is obviously also injective. Therefore (see Proposition 26.24) unis isometric and u is automatically a complete isometry.

Definition 1.7. Let E ⊂ B(H) and G ⊂ B(K) be operator spaces. We have a natural embedding

G⊗ E ⊂ B(K ⊗2 H)

that allows us to defineG⊗min E = G⊗ Enorm ⊂ B(K ⊗2 H).

The space G⊗min E is then called the minimal tensor product of G and E.

In particular, in the case G = B(`n2 ) = Mn, we have an obvious completely isometric identifica-tion

(1.4) Mn(E) = Mn ⊗min E.

Indeed, by Remark 1.5 this simply follows from the Hilbert space identification `n2 (H) ' `n2 ⊗2 H.

Remark 1.8 (Associativity of the minimal tensor product). Let Ej ⊂ B(Hj) be operator spaces(1 ≤ j ≤ n). We define similarly

E1 ⊗min · · · ⊗min En = E1 ⊗ · · · ⊗ En ⊂ B(H1 ⊗2 · · · ⊗2 Hn).

Since we have H1 ⊗2 H2 ⊗2 H3 ' (H1 ⊗2 H2) ⊗2 H3 ' H1 ⊗2 (H2 ⊗2 H3), by Remark 1.5 we alsohave completely isometrically

(1.5) E1 ⊗min E2 ⊗min E3 ' (E1 ⊗min E2)⊗min E3 ' E1 ⊗min (E2 ⊗min E3).

Thus we may also view E1 ⊗min · · · ⊗min En = E1 ⊗ · · · ⊗ En as obtained from successive minimaltensor products of suitable pairs, and we may suppress the parentheses since they become irrelevant.

Remark 1.9 (Commutativity of the minimal tensor product). Since we have K ⊗2 H ' H ⊗2 K,by Remark 1.5 we also have completely isometrically

(1.6) G⊗min E ' E ⊗min G,

via x⊗ y 7→ y ⊗ x.

10

Remark 1.10 (Injectivity of the minimal tensor product). From the preceding definition the follow-ing property is obvious: Let E1 ⊂ E2 ⊂ B(H) and G1 ⊂ G2 ⊂ B(K) be operator subspaces, sothat E1 ⊗G1 ⊂ E2 ⊗G2. Then for any t ∈ E1 ⊗G1 we have

(1.7) ‖t‖E1⊗minG1 = ‖t‖E2⊗minG2 .

Proposition 1.11. Let u : E → F be a c.b. map between two operator spaces. Then for anyother operator space G the mapping IdG ⊗ u : G ⊗ E → G ⊗ F extends to a bounded mappinguG : G⊗min E → G⊗min F and we have

(1.8) ‖u‖cb = supG ‖uG‖ = supG ‖uG‖cb,

where the suprema run over all possible G’s.

Proof. First observe that the choice of G = Mn shows that

sup‖uG‖ : G an operator space ≥ ‖u‖cb .

To prove the converse assume G ⊂ B(K). For notational simplicity, assume K = `2 and lett =

∑rk=1 ak ⊗ bk ∈ G ⊗ E. Consider the natural embeddings `2n ' [e1, · · · , en] ⊂ `2 (n ≥ 1) with

respect to some choice ej : j ≥ 1 of orthonormal basis for `2 and the corresponding orthogonal

projections Pn : K ⊗2 H → `n2 ⊗2 H. Then ∪n`2n ⊗H = K ⊗2 H and hence

‖t‖min = supn ‖Pn t|`n2⊗H : `n2 ⊗2 H → `n2 ⊗2 H‖.

Let ak(i, j) = 〈ei, akej〉. It is not hard to see that Pn t|`n2⊗Hcan be identified with the matrix

tn ∈Mn(E) given by tn(i, j) =∑

k ak(i, j)bk. This shows that

(1.9) ‖t‖G⊗minE = supn ‖tn‖Mn(E),

We have un(tn) = [∑

k ak(i, j)u(bk)] ∈ Mn(F ). Applying (1.9) to uG(t) =∑ak ⊗ u(bk) gives

us ‖uG(t)‖min = supn ‖un(tn)‖Mn(F ) and hence ‖uG(t)‖min ≤ ‖u‖cb supn ‖tn‖Mn(E) = ‖u‖cb‖t‖min,which shows ‖uG‖ ≤ ‖u‖cb. Thus ‖u‖cb = supG ‖uG‖. Then, substituting Mn(G) for G, we easilydeduce that supG ‖uG‖ = supG ‖uG‖cb.

Corollary 1.12. Let E1, F1, E2, F2 be operator spaces. Let u1 ∈ CB(E1, F1) and u2 ∈ CB(E2, F2).Then u1 ⊗ u2 continuously extends by density to a c.b. map u1 ⊗ u2 : E1 ⊗min E2 −→ F1 ⊗min F2

such that

(1.10) ‖u1 ⊗ u2‖cb ≤ ‖u1‖cb‖u2‖cb.

Proof. The argument is based on the obvious identity u1⊗u2 = (u1⊗IdF2)(IdE1⊗u2), which givesus ‖u1 ⊗ u2‖ ≤ ‖u1 ⊗ IdF2‖‖IdE1 ⊗ u2‖. By (1.8) we have ‖IdE1 ⊗ u2‖ ≤ ‖u2‖cb and using (1.6)we also find ‖u1 ⊗ IdF2‖ ≤ ‖u1‖cb. This gives us ‖u1 ⊗ u2‖ ≤ ‖u1‖cb‖u2‖cb. Now replacing u1 byIdMn ⊗ u1, by (1.5) and (1.4) we obtain the announced (1.10) after taking the sup over n.

It is an easy exercise to show that (1.10) is actually an equality but we do not use this in the sequel.We will now generalize (1.9).

11

Proposition 1.13. For any t =∑ak ⊗ bk ∈ G⊗ E we have

(1.11) ‖t‖min = supn≥1

∥∥∥∑ v(ak)⊗ bk∥∥∥Mn(E)

| v ∈ CB(G,Mn), ‖v‖cb ≤ 1

.

Furthermore

(1.12) ‖t‖min = sup

∥∥∥∑ v(ai)⊗ w(bi)∥∥∥Mnm

where the supremum runs over n,m ≥ 1 and all pairs v : G → Mn, w : E → Mm with ‖v‖cb ≤ 1and ‖w‖cb ≤ 1. (We can of course restrict to n = m if we wish.)

Proof. By (1.8) and (1.6) we have ‖v⊗ IdE‖ ≤ ‖v‖cb, so the left-hand side of (1.11) is ≥ the right-hand side. But by (1.9) we see that equality holds: indeed just observe that tn = (vn ⊗ IdE)(t)with vn(·) = a∗n · an where an : `n2 → H denote the inclusion. To check (1.12) we again invoke (1.6)that allows us to apply (1.11) one more time on the second factor.

Remark 1.14. The preceding proposition shows that the min-norm on G⊗E depends only on thesequences of norms on Mn(G) and Mn(E) and not on the particular embeddings G ⊂ B(K) andE ⊂ B(H). Indeed, the latter sequences suffice to determine the norms of the spaces CB(G,Mn)and CB(E,Mn) (see Proposition 1.19 for more precision).More generally, the same remark holds for the norm in Mn(G⊗min E) and hence the whole sequenceof the norms ‖ · ‖Mn(G⊗minE) depends only on the sequences of norms on Mn(G) and Mn(E).

Corollary 1.15. If an element t ∈ G ⊗min E is such that (v ⊗ w)(t) = 0 for any v ∈ G∗ andw ∈ E∗, then t = 0.

Proof. This is immediate from (1.12). Indeed, the assumption remains obviously true for anyv : G→Mn and w : E →Mm.

Warning: The reason we emphasize the rather simple fact in Corollary 1.15 is that the analogousfact for the maximal tensor product of two C∗-algebras fails in general.

Remark 1.16 (Direct sum of operator spaces). Let Ei ⊂ B(Hi) (i ∈ I) be a family of operatorspaces. Let E = (⊕

∑i∈I Ei)∞. Note that E ⊂ (⊕

∑i∈I B(Hi))∞, and that (⊕

∑i∈I B(Hi))∞ is

a C∗-algebra naturally embedded in B(H) with H = (⊕∑

i∈I Hi)2. This allows us to equip Ewith an operator space structure as a subspace of B(H). Let n ≥ 1. Any matrix a ∈ Mn(E) isdetermined by a family (ai)i∈I with ai ∈Mn(Ei) for all i ∈ I. It is then easy to check that for anyn ≥ 1 and any a ∈Mn(E) we have

(1.13) ‖a‖Mn(E) = supi∈I ‖ai‖Mn(Ei).

More generally, for any operator space F ⊂ B(K), we have a natural isometric embedding

(1.14) F ⊗min (⊕∑

i∈IEi)∞ ⊂ (⊕

∑i∈I

F ⊗min Ei)∞,

which is an isomorphism if dim(F ) <∞ since both sides are then setwise identical.The equality (1.13) shows furthermore that for any operator space D, a linear map u : D → E isc.b. if and only if the coordinates ui : D → Ei are c.b. with supi∈I ‖ui‖cb <∞ and we have

(1.15) ‖u‖cb = supi∈I ‖ui‖cb.

12

1.2 Extension property of B(H)

We first recall that the spaces L∞ are the injective objects in the category of Banach spaces.

Theorem 1.17 (Nachbin’s Hahn-Banach theorem). Let (Ω, µ) be any measure space, and let E ⊂ Xbe any subspace of a Banach space X. Then any u ∈ B(E,L∞(µ)) admits an extension u ∈B(X,L∞(µ)) such that ‖u‖ = ‖u‖.

X

u

##E?

OO

u // L∞(µ)

The proof of Nachbin’s theorem relies on several identifications. First we note the elementaryisometric isomorphisms

B(E,F ∗) ∼= B(F,E∗) ∼= Bil(E × F )

where Bil(E × F ) is the Banach space of all bounded bilinear forms on E × F . Then we have anisometric identification

(1.16) B(E,F ∗) ∼= (E∧⊗F )∗

where∧⊗ is the projective tensor product, i.e. the completion of the algebraic tensor product E⊗F

with respect to the so-called “projective” norm (see §26.1)

‖t‖∧ = inf∑n

1‖aj‖‖bj‖ : t =

∑n

1aj ⊗ bj.

Note that E∧⊗F and F

∧⊗E can obviously be (isometrically) identified. The duality between tensors

t ∈ E∧⊗F and operators u ∈ B(E,F ∗) is defined first on rank one tensors by setting

〈u, a⊗ b〉 = 〈u(a), b〉,

then this can be extended to unambiguously define 〈u, t〉 for t ∈ E⊗F by linearity. Then by density

we define 〈u, t〉 for t ∈ E∧⊗F , and (1.16) holds for this duality.

By a classical result (due to Grothendieck) when F = L1(Ω, µ) (on some measure space (Ω, µ)),

the space E∧⊗F (or equivalently F

∧⊗E) can be identified isometrically to the (Bochner sense) vector

valued L1-space L1(µ;E).

Sketch of Proof of Nachbin’s Theorem. Taking F = L1(µ) in the preceding, we find

B(E,L∞(µ)) = L1(µ;E)∗ and B(X,L∞(µ)) = L1(µ;X)∗.

Then since we have an isometric inclusion

L1(µ;E) ⊂ L1(µ;X)

Nachbin’s Theorem can be deduced from the classical Hahn-Banach theorem.

We will follow the same approach to prove the non-commutative version of Nachbin’s Theorem,due to Arveson.

13

Theorem 1.18 (Arveson’s Hahn-Banach theorem). Let H be any Hilbert space, and let X ⊂ B(H)be any operator space and let E ⊂ X be any subspace. Then any u ∈ CB(E,B(H)) admits anextension u ∈ B(X,B(H)) such that ‖u‖cb = ‖u‖cb.

X

u

""E?

OO

u // B(H)

The projective tensor norm ‖ ‖∧ on L1(µ)⊗E will be replaced by the following one on K⊗E⊗Hwhere H,K are Hilbert spaces:For any t ∈ K ⊗ E ⊗H, we define (recall ‖k‖ = ‖k‖ for all k ∈ K)

γE(t) = inf

(∑m

i=1‖ki‖2

)1/2‖[aij ]‖Mm×n(E)

(∑n

j=1‖hj‖2

)1/2

where the infimum runs over all representations of t of the form

t =∑m

i=1

∑n

j=1ki ⊗ aij ⊗ hj .

In analogy with Nachbin’s Theorem, we will show that this norm satisfies:

(i) The dual space (K ⊗ E ⊗H, γE)∗ can be identified with CB(E,B(H,K)).

(ii) The natural inclusion (K ⊗ E ⊗H, γE) ⊂ (K ⊗X ⊗H, γX) is isometric.

Proof of Arveson’s Hahn-Banach theorem. Using (i) and (ii) the proof of Theorem 1.18 can becompleted exactly as in the case of Banach spaces: We simply take K = H and apply the Hahn-Banach theorem to the subspace (K ⊗ E ⊗H, γE) ⊂ (K ⊗X ⊗H, γX).

Thus the proof now reduces to the verification of (i) and (ii). It is easy to check that γE is anorm by arguing as follows:Let t =

∑mi=1

∑nj=1 ki ⊗ aij ⊗ hj and t′ =

∑m′

p=1

∑n′

q=1 k′p ⊗ a′pq ⊗ h′q be elements of K ⊗E ⊗H. We

have obviously (consider the block diagonal matrix with blocks a and a′)

γE(t+t′) ≤ (∑‖ki‖2+

∑‖k′p‖2)1/2 max‖[aij ]‖Mm×n(E), ‖[a′pq]‖Mm′×n′ (E)(

∑‖hj‖2+

∑‖h′q‖2)1/2.

But by homogeneity, for any ε > 0 there are suitable representations of t, t′ such that

‖[aij ]‖Mm×n(E) = 1 and∑m

i=1‖ki‖2 =

∑n

j=1‖hj‖2 < γE(t) + ε,

as well as

‖[a′pq]‖Mm′×n′ (E) = 1 and∑m

p=1‖k′p‖2 =

∑n

q=1‖h′q‖2 < γE(t′) + ε.

Then we find for any ε > 0

(1.17) γE(t+ t′) ≤ γE(t) + γE(t′) + 2ε,

which shows that γE is subadditive and hence (since γE(t) dominates the norm of t as a boundedtrilinear form on K∗ × E∗ ×H∗) it is a norm.Consider the C-linear correspondence

CB(E,B(H,K)) 3 u 7→ ϕu ∈ (K ⊗ E ⊗H)∗

14

defined by ϕu(t) =∑

i,j〈ki, u(aij)hj〉, if t =∑

i,j ki ⊗ aij ⊗ hj . Note that

‖ϕu‖γ∗E = supγE(t)<1

|ϕu(t)| = sup|∑

i,j〈ki, u(aij)hj〉| |

∑‖hj‖2 =

∑‖ki‖2 < 1, ‖[aij ]‖Mn(E) = 1

= sup‖u(aij)‖Mn(B(H,K)) | ‖[aij ]‖Mn(E) = 1 = ‖u‖cb.Hence ‖ϕu‖γ∗E = ‖u‖cb and (i) is proved.

To verify (ii) consider t ∈ K ⊗ E ⊗ H. Note that obviously γX(t) ≤ γE(t) since there are morerepresentations allowed in the definition of γX . To establish (ii) it suffices to prove that converselyγE(t) ≤ γX(t), or equivalently, by homogeneity, that γX(t) < 1 implies γE(t) < 1. The assumptionγX(t) < 1 implies that there exists a decomposition (a priori with aij ∈ X)

t =∑m

i=1

∑n

j=1ki ⊗ aij ⊗ hj ,

such that(∑m

i=1‖ki‖2)1/2 · ‖[aij ]‖Mm×n(X) · (

∑n

j=1‖hj‖2)1/2 < 1 .

To conclude it suffices to show that there is a (possibly different) representation of t with the samebounds but with aij ∈ E. We will use the following simple linear algebraic fact.Claim: Let h1, · · · , hn ∈ H, and let L be their linear span. Let r = dim(L) ≤ n. Then there existr linearly independent vectors h′q : 1 ≤ q ≤ r in L and a rectangular matrix C = [cqj ] ∈ Mr×nsuch that ‖C‖Mr×n ≤ 1, hj =

∑rq=1 cqjh

′q for 1 ≤ j ≤ n, and

∑nj=1 ‖hj‖2 =

∑rq=1 ‖h′q‖2. We denote

by tC ∈Mn×r the transposed matrix.Let s be the dimension of the span of the vectors (ki) in K. It follows from the claim applied

to (ki) that there also exists a linearly independent set k′p : 1 ≤ p ≤ s and a matrix D ∈ Ms×msuch that ‖D‖ ≤ 1, ki =

∑sp=1 dpik

′p, and

∑mi=1 ‖ki‖2 =

∑sp=1 ‖k′p‖2. Then, note that

t =∑m

i=1

∑n

j=1ki ⊗ aij ⊗ hj =

∑s

p=1

∑r

q=1k′p ⊗ a′pq ⊗ h′q ,

where we have denoted a = [aij ] and a′ = [a′pq] = DatC, so that ‖a′‖Ms×r(X) ≤ ‖a‖Mm×n(X).But now, since k′p : 1 ≤ p ≤ s and h′q : 1 ≤ q ≤ r are linearly independent sets, we have

t ∈ K ⊗ E ⊗H ⇒ a′pq ∈ E ∀p, q,

so that a′ ∈Ms×r(E) and ‖a′‖Ms×r(E) = ‖a′‖Ms×r(X) ≤ ‖a‖Mm×n(X). Thus we may write

γE(t) ≤ (∑s

p=1‖k′p‖2)1/2 · ‖a′‖Ms×r(E) · (

∑r

q=1‖h′q‖2)1/2

and henceγE(t) ≤ (

∑m

i=1‖ki‖2)1/2 · ‖a‖Mm×n(X) · (

∑n

j=1‖hj‖2)1/2 < 1,

which completes the proof of (ii).Proof of the Claim: Consider the linear operator T : `n2 → H defined by Tei = hi for each1 ≤ i ≤ n, where ei : 1 ≤ i ≤ n is the canonical basis of `n2 . Let fq : 1 ≤ q ≤ r denote anyorthonormal basis for N = ker(T )⊥ and let P denote the orthogonal projection of `n2 onto N . Seth′q = Tfq for 1 ≤ q ≤ r. Clearly, ‖T‖2HS = ‖TP‖2HS =

∑rq=1 ‖Tfq‖2 and hence∑n

i=1‖hi‖2 =

∑r

q=1‖h′q‖2.

Let C ∈ Mr×n be the matrix representing P with respect to the bases (ej) and (fq), so thatPej =

∑q cqjfq. Then hj = T (ej) = TP (ej) =

∑q cqjh

′q. This proves the claim.

15

The following variant due to Roger Smith is often useful.

Proposition 1.19. Consider operator spaces E,X with E ⊂ X and u : E →Mn. Then we have

(1.18) ‖u‖cb = ‖un‖Mn(E)→Mn(Mn).

Moreover, u : E →Mn admits an extension u : X →Mn such that ‖un‖ = ‖un‖.

Proof. For any m ≥ 1 and any x1, . . . , xm in `n2 with∑m

1 ‖xi‖2 ≤ 1 there are an m × n scalarmatrix b = [bjk] with ‖[bjk]‖ ≤ 1 and vectors x1, . . . , xn in `n2 such that

∑n1 ‖xi‖2 ≤ 1 and

∀j ≤ m xj =∑n

k=1bjkxk.

This follows from the claim in the preceding proof. Similarly, for any y1, . . . , ym in `n2 there are ascalar matrix c = [cil] with ‖[cil]‖ ≤ 1 and y1, . . . , yn in `n2 such that

∑n1 ‖yi‖2 ≤ 1 and

∀i ≤ m yi =∑n

l=1cilyl.

Hence for any m×m matrix [aij ] in Mm(E) we have∑m

i,j=1〈yi, u(aij)xj〉 =

∑n

k,l=1〈yl, u(αlk)xk〉

where [αlk] ∈Mn(E) is defined by [αlk] = c∗.[aij ].b (matrix product). Therefore:

‖[u(aij)]‖Mm(Mn) ≤ ‖[u(αkl)]‖Mn(Mn) ≤ ‖un‖Mn(E)→Mn(Mn)‖[αlk]‖Mn(E)

≤ ‖un‖Mn(E)→Mn(Mn)‖[aij ]‖Mn(E).

Thus we obtain ‖u‖cb ≤ ‖un‖ and hence ‖u‖cb = ‖un‖. The second assertion is then a corollary ofTheorem 1.18.

1.3 Completely positive maps

We start with the definition of complete positivity.

Definition 1.20. Let E ⊂ A, F ⊂ B be linear subspaces of C∗-algebras A,B. We set Mn(E)+ =Mn(E)∩Mn(A)+ and similarly for Mn(F )+. We will say that a linear map u : E → F is completelypositive (c.p. in short) if for any n we have un(Mn(E)+) ⊂Mn(F )+. We will denote by CP (E,F )the set of all such linear maps.

Actually, to get interesting examples this framework is too general and most of the time we needto work not only with operator subspaces, but either with C∗-algebras (the case E = A,F = B) orwith operator systems as defined later on in Definition 1.36.

Remark 1.21. Let v ∈ CP (E,B(H)). Then for any bounded V : K → H (H,K Hilbert) themapping x 7→ V ∗v(x)V clearly belongs to CP (E,B(K)). Since any ∗-homomorphism π : A → Bis clearly c.p., this shows that any u of the form u(x) = V ∗π(x)V is also c.p.

By a classical dilation result due to Stinespring, it turns out that the example we just describedis actually the general form of a c.p. map.

16

Theorem 1.22 (Stinespring’s Theorem). Let A be a C∗-algebra. Then a linear map u : A→ B(H)is c.p. if and only if there is a Hilbert space H, an operator V : H → H and a ∗-homomorphismπ : A→ B(H) (unital if A is unital) such that

(1.19) ∀a ∈ A u(a) = V ∗π(a)V.

If A is unital and u(1) = 1 then V is an isometry. Moreover, if u is c.p. and A is unital, we have

(1.20) ‖u‖ = ‖u‖cb = ‖u(1)‖

and in any case (unital or not)

(1.21) ‖u‖ = sup‖u(a)‖ | a ≥ 0 ‖a‖ ≤ 1.

Let (xi) be an increasing net in BA ∩ A+ such that xi ≥ 0, ‖xi‖ ≤ 1 and ‖xix − x‖ → 0 for anyx ∈ A (thus (xi) is an approximate unit of A in the sense of §26.15), then

(1.22) ‖u‖ = ‖u‖cb = lim ‖u(xi)‖.

Proof. Assume u ∈ CP (A,B(H)). We introduce a scalar product on A⊗H by setting

∀t =∑

aj ⊗ bj ∈ A⊗H, 〈t, t〉 =∑

ij〈bi, u(a∗i aj)bj〉,

so that t → 〈s, t〉 is linear and s → 〈s, t〉 is antilinear. The complete positivity of u shows that〈t, t〉 ≥ 0. Therefore after passing to the quotient and completing we obtain a Hilbert space H.The ∗-homomorphism π defined by π(a)(t) =

∑aaj⊗ bj extends to a continuous ∗-homomorphism

(unital if A is unital) from A to B(H). If A is unital, let V h = 1 ⊗ h. Then ‖V ‖ ≤ 1 and〈1⊗ h, π(a)(1⊗ h)〉 = 〈h, u(a)h〉. Therefore 〈V h, π(a)V h〉 = 〈h, u(a)h〉 or equivalently V ∗π(a)V =u(a). If u(1) = 1, then V ∗V = I and V is an isometry. If A is unital, we have clearly ‖u‖ = ‖V ∗V ‖ =‖u(1)‖. Moreover, it is easy to check that ‖u‖cb ≤ ‖V ‖2‖π‖cb and since ‖π‖cb = 1 (indeed, πn is a∗-homomorphism for any n, and hence ‖πn‖ = 1 by (26.17)), we find ‖u‖cb ≤ ‖u(1)‖, which proves(1.20).

If A is not unital, it admits an approximate unit (see §26.15). This will allow us to extend u toa c.p. map u defined on a larger unital C∗-algebra A containing A. Then the already proved unitalcase applied to u will give us the factorization (1.19). Assume A separable for simplicity. Then we

have a sequence 0 ≤ · · · ≤ χn ≤ χn+1 ≤ · · · in A+ with ‖χn‖ < 1 such that ‖χ1/2n aχ

1/2n − a‖ → 0

for any a ∈ A. Assume A ⊂ B(H) and let A = A+ CI ⊂ B(H) be the unitization of A. We claimthat there is a c.p. map u : A → B(H) extending u. Note that, since u is positive, u(χn) is orderincreasing and bounded (which means the same for 〈h, u(χn)h〉,∀h ∈ H), and hence converges inthe weak operator topology to an operator T ∈ B(H) with ‖T‖ ≤ ‖u‖. We then define (assumingI /∈ A)

u(a+ λI) = u(a) + λT.

Since ‖u(χ1/2n aχ

1/2n − a)‖ → 0 ∀a ∈ A, we have (in the weak operator topology)

∀x ∈ A u(x) = limn→∞

u(χ1/2n xχ1/2

n ),

from which it is easy to see that u is c.p. Moreover, we have

‖u‖ ≤ ‖u‖cb ≤ ‖u‖cb = ‖u(1)‖ = ‖T‖ ≤ ‖u‖.

17

Thus we conclude

‖u‖ = ‖u‖cb = ‖u(1)‖ = sup‖u(χ)‖ | χ ≥ 0 ‖χ‖ < 1.

For any approximate unit (xi) in BA ∩A+ we have ‖xiχxi − χ‖ → 0 and hence

‖u(χ)‖ ≤ lim ‖u(xiχxi)‖

and, if χ ≥ 0, ‖χ‖ < 1, we have 0 ≤ xiχxi ≤ x2i ≤ xi and hence 0 ≤ u(xiχxi) ≤ u(x2

i ) ≤ u(xi) sothat we obtain (1.22).

Remark 1.23. In Theorem 1.22 we may clearly replace H by H0 = π(A)(V H) and a 7→ π(a) bya 7→ π(a)|H0

. Then we obtain (1.19) with in addition H = π(A)(V H). In the latter case π is called

a minimal dilation of u. It is easy to check that it is unique up to conjugation by a unitary.

Remark 1.24. Any positive linear form ϕ : E → C on a subspace E ⊂ A of a C∗-algebra is c.p.This can be checked directly using ϕ(

∑ij xiaijxj) =

∑ij xiϕ(aij)xj (xi ∈ C) and the fact that

for b ∈ Mn(C) we have [bij ] ≥ 0 if and only if∑

ij xibijxj ≥ 0 ∀n, ∀(xi) ∈ Cn, together with theobservation that a ∈Mn(E) ∩Mn(A)+ implies

∑ij xiaijxj ∈ E ∩A+.

Remark 1.25. The classical GNS factorization (see §26.13) of a positive linear form ϕ ∈ A∗ rewritesϕ as ϕ(a) = 〈ξ, π(a)ξ〉 for some Hilbert space H, some ∗-homomorphism π : A→ B(H) and someξ ∈ H. This can be rewritten as (1.19) with V : C→ H defined by V (λ) = λξ (λ ∈ C). Thus thiscan be viewed as a particular case (namely the case B(H) = C) of Theorem 1.22.

Corollary 1.26. Let A,B,G be C∗-algebras. For any u ∈ CP (A,B) the mapping uG = IdG ⊗ uis a c.p. map from G⊗min A to G⊗min B.

Proof. Assume G ⊂ B(K). By Theorem 1.22 we have u(·) = V ∗π(·)V . Therefore uG(·) = (IdK ⊗V )∗πG(·)(IdK ⊗ V ) and πG : G ⊗min A → G ⊗min B(H) is a ∗-homomorphism. A more directalternative proof can be given following the same idea as for Proposition 1.11.

By composition as for (1.10) we obtain as an immediate consequence:

Corollary 1.27. Let Aj , Bj be C∗-algebras (j = 1, 2). Let uj ∈ CP (Aj , Bj) (j = 1, 2). Thenu1 ⊗ u2 extends to a c.p. map from A1 ⊗min A2 to B1 ⊗min B2.

Remark 1.28 (Positivity with commutative range). Let A,B and u : E → F ⊂ B be as in Definition1.20. Assume B commutative and unital. Then u positive ⇒ u completely positive.Indeed, B ' C(T ) for some compact set T (see §26.12), and Mn(C(T ))+ consists of the functionsa : T → Mn such that a(t) ∈ (Mn)+ for any t ∈ T . Therefore, u is c.p. if and only if x 7→ u(x)(t)is c.p. for any t ∈ T , or equivalently, by the preceding Remark 1.24, if and only if x 7→ u(x)(t) is apositive linear form on E for any t ∈ T . Thus, if u is positive, it is “automatically” c.p.

Remark 1.29. A simple example of a positive (unital) linear map that is not c.p. is the transpositionon Mn for n > 1 or on B(`2) or K(`2) (see also Remark 2.2).

Remark 1.30. Consider a linear mapping u : Mn → A into a C∗-algebra A. Let a ∈Mn(A) be thematrix defined by aij = u(eij). Then u ∈ CP (Mn, A) if and only if a ∈Mn(A)+.Indeed, note that a = un(ξ) with ξ ∈ Mn(Mn)+ defined by ξij = eij . From this the “onlyif” part follows. Conversely, if a ∈ Mn(A)+ then we have a = b∗b for some b ∈ Mn(A), andhence u(x) =

∑k b∗kixijbkj (x ∈ Mn) from which it is easy to deduce that u is of the form (1.19)

and hence is c.p. We leave the details as an exercise. Note that if u ∈ CP (Mn, A) we have‖u‖ = ‖u(1)‖ = ‖

∑aii‖ = ‖

∑ik b∗ikbik‖.

18

Remark 1.31. In the case when u(eij) = 0 for any i 6= j, u can be identified with a mapping from`n∞ to A, and the associated matrix a is diagonal. Then u : `n∞ → A is c.p. if and only if u (orequivalently a) is positive.

Remark 1.32 (Positivity with commutative domain). Consider C∗-algebras A,B and assume nowthat A is commutative. Then any positive linear map u : A → B is c.p. A simple way to see thisis to observe that the identity of A is the pointwise limit of a net of maps ui : A→ A of the form

ui : Avi−→`n(i)

∞wi−→A

where n(i) are integers and vi, wi are positive contractions (see Remark 26.20). By Remark 1.28 vi

is c.p. Using this we are reduced to show that uwi : `n(i)∞ → B is c.p., and this case is covered by

the end of the preceding Remark 1.31.

Remark 1.33 (Positivity for unital forms). Let A a unital C∗-algebra and f ∈ A∗. Then

f ≥ 0⇔ f(1) = ‖f‖.

Indeed, if A is commutative, say A = C(T ) with T compact, this is a well known characterizationof positive measures on T . But if we fix x ∈ A+, and let Ax denote the (commutative) unitalC∗-algebra generated by x,

f(1) = ‖f‖ ⇒ f(1) = ‖f|Ax‖ ⇒ f|Ax ≥ 0⇒ f(x) ≥ 0,

which proves the implication from right to left. The converse direction can be proved easily usingCauchy-Schwarz for the inner product 〈y, x〉 = f(y∗x), applied with y = 1 and x ∈ BA. We actuallyalready proved a more general fact when we proved (1.20).We note in passing that for any a ∈ A

(1.23) a ≥ 0⇔ ϕ(a) ≥ 0 ∀ϕ ∈ A∗+.

Remark 1.34 (Positivity for unital maps). More generally let u : A → B be a linear map withvalues in a C∗-algebra B. If u(1) = 1 and ‖u‖ = 1 then u is positive. Indeed, by the precedingremark, for any F ∈ B∗+, the linear form f : x 7→ F (u(x)) satisfies f(1) = ‖f‖ and hence is positive.Conversely, if u is positive and unital then ‖u‖ = 1 by Corollary 26.19.

The next statement, which is a sort of recapitulation, introduces an important “bridge” betweenc.p. and c.b. maps.

Theorem 1.35. Let E ⊂ A be a subspace of a unital C∗-algebra such that 1 ∈ E. Let u : E → B(H)be a unital linear map (i.e. such that u(1) = 1). Then ‖u‖cb = 1 if and only if u extends to a c.p.map u : A→ B(H).

Proof. By the injectivity of B(H) (see Theorem 1.18), if ‖u‖cb = 1 there is an extension u : A →B(H) with ‖u‖cb = ‖u‖cb = 1. Since u(1) = 1, the preceding remark shows that u is positive, butsince the same can be applied to un = IdMn ⊗ u, we conclude that u is actually c.p. Conversely, ifu admits a c.p. extension u then we have ‖u‖ = ‖u(1)‖ = 1 and hence ‖u‖ = 1. Again applyingthis to un : Mn(E)→Mn(B(H)) we obtain ‖un‖ = 1 ∀n and hence ‖u‖cb = 1.

Definition 1.36. A (not necessarily closed) subspace S ⊂ B(H) for some Hilbert space H is calledan operator system, if it is self-adjoint and unital.

19

Clearly, if S is an operator system, Mn(S) ⊂Mn(B(H)) also is one.Let S+ = S ∩ B(H)+. If S ⊂ B(H) is an operator system, S coincides with the linear span of

S+. Indeed, if a ∈ S and a = a∗, then a = ‖a‖1 − (‖a‖1 − a) ∈ S+ − S+. This explains why c.p.maps can be used efficiently on operator systems.

The elementary proof of the next Lemma is left to the reader, to whom we recall the arith-metic/geometric mean inequality

√st ≤ (s+ t)/2, ∀s, t ≥ 0.

Lemma 1.37.

(i) Let s1, s2, a ∈ B(H). Assume s1, s2 ≥ 0. Then

(1.24)

(s1 aa∗ s2

)≥ 0⇔ |〈y, ax〉| ≤ (〈y, s1y〉〈x, s2x〉)1/2 ≤ (〈y, s1y〉+ 〈x, s2x〉)/2 ∀x, y ∈ H.

When this holds, we have

(1.25) ‖a‖ ≤ (‖s1‖‖s2‖)1/2 ≤ (‖s1‖+ ‖s2‖)/2.

(ii) In particular, (1 aa∗ 1

)≥ 0⇔ ‖a‖ ≤ 1.

Lemma 1.38. Let w : E → B(H) be a map defined on an operator space E ⊂ B(K). Let

S ⊂M2(B(K)) be the operator system consisting of all matrices

(λ1 ab∗ µ1

)with λ, µ ∈ C, a, b ∈ E,

and let W : S →M2(B(H)) be the (unital) mapping defined by

W

((λ1 ab∗ µ1

))=

(λ1 w(a)w(b)∗ µ1

).

Then ‖w‖cb ≤ 1 if and only if W is c.p.

Proof. The easy direction is W c.p. ⇒ ‖w‖cb ≤ 1. Indeed, if W is c.p. we have ‖W‖cb = ‖W (1)‖ =1, and a fortiori ‖w‖cb ≤ 1.We now turn to the converse. Assume ‖w‖cb ≤ 1. Consider an element s ∈ Mn(S), say s =(λ ab∗ µ

), λ, µ ∈ Mn(C1), a, b ∈ Mn(E). Assume s ≥ 0, then necessarily λ, µ ∈ Mn(C1)+ and

a = b. Fix ε > 0 and let λε = λ + ε1 and µε = µ + ε1 (invertible perturbations of λ and µ). Let

sε = s+ ε1. Let us denote xε = λ−1/2ε aµ

−1/2ε and let Wn = IdMn ⊗W , wn = IdMn ⊗ w. We have

(1.26)

(1 xεx∗ε 1

)=

(λ−1/2ε 0

0 µ−1/2ε

)sε

(λ−1/2ε 0

0 µ−1/2ε

),

and hence the left-hand side of the preceding equation is ≥ 0, which implies by part (ii) in Lemma1.37 that ‖xε‖ ≤ 1. Therefore if ‖w‖cb ≤ 1, we have ‖wn(xε)‖ ≤ 1, which implies that 1 wn(xε)

wn(xε)∗ 1

≥ 0.

20

But this last matrix is the same as Wn

(1 xεx∗ε 1

). Now, applying Wn to both sides of (1.26) and

using the linearity of w, we find

Wn

(1 xεx∗ε 1

)=

(λ−1/2ε 0

0 µ−1/2ε

)Wn(sε)

(λ−1/2ε 0

0 µ−1/2ε

).

Thus

Wn(sε) =

(λ

1/2ε 0

0 µ1/2ε

) 1 wn(xε)

wn(xε)∗ 1

(λ1/2ε 0

0 µ1/2ε

).

Since the right-hand side is ≥ 0, we have Wn(sε) ≥ 0 and letting ε→ 0 we conclude that Wn(s) ≥ 0,whence that W is c.p.

We now give the c.p. version of Arveson’s extension Theorem:

Theorem 1.39 (Arveson’s Extension Theorem/C.P. version). Let E ⊂ A ⊂ B(K) be an operatorsystem included in a unital C∗-subalgebra of B(K). Any c.p. map u : E → B(H) satisfies

(1.27) ‖u‖cb = ‖u‖ = ‖u(1)‖.

Moreover, u : E → B(H) extends to a c.p. map u : A→ B(H) such that ‖u‖cb = ‖u(1)‖.

Proof. We first establish (1.27). By part (ii) in Lemma 1.37, we see that ‖x‖ ≤ 1 ⇒(

1 xx∗ 1

)≥

0⇒(u(1) u(x)u(x∗) u(1)

)≥ 0. Hence by part (i) in Lemma 1.37, we find that ‖u(x)‖ ≤ ‖u(1)‖ ⇒ ‖u‖ =

‖u(1)‖.

Similarly, x ∈ Mn(S), ‖x‖ ≤ 1 ⇒(

1 xx∗ 1

)≥ 0 ⇒ ‖un(x)‖ ≤ ‖un(1)‖ = ‖u(1)‖. Hence ‖u‖cb =

supn ‖un‖ ≤ ‖u(1)‖, i.e. ‖u‖cb = ‖u(1)‖, proving (1.27).Let ϕ ∈ A∗ be any state on A. Since u is positive, we know that u(1) ≥ 0. We may assume

that ‖u(1)‖ = 1, so that 0 ≤ u(1) ≤ 1. Consider E ⊕ E ⊂ A⊕A ⊂ B(K)⊕B(K) ⊂ B(K ⊕K).Let v : E ⊕ E → B(H) be the mapping defined by

v(x⊕ y) = u(x) + (1− u(1))ϕ(y).

Then v is clearly c.p. and v(1 ⊕ 1) = 1. By (1.27) ‖v‖cb = 1. Therefore, by Theorem 1.35 vadmits a c.p. extension v : A ⊕ A → B(H) with ‖v‖cb = ‖v(1 ⊕ 1)‖ = 1. Then the map udefined by u(x) = v(x ⊕ 0), is a c.p. extension of u and ‖u‖cb ≤ ‖v‖cb = 1 = ‖u(1)‖. Thus (since1 = ‖u(1)‖ = ‖u(1)‖ ≤ ‖u‖cb) we obtain ‖u‖cb = ‖u(1)‖.

Lemma 1.40 (Schur product of c.p. mappings). Let B,C be C∗-algebras (or merely linear sub-spaces of C∗-algebras). Fix a number n ≥ 1. Consider u ∈ CP (A,Mn(B)) and v ∈ CP (B,Mn(C)).Then the mapping x 7→ [vij(uij(x))] is in CP (A,Mn(C)).

Proof. We identify Mn(B) with Mn⊗minB, so that u(x) =∑eij⊗uij(x). Consider the composition

w = (IdMn ⊗ v) u : A→Mn ⊗min Mn ⊗min C.

21

Since complete positivity is obviously preserved under composition, w ∈ CP (A,Mn⊗minMn⊗minC).We have

w(x) =∑

ijeij ⊗ v uij(x) =

∑ij

∑k`eij ⊗ ek` ⊗ vk` uij(x).

Let S : `n2 → `n2 ⊗ `n2 be the isometry defined by Sej = ej ⊗ ej . We may assume C ⊂ B(H). Thenwe have

(S ⊗ IdH)∗w(x)(S ⊗ IdH) =∑

eij ⊗ vij(uij(x)),

and hence the latter mapping is c.p.

1.4 Normal c.p. maps on von Neumann algebras

Recall that a bounded linear map u : M → N between von Neumann algebras is called “normal”if it is continuous when M and N are both equipped with the weak* topology, or equivalently ifu∗(N∗) ⊂M∗. We wish to record here the following variant of the extension theorem.

Theorem 1.41. Let M ⊂ B(K) be a von Neumann algebra. Let u : M → B(H) be a normal c.p.map with ‖u‖ = 1. Then there is a normal c.p. map u : B(K)→ B(H) extending u with ‖u‖ = 1.More precisely, there are H, a normal ∗-homomorphism π : B(K) → B(H) and a contractionW : H → H such that u(x) = W ∗π(x)W for all x ∈ M . Moreover, we can obtain the latter withH of the form H = L⊗2 K and with π(b) = IdL ⊗ b (b ∈ B(K)) for some Hilbert space L.

Proof. Consider a minimal dilation of u of the form u(x) = V ∗π(x)V for all x ∈ M as in Remark1.23. Let ξ, ξ′ ∈ π(M)(V H), say ξ = π(m)(V h), ξ′ = π(m′)(V h′) (m,m′ ∈M,h, h′ ∈ H). We have〈ξ′, π(x)ξ〉 = 〈h′, u(m′∗xm)h〉, which shows that x 7→ 〈ξ′, π(x)ξ〉 is normal on M . By the density ofthe linear span of π(M)(V H) in H this implies that π : M → B(H) is normal (see Remark 26.42).Then the result follows immediately by the special form of the normal ∗-homomorphisms describedin Theorem 26.61. Indeed, the latter says that we can find a Hilbert space L, a subspace E ⊂ L⊗2K(invariant under IdL ⊗M) and a unitary U : H → E such that π(x) = U∗PE(IdL ⊗ x)|EU for anyx ∈M . Let jE : E → L⊗2 H be the inclusion, let π(b) = IdL ⊗ b and W = jEUV : H → L⊗2 K.Then the mapping u defined by u(b) = W ∗π(b)W for any b ∈ B(H) is a c.p. extension of u with‖u‖ = 1.

Remark 1.42. In the preceding situation if u is unital the extension u is unital and W is an isometry.

In the case of the inclusion A ⊂ A∗∗ we also have a simple extension property, as follows.

Lemma 1.43. Let A be a C∗-algebra, M a von Neumann algebra. Then for any u ∈ CP (A,M)the mapping u : A∗∗ →M is a normal c.p. map extending u with ‖u‖ = ‖u‖.

Proof. We recall that, by density, u can be viewed as the unique (σ(A∗∗, A∗), σ(M,M∗))-continuousextension of u (see (26.32)). By the weak*-density of BA ∩ A+ in BA∗∗ ∩ (A∗∗)+ (see (26.36)),it follows that u is positive. Applying that to un and observing (see Proposition 26.58) thatMn(A)∗∗ = Mn(A∗∗) shows that un is positive, and hence u is completely positive.

1.5 Injective operator algebras

Definition 1.44. A C∗-algebra (or an operator space) A is called injective if there exists a com-pletely isometric embedding A ⊂ B(H) and a projection P : B(H)→ A with ‖P‖cb = 1.

22

Of course, B(H) is the fundamental example of an injective C∗-algebra. We will focus onC∗-algebras (see Remark 1.49 for the case of operator spaces).

The next result is classical.

Theorem 1.45 (Tomiyama’s Theorem [246]). If A ⊂ B is a C∗-subalgebra of a C∗-algebra B,any linear projection P : B → A with ‖P‖ = 1 is automatically completely positive and completelybounded with ‖P‖cb = 1. Moreover, P is a conditional expectation in the sense that

(1.28) ∀a1, a2 ∈ A, ∀b ∈ B P (a1ba2) = a1P (b)a2.

Proof. The (nontrivial) proof can be found in Takesaki’s book (see [241, p. 131]) or in [39, p. 13].We skip it here.

Proposition 1.46 (Extension property). Consider a C∗-subalgebra A ⊂ B(H). The following areequivalent.

(i) A is injective.

(ii) For any completely isometric embedding A ⊂ B into a C∗-algebra B there is a projectionQ : B → A with ‖Q‖cb = 1.

(iii) For any pair of operator spaces with X1 ⊂ X2 (completely isometrically) any u ∈ CB(X1, A)admits an extension u ∈ CB(X2, A) with ‖u‖cb = ‖u‖cb.

(iv) For any pair B1, B2 of C∗-algebras with B1 ⊂ B2 (C∗-subalgebra), any u in CB(B1, A) admitsan extension u ∈ CB(B2, A) with ‖u‖cb = ‖u‖cb.

Proof. Let j1 : A → B(H) and j2 : A → B denote completely isometric embeddings. By theextension Theorem 1.18, ∃j1 ∈ CB(B,B(H)) such that ‖j1‖cb = ‖j1‖cb and j1j2 = j1. Assume(i). Let P : B(H) → j1(A) be a projection such that ‖P‖cb = 1. Then Q = j2j

−11 |j1(A)P j1 is a

projection onto j2(A) with ‖Q‖cb = 1. This shows (i)⇒(ii).Assume (ii). Consider A ⊂ B(H) and a projection P : B(H) → A with ‖P‖cb = 1. By

the extension Theorem 1.18, any u ∈ CB(X1, A) admits an extension v ∈ CB(X2, B(H)) with‖v‖cb = ‖u‖cb. Then u = P v ∈ CB(X2, A) satisfies (iii). This shows (ii)⇒(iii). (iii)⇒(iv) is trivial.

Assume (iv). Consider A ⊂ B(H). Let B1 = A, B2 = B(H) and u = IdA. Then u : B(H)→ Ais a projection with cb norm 1, and hence (i) holds.

Corollary 1.47. Let A be a C∗-algebra and A1 ⊂ A a C∗-subalgebra. Assume that there is aprojection P : A→ A1 with ‖P‖ = 1. If A is injective then A1 is also injective.

We end this section with a simple stability property of injective C∗-algebras.

Proposition 1.48. Let (Ai)i∈I be a family of C∗-algebras. Then (⊕∑

i∈I Ai)∞ is injective if andonly if Ai is injective for all i ∈ I.

Proof. If Ai ⊂ B(Hi) and Pi : B(Hi) → Ai are projections with ‖Pi‖cb = 1 then the mappingP : (⊕

∑i∈I B(Hi))∞ → (⊕

∑i∈I Ai)∞ taking (xi) to (Pixi) is clearly c.b. with ‖P‖cb ≤ 1. Let

H = (⊕∑

i∈I Hi)2. If we denote by Vi : Hi → H the natural (isometric) inclusion, and if we defineQi : B(H)→ B(Hi) by Qi(T ) = V ∗i TVi and Q : B(H)→ (⊕

∑i∈I B(Hi))∞ by

Q(T ) = (Qi(T ))i∈I ,

23

then Q is clearly c.b. with ‖Q‖cb = supi ‖Qi‖cb = 1. We may identify Hi with a subspace of Hso that Vi becomes the inclusion. Then (⊕

∑i∈I Ai)∞ can be naturally identified with the C∗-

subalgebra of B(H) formed of all operators T ∈ B(H) such that THi ⊂ Hi and T|Hi ∈ Ai for alli ∈ I. With this convention, PQ is a projection from B(H) onto (⊕

∑i∈I Ai)∞ with ‖PQ‖cb = 1.

This proves the “if” part. The converse follows easily from the fact that the canonical projectionfrom (⊕

∑i∈I Ai)∞ to Ai (which is a ∗-homomorphism) has cb-norm = 1 for all i ∈ I.

Remark 1.49. For operator spaces, Tomiyama’s theorem does not hold. Therefore, it is more naturalto say that an operator space E ⊂ B(H) is c-injective if there is a projection P : B(H)→ E with‖P‖cb ≤ c. With this terminology, we call E injective if it is 1-injective. The reader will easilycheck that Proposition 1.46 and Proposition 1.48 remain valid for injective (i.e. 1-injective) operatorspaces.

See [185] for an example of a bounded linear map u : A → B(H) defined on a C∗-subalgebraA ⊂ B(H) that does not extend to a bounded map on B(H).

We return to injectivity for von Neumann algebras in §8.3. See [80, §6] for more informationon injective operator spaces.

1.6 Factorization of completely bounded (c.b.) maps

We now turn to the factorization of c.b. maps. This important result, proved independently byWittstock, Haagerup and Paulsen in the early 1980’s, can be viewed as the “linearization” of theStinespring factorization Theorem 1.22.

Theorem 1.50 (Factorization of c.b. maps). Let H,K be Hilbert spaces. Consider an operatorspace E ⊂ B(K). Let B ⊂ B(K) be a unital C∗-algebra such that E ⊂ B ⊂ B(K). Consider a c.b.map

B∪E

u−→ B(H)

Then there is a Hilbert space H, a unital ∗-homomorphism π : B −→ B(H) and operators V1, V2 ∈B(H, H) such that ‖V1‖ ‖V2‖ = ‖u‖cb and

(1.29) ∀x ∈ E u(x) = V ∗2 π(x)V1.

Conversely, if (1.29) holds then u is c.b. and

(1.30) ‖u‖cb ≤ ‖V1‖ ‖V2‖.

In addition, if V1 = V2, then u is completely positive.

The corresponding diagram is as follows:

Bπ // B(H)

b7→V ∗2 bV1

E?

OO

u // B(H)

24

Proof. It suffices to prove this when B = B(K). We may assume ‖u‖cb = 1. Consider the operatorsystem S ⊂M2(B(K)) defined in Lemma 1.38 and let W ∈ CP (S,M2(B(H))) be defined by

W

((λ1 ab∗ µ1

))=

(λ1 u(a)u(b)∗ µ1

).

By (Arveson’s extension) Theorem 1.39, there is W ∈ CP (M2(B(K)),M2(B(H))) extending W , towhich we may apply (Stinespring’s) Theorem 1.22, with A = M2(B(K)) and, setting H = H ⊕H,we have M2(B(H)) = B(H). This gives us a Hilbert space H, an operator V : H → H and a∗-homomorphism σ : M2(B(K))→M2(B(H)) such that for any x ∈ E we have(

0 u(x)0 0

)= V ∗σ

(0 x0 0

)V

or equivalently (0 u(x)0 0

)= V ∗σ

(x 00 x

)σ

(0 10 0

)V

and hence if P1 : H ⊕ H → H (resp. P2 : H ⊕ H → H) is the first (resp. second) coordinateprojection, we have

u(x) = P1V∗σ

(x 00 x

)σ

(0 10 0

)V P ∗2

and we obtain (1.29) with π(x) = σ

(x 00 x

), V1 = σ

(0 10 0

)V P ∗2 and V ∗2 = P1V

∗. Note that

‖V1‖‖V2‖ ≤ ‖V ‖2 = 1.Conversely, if (1.29) holds we obviously have 1 = ‖u‖cb ≤ ‖V1‖‖V2‖.Moreover, if V1 = V2, u is clearly c.p.

Remark 1.51. In (1.29) we may without loss of generality replace H by

H0 = spanπ(b)V1h | b ∈ B, h ∈ H ⊂ H.

Indeed, since H0 is an invariant subspace for all π(b)’s for b ∈ B, the mapping b 7→ PH0π(b)|H0

is

a representation of B on H0 and we may define W1,W2 ∈ B(H, H0) by W1h = V1h and W2h =PH0V2h, so that ‖W1‖‖W2‖ ≤ ‖u‖cb and

∀x ∈ E u(x) = W ∗2 π(x)W1.

The point of this remark is that if H,B are both separable, we can ensure that H also is. Since itis a trivial matter to enlarge H if necessary, this shows that if H = `2 and B is separable we canalways take H = `2. Note that if E is separable, we may replace B by the separable C∗-algebragenerated by E. Thus it suffices to assume E separable.Similarly, if dim(B) <∞ and dim(H) =∞, we can ensure that H and H have the same Hilbertiandimension and hence by unitary equivalence we can take H = H.

For emphasis and for later reference, we state as separate corollaries parts of Theorem 1.50 thatwill be used frequently in the sequel. The first one repeats part of Theorem 1.35.

Corollary 1.52. Let E ⊂ B(H) be an operator space containing I. Consider a map u : E → B(K).If u(I) = I and ‖u‖cb = 1, then there is a Hilbert space H with K ⊂ H and a unital representationπ : B(H)→ B(H) such that

∀ x ∈ E u(x) = PKπ(x)|K .

In particular, u is completely positive.

25

Proof. By Theorem 1.50, we have u(·) = V ∗2 π(·)V1. By homogeneity, we may assume ‖V1‖ =‖V2‖ = 1. Since I = u(I) = V ∗2 π(I)V1 = V ∗2 V1, we have 〈V2h, V1h〉 = ‖h‖2 for any h ∈ H. This

forces ‖V1h‖ = ‖V2h‖ = ‖h‖, and also V1h = V2h. Thus V1 is an isometric embedding of K into H.Identifying K with V1(K), u(·) = V ∗2 π(·)V1 becomes u(·) = PKπ(·)|K .

The second corollary is the decomposability of c.b. maps into B(H) as linear combinations ofc.p. maps. We should emphasize that while this holds for c.b. maps with range in B(H), it usuallyfails for c.b. maps with range in a C∗-subalgebra B ⊂ B(H): in general we cannot get the c.p.maps uj ’s to be B-valued. See chapter 6 for more on the decomposability theme.

Corollary 1.53. Any c.b. map u : E → B(K) can be decomposed as u = u1 − u2 + i(u3 − u4)where u1, u2, u3, u4 are c.p. maps with ‖uj‖cb ≤ ‖u‖cb. More precisely, we have ‖u1 + u2‖ ≤ ‖u‖cband ‖u3 + u4‖ ≤ ‖u‖cb.

Proof. By Theorem 1.50, we have u(·) = V ∗2 π(·)V1. By homogeneity, we may assume ‖V1‖ =

‖V2‖ = ‖u‖1/2cb = 1. Let us denote V = V1 and V2 = W , so that u(·) = W ∗π(·)V . Then the resultsimply follows from the polarization formula: we define u1, u2, u3, u4 by

u1(·) = 4−1(V +W )∗π(·)(V +W ), u2(·) = 4−1(V −W )∗π(·)(V −W ),

u3(·) = 4−1(V + iW )∗π(·)(V + iW ), u4(·) = 4−1(V − iW )∗π(·)(V − iW ).

Then, by (1.30), ‖uj‖cb ≤ 1 for j = 1, 2, 3, 4 and u = u1 − u2 + i(u3 − u4). Note that actually(u1+u2)(·) = 2−1(V ∗π(·)V +W ∗π(·)W ) and hence again by (1.30) ‖u1+u2‖cb ≤ (‖V ‖2+‖W‖2)/2 ≤1. Similarly ‖u3 + u4‖cb ≤ 1.

Remark 1.54 (GNS and Hahn decomposition). Let A be a C∗-algebra and let f ∈ A∗. Then it iswell known that that there are a ∗-homomorphism π : A → B(H) and vectors η, ξ ∈ H such thatfor any x ∈ A

f(x) = 〈η, π(x)ξ〉

and ‖ξ‖‖η‖ = ‖f‖A∗ . This can be derived from the classical GNS factorization (see (26.16)).We would like to point out to the reader that we have proved it in passing. Indeed, we can viewthis fact as a particular case of Theorem 1.50 applied with E = A to the linear map f : A → C,since by Proposition 1.3 we have ‖f‖cb = ‖f‖ in this case.Moreover, if f is a self-adjoint form i.e. if f(x) = f(x∗) for any x ∈ A then we recover the classicalHahn decomposition: there are f+, f− ∈ A∗+ such that f = f+−f− and ‖f+‖+‖f−‖ = ‖f‖. Indeed,we can take f±(x) = 4−1〈η±ξ, π(x)(η±ξ)〉 and by homogeneity we may assume ‖η‖ = ‖ξ‖ = ‖f‖1/2.Then

‖f+‖+ ‖f−‖ ≤ 4−1(‖η + ξ‖2 + ‖η − ξ‖2) = 2−1(‖η‖2 + ‖ξ‖2) = ‖f‖.

We already saw in (1.3) that every finite rank map u : E → F (between arbitrary operatorspaces) is c.b. Let α(n) be the best constant C such that, for any E,F , any map u : E → F ofrank n satisfies

‖u‖cb ≤ C‖u‖.

To majorize α(n), we will need the following classical lemma.

Lemma 1.55 (Auerbach’s Lemma). Let E be an arbitrary n-dimensional normed space. Thereis a biorthogonal system xj ∈ E, ξj ∈ E∗ (j = 1, 2, . . . , n) such that ‖xj‖ = ‖ξj‖ = 1 for allj = 1, . . . , n.

26

Proof. Choose x1, . . . , xn in the unit sphere of E on which the function x→ |det(x1, . . . , xn)| attainsits maximum, supposed equal to C > 0. Then let ξj(y) = C−1 det(x1, . . . , xi−1,y, xi+1, . . . , xn). The desired properties are easy to check.

Remark 1.56. We may as well assume dim(F ) = n. Then by Auerbach’s Lemma we can write IdFas the sum of n rank one maps of unit norm. This immediately implies

α(n) ≤ n.

However, this is not best possible: It is known (due to Eric Ricard, see [208, p. 145]) that α(n) ≤n/21/4, but the exact value of α(n), or of lim supα(n)/n, does not seem to be known although it isknown that the latter limit is ≥ 1/2, by an argument due to Paulsen (see [196] or [208, p. 75]).

To illustrate the use of the factorization theorem, we end by a well known characterization ofcomplete boundedness for Schur mulipliers.Let I be any set. We denote by (ei)i∈I the canonical basis of `2(I) and by eij (i, j ∈ I) the matrixunits in B(`2(I)). To any a ∈ B(`2(I)) we associate the matrix [aij ] (i, j ∈ I) defined as usualby aij = 〈ei, a(ej)〉. Let Φ : I × I → C be any function. Any linear mapping that takes [aij ]to [aijΦ(i, j)] is commonly called a “Schur multiplier”. The next result characterizes the Schurmulipliers that are c.b. linear maps from B(`2(I)) to itself.

Proposition 1.57 (Schur multipliers on B(`2)). Let C ≥ 0 be a constant. The following areequivalent:

(i) There is a c.b. map u : B(`2(I))→ B(`2(I)) such that u(eij) = Φ(i, j)eij for any i, j ∈ I × Iwith ‖u‖cb ≤ C.

(ii) There are a Hilbert space H and bounded functions y : I → H, x : I → H such thatsupI ‖y‖H supI ‖x‖H ≤ C and

∀i, j ∈ I × I Φ(i, j) = 〈y(i), x(j)〉.

(iii) The Schur multiplier u : B(`2(I)) → B(`2(I)) that takes [aij ] to [aijΦ(i, j)] is c.b. with‖u‖cb ≤ C.

Proof. Assume (i). By Theorem 1.50 there is π : B(`2(I)) → B(H) and V,W : `2(I) → H with‖V ‖‖W‖ ≤ C such that Φ(i, j)eij = V ∗π(eij)W . This implies Φ(i, j) = 〈ei, (V ∗π(eij)W )ej〉. Fix anelement o ∈ I. Note eij = eioeoj . Therefore we have Φ(i, j) = 〈y(i), x(j)〉 where x(j) = π(eoj)Wejand y(i) = π(eio)

∗V ei, and (ii) follows.Assume (ii). Define π : B(`2(I))→ B(H) by π(x) = x⊗IdH and let Vx : `2(I)→ `2(I)⊗2H be themap taking ei to ei ⊗ x(i) (i ∈ I). Note ‖Vx‖ = supI ‖x‖H . Let u(·) = V ∗y π(·)Vx. Then u coincideswith the Schur multiplier in (iii) and ‖u‖cb ≤ C. Thus (ii) ⇒ (iii). (iii) ⇒ (i) is trivial.

Remark 1.58. Actually, it is known that (iii) ⇒ (ii) holds even if we merely assume that u is abounded Schur multiplier with ‖u‖ ≤ C. In other words the c.b. norm and the norm of a Schurmultiplier on B(`2(I)) are equal. We prove this as a consequence of a more general phenomenonin Corollary 2.7.

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1.7 Normal c.b. maps on von Neumann algebras

It is well known that the Banach space X = L1(µ) has the property that there is a contractiveprojection from X∗∗ to X. This corresponds to the decomposition into absolutely continuous andsingular parts (with respect to µ) in the abstract L1-space X∗∗.The non-commutative analogue is also true, as follows. Let M be a von Neumann algebra withpredual M∗. We will work with M∗∗ that we assume realized as a von Neumann subalgebra ofB(K). Since this is a source of mistakes, as a preliminary precaution Remark 26.60 is recommendedreading.

For any linear map u : M → B(H), we will use the notation (see §26.16)

u = (u∗|B(H)∗)∗ : M∗∗ → B(H).

Note that u is normal and for any f ∈ B(H)∗ and any z ∈M∗∗ we have

(1.31) 〈f, u(z)〉 = 〈u∗(f), z〉.

If u is a unital ∗-homomorphism, u is also one. In particular taking u = IdM we find a normalunital ∗-homomorphism π : M∗∗ →M that extends the identity on M .

Consider then the set I0 ⊂ M∗∗ that is the annihilator of M∗ ⊂ M∗. It is immediate that I0

is a weak* closed two-sided ideal in M∗∗ because I0 = ker(π). It follows (see Remark 26.34) thatthere is a central projection Q0 ∈M∗∗ such that I0 = Q0M

∗∗.

Lemma 1.59. For any bounded linear map u : M → B(H) we define

∀x ∈M uN (x) = u((1−Q0)x) and uS(x) = u(Q0x).

Then uN : M → B(H) is normal with ‖uN ‖ ≤ ‖u‖ and in the c.b. case ‖uN ‖cb ≤ ‖u‖cb. If u isnormal we have uN = u. Moreover if u is a ∗-homomorphism (resp. is c.p.) so is uN .

Proof. If u is a ∗-homomorphism (resp. is c.p.) so is u, thus (Q0 being central) the last assertionis obvious, as well as the norm inequalities. To show that uN is normal it suffices to show thatu∗N (B(H)∗) ⊂ M∗. A priori we only know u∗N (B(H)∗) ⊂ M∗. By the bipolar criterion (appliedto the duality between M∗ and M∗∗) it suffices to show that u∗N (B(H)∗) ⊂ M∗

⊥⊥ = I0⊥, or

equivalently to show that 〈u∗N (f), z〉 = 0 for any f ∈ B(H)∗ and any z ∈ I0. This can be checkedas follows. Let (zi) be a bounded net in M tending weak* to z ∈ I0. Then (see Remark 26.37)

(1−Q0)ziσ(M∗∗,M∗)−→ (1−Q0)z = 0.

Since uN (zi) tends weak* to u∗∗N (z) we have

〈u∗N (f), z〉 = 〈f, u∗∗N (z)〉 = lim〈f, uN (zi)〉 = lim〈f, u((1−Q0)zi)〉 = lim〈u∗(f), (1−Q0)zi〉 = 0,

where the last step uses (1.31). This completes the proof that uN is normal.Assume u normal. We will show that 〈f, uS(x)〉 = 0 for any f ∈ B(H)∗ and x ∈ M . Indeed, wehave 〈f, uS(x)〉 = 〈u∗(f), Q0x〉 and Q0x ∈ I0 = M⊥∗ while u∗(f) ∈ M∗ therefore 〈f, uS(x)〉 = 0.Since B(H)∗ separates the points of B(H) this implies uS = 0 and hence uN = u.

Theorem 1.60. Let M ⊂ B(K) be a von Neumann algebra. Let u : M → B(H) be a normal c.b.map. Then there is a normal c.b. map u : B(K)→ B(H) extending u with ‖u‖cb = ‖u‖cb.There are H, a normal ∗-homomorphism π : B(K)→ B(H) and operators V1 : H → H, V2 : H →H with ‖V1‖‖V2‖ = ‖u‖cb such that u(x) = V ∗2 π(x)V1 for all x ∈M . Moreover, we can obtain this

with H of the form H = L⊗2 K and with π(x) = IdL ⊗ x (x ∈ B(K)) for some Hilbert space L.

28

Proof. By Theorem 1.50 we can find H, π : M → B(H) and V1, V2 such that u(·) = V ∗2 π(·)V1. Wehave clearly u(·) = V ∗2 π(·)V1 and hence uN (·) = V ∗2 πN (·)V1 and since u is normal u = uN . Thus wemay replace π by πN and assume that π is normal. Then the proof can be completed by applyingTheorem 1.41 to u = π (or, more directly, by invoking Theorem 26.61).

We end by a very simple observation for later use.

Lemma 1.61. Let A be a C∗-algebra, M a von Neumann algebra. Then for any u ∈ CB(A,M)the mapping u : A∗∗ →M is a normal c.b. map extending u with ‖u‖cb = ‖u‖cb.

Proof. Recall that u is the unique (σ(A∗∗, A∗), σ(M,M∗))-continuous extension of u. SinceMn(A)∗∗ =Mn(A∗∗) (see Proposition 26.58), it follows from the weak*-density of BMn(A) in BMn(A∗∗) that‖un‖ = ‖un‖.

1.8 Notes and Remarks

The history of complete positivity starts with Stinespring’s 1955 paper where he proves his factor-ization theorem for c.p. maps on a C∗-algebra A. The case when A is commutative was alreadyknown due to Naimark’s work on spectral measures. In two major very influential Acta Mathemat-ica papers in 1969 and 1972 Arveson [12] considerably expanded on Stinespring’s breakthrough. Hemade the crucial step of considering complete positivity for maps defined only on operator systems,and he proved his extension theorem. Later on, Choi and Effros [47] made a deep study of injectiv-ity for operator systems, that somehow opened the way for the later development of operator spacetheory. While it seems a bit surprising in retrospect, the factorization of completely bounded mapsemerged only in the early 1980’s through independent works by Wittstock, Haagerup (unpublished)and Paulsen. We refer the reader to Paulsen’s book [196] for details and more proper credit on thegenesis of that important result, which is fundamental for operator space theory. The latter wasignited by Ruan’s 1987 PhD thesis where his abstract characterization of operator spaces is proved.This opened the way to the study of duality for operator spaces (see §2.4), which was thoroughlyinvestigated independently by Effros-Ruan and Blecher-Paulsen. The books by Effros and Ruan[80], by Paulsen [196], by Blecher and Le Merdy [27], as well as our own [208], provide multiplecomplements to our presentation of complete positivity and complete boundedness. See Størmer’s[236] for more on the comparison between positivity and complete positivity. Theorem 1.57 andRemark 1.58 about Schur multipliers have a long history: some essentially equivalent formulation(up to some factor 2) can be traced back to Grothendieck’s [98]. Later on the result was redis-covered independently by J. Gilbert and U. Haagerup (see [210] or [207, p. 100] for details). Theappendix of Haagerup’s manuscript dating from 1986 but recently published as [108] contains moreresults on Schur multipliers. See also Corollary 2.7.

2 Completely bounded and completely positive maps: a tool kit

In this follow-up chapter on c.b. and c.p. maps we include a variety of related topics that willlater allow us to better illustrate the C∗-algebraic tensor product theory, and its significance forspaces of linear mappings between operator spaces or C∗-algebras. We advise the reader to browsethrough this chapter on first reading and return to each specific topic whenever needed.

2.1 Rows and columns. Operator Cauchy-Schwarz inequality

Let (xj) be an n-tuple in a C∗-algebra A. Let r ∈Mn(A) be the “row matrix” that has (xj) on itsfirst row and zero everywhere else. In other words r1j = xj and rij = 0 for any i > 1. Equivalently,

29

in tensor product notation r =∑n

j=1 e1j ⊗ xj ∈Mn ⊗A. Then

‖r‖Mn(A) = ‖∑

xjx∗j‖1/2.

Indeed, this follows from ‖r‖Mn(A) = ‖rr∗‖1/2Mn(A). Motivated by this, we will frequently use thenotation

‖x‖R = ‖∑

xjx∗j‖1/2.

Analogously, if c ∈ Mn(A) is the “column matrix” that has (xj) on its first column and zeroeverywhere else, or in tensor product notation if c =

∑nj=1 ej1 ⊗ xj ∈Mn ⊗A, then

‖c‖Mn(A) = ‖∑

x∗jxj‖1/2.

This follows either from ‖c‖Mn(A) = ‖c∗c‖1/2Mn(A) or from the fact that r = c∗ is a row matrix. Wewill use the notation

‖x‖C = ‖∑

x∗jxj‖1/2.

These simple remarks lead us to the following useful test to check whether a map is c.b. or toevaluate its c.b. norm.

Proposition 2.1. Let E ⊂ B(H), F ⊂ B(K) be operator spaces. Then any u ∈ CB(E,F ) satisfiesthe following inequalities for any n and any n-tuple (xj) in E:

‖∑

u(xj)u(xj)∗‖1/2 ≤ ‖u‖cb‖

∑xjx∗j‖1/2,

‖∑

u(xj)∗u(xj)‖1/2 ≤ ‖u‖cb‖

∑x∗jxj‖1/2.

Proof. Just observe that un takes a row (resp. column) matrix to a row (resp. column) matrix.

Remark 2.2. Let Rn = span[e1j ] ⊂ Mn and Cn = span[ej1] ⊂ Mn. Using the preceding test thereader can check as an easy exercise that the linear mapping u : Rn → Cn defined by u(e1j) = ej1(which is isometric) satisfies ‖u‖cb ≥

√n, and in fact ‖u‖cb =

√n.

Another classical exercise consists in checking that the c.b. norm of transposition viewed as a linearmap on Mn is equal to n (hint: compute ‖

∑ij eij ⊗ eij‖ and ‖

∑ij eij ⊗ eji‖).

It will be convenient to record here several simple consequences of the classical Cauchy-Schwarzinequality.

Lemma 2.3. Let (ai)i∈I and (bi)i∈I be finitely supported families of operators in B(H). We havefor any bounded family (xi) in B(H)

(2.1)∥∥∥∑

i∈Iaixibi

∥∥∥ ≤ ∥∥∥∑ aia∗i

∥∥∥1/2sup ‖xi‖

∥∥∥∑ b∗i bi

∥∥∥1/2.

In particular

(2.2)∥∥∥∑

i∈Iaibi

∥∥∥ ≤ ∥∥∥∑ aia∗i

∥∥∥1/2 ∥∥∥∑ b∗i bi

∥∥∥1/2.

Proof. See (26.12).

More generally:

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Lemma 2.4. Let A ⊂ B(H) be a C∗-algebra. Let (ai)1≤i≤n and (bj)1≤j≤n be operators in A, andlet x = [xij ] ∈Mn(A). Then

(2.3)∥∥∥∑n

i,j=1aixijbj

∥∥∥ ≤ ∥∥∥∑ aia∗i

∥∥∥1/2‖x‖Mn(A)

∥∥∥∑ b∗jbj

∥∥∥1/2.

Proof.

‖∑n

i,j=1aixijbj‖ = sup

ξ,η∈BH

∣∣∣∑〈η, aixijbjξ〉∣∣∣ = supξ,η∈BH

∣∣∣∑〈a∗i η, xijbjξ〉∣∣∣≤ ‖[xij ]‖Mn(B(H)) sup

ξ∈BH

(∑‖bjξ‖2

)1/2supη∈BH

(∑‖a∗i η‖2

)1/2

≤ ‖x‖Mn(E)

∥∥∥∑ b∗jbj

∥∥∥1/2 ∥∥∥∑ aia∗i

∥∥∥1/2.

In particular, we quote the following variant for future reference.

Lemma 2.5. Let A ⊂ B(H) be a C∗-algebra and let (aj), (bj) be finite sequences in A such that‖∑aja∗j‖ ≤ 1 and ‖

∑b∗jbj‖ ≤ 1. For any ϕ ∈ A∗ let ϕj be defined by ϕj(x) = ϕ(ajxbj). Then

(2.4)∑‖ϕj‖A∗ ≤ ‖ϕ‖A∗ .

Proof. We have∑‖ϕj‖A∗ = sup|

∑ϕj(xj)| where the sup runs over all xj ∈ BA. Then by (2.1)

we havesup|

∑ϕj(xj)| = |ϕ(

∑ajxjbj)| ≤ ‖ϕ‖A∗ ,

whence (2.4).

2.2 Automatic complete boundedness

Let u : E → B be a bounded linear map from an operator space E to a C∗-algebra B. Wealready saw (see Proposition 1.3) that if B is commutative then u is automatically c.b. and‖u‖cb = ‖u‖. Essentially, this phenomenon reduces to the case B = C. We will now show a veryuseful generalization of the latter fact involving “cyclicity”. We recall that a ∗-homomorphismπ : A → B(H) on a C∗-algebra is called cyclic if there is a vector ξ ∈ H (itself called cyclic) suchthat π(A)ξ = H. Note that when dim(H) = 1 any nonzero π is trivially cyclic.

Theorem 2.6 ([233]). Let E ⊂ B(H) be an operator space. Let u : E → B(H) be a boundedlinear map. Assume that there are unital C∗-subalgebras A1, A2 ⊂ B(H) and ∗-homomorphismsπ1 : A1 → B(H) and π2 : A2 → B(H) with respect to which E is a bimodule and u is bimodular,meaning that for all aj ∈ Aj and all x ∈ E we have

a1xa2 ∈ E and u(a1xa2) = π1(a1)u(x)π2(a2).

If π1 and π2 are cyclic then u is c.b. and ‖u‖cb = ‖u‖.

Proof. We may assume ‖u‖ = 1. Let ξj be a cyclic unit vector for πj . Let n ≥ 1 and x = [xij ] ∈Mn(E) with ‖x‖Mn(E) ≤ 1. To complete the proof it suffices to show that ‖[u(xij)]‖Mn(B(H)) ≤ 1,or that |

∑ij〈ki, u(xij)hj〉| ≤ 1 for any (ki), (hj) ∈ Hn such that

∑‖ki‖2 < 1 and

∑‖hj‖2 < 1. By

cyclicity, we may assume that ki = π1(ai)ξ1 and hj = π2(bj)ξ2 for some ai ∈ A1, bj ∈ A2. Assume

31

for the moment that a = (∑a∗i ai)

1/2 and b = (∑b∗jbj)

1/2 are invertible. We may then factorize

ai,bj as ai = a′ia and bj = b′jb where a′i = aia−1 and b′j = bjb

−1. Let ξ′1 = π1(a)ξ1 and ξ′2 = π2(b)ξ2.

A simple verification shows that∑a′i∗a′i = 1,

∑b′j∗b′j = 1 and also that ‖ξ′1‖2 =

∑‖ki‖2 < 1 and

‖ξ′2‖2 =∑‖hj‖2 < 1. Therefore using the modular assumptions we have∣∣∣∣∣∣

∑ij

〈ki, u(xij)hj〉

∣∣∣∣∣∣ =∣∣∣∑〈ξ′1, π1(a′i

∗)u(xij)π2(b′j)ξ

′2〉∣∣∣ =

∣∣∣〈ξ′1, u(∑

a′i∗xijb

′j)ξ′2〉∣∣∣ ≤ ‖u‖ ∥∥∥∑ a′i

∗xijb

′j

∥∥∥and hence by (2.3) ≤ ‖u‖. This proves the result assuming a, b invertible. The general case requiresa minor adjustment: fixing ε > 0 we set a = (ε1+

∑a∗i ai)

1/2 and b = (ε1+∑b∗jbj)

1/2. Then a, b areinvertible and a simple modification of the preceding argument leads to the same conclusion.

Corollary 2.7 (Schur multipliers as module maps). For any Schur multiplier u : B(`2(I)) →B(`2(I)) as in Proposition 1.57 we have ‖u‖cb = ‖u‖.

Proof. We apply Theorem 2.6 taking for A1 and A2 the algebra of diagonal operators on `2(I). Wemay reduce to the case when I is countable, in which case the latter algebra has a cyclic vector.

2.3 Complex conjugation

Let E be a Banach space. We will denote by E the complex conjugate of E, i.e. the vector space Ewith the same norm but with the conjugate multiplication by a complex scalar. We will denote byx→ x the identity map from E to E. Thus, x and x are the same element but we “declare” that

∀λ ∈ C λx = λx.

The space E is anti-isometric to E. Perhaps, we should warn the reader that although this notionis very simple, it is easy to get confused by it.

Remark 2.8. For any Hilbert space H the dual H∗ is a Hilbert space that can be canonicallyidentified with H using the (sesquilinear) scalar product. Let (ej) (resp. (fi)) be othonormal basesin a Hilbert space H (resp. K). The spaces H∗ (resp. K∗) can be equipped with the biorthogonalorthonormal bases (that can be identified with (ej) and (fi) in H and K). Let a ∈ B(H,K). We

define a : H → K by setting a(h) = a(h) for any h ∈ H. It is easy to see that a 7→ a is anisomorphism from B(H,K) to B(H,K).We can associate a biinfinite matrix [aij ] in the usual way so that aij = 〈fi, u(ej)〉. Then on onehand [aij ] is the matrix associated to a, and on the other hand, the Banach space sense adjointoperator from K∗ to H∗ admits the transposed matrix [aji] as associated matrix. We choose todenote it by Ta : K∗ → H∗ to avoid the conflict with the usual Hilbert sense adjoint a∗ : K → H.Note that a 7→ Ta is linear while a 7→ a∗ is antilinear. A moment of thought shows that themapping Ta : K∗ → H∗ can be canonically identified with a∗ : K → H.Incidentally, it is perhaps worthwhile to remind the reader that while H∗ ' H is canonical, theidentifications H∗ ' H and H ' H depend on the choice of an orthonormal basis.

In sharp contrast, for general Banach spaces E is not (C-linearly) isomorphic to E (see [33]).When A is a C∗-algebra, A is also a C∗-algebra for the same product and involution. This

allows us to extend the notion of complex conjugate to operator spaces: for any operator spaceE ⊂ A, we define E ⊂ A as the corresponding subspace of A.

Equivalently, assuming E ⊂ B(H), by what precedes B(H) can be canonically identified withB(H), thus the embedding

E ⊂ B(H) = B(H)

32

allows us to equip E with an operator space structure. Moreover, if E ⊂ B(H) is a C∗− (resp. vonNeumann) subalgebra, so is E ⊂ B(H).

To be more “concrete”, if an operator space E is given as a collection of bi-infinite matrices[aij ] | a ∈ E (representing operators acting on `2), then the space formed of all operators withcomplex conjugate matrices [aij ] | a ∈ E is C-linearly (completely) isometrically isomorphic toE.

Remark 2.9. It is an easy exercise to check that the injectivity of E is equivalent to that of E.

Remark 2.10. Let (ei)i∈I be an orthonormal basis of H. While the isomorphism B(H) ' B(H)is canonical (which means independent of the choice of orthonormal basis), there is also a non-canonical isomorphism B(H) ' B(H), associated to the usual (basis dependent) isometric isomor-phism H ' H that takes ei to ei (i ∈ I). In the case of Mn = B(`n2 ), the (completely isometric)isomorphism π : Mn → Mn is the linear map defined by π([aij ]) = [aij ] or in tensor product no-

tation π(∑aijeij) =

∑aijeij =

∑aijeij . The underlying map from Mn to itself is the complex

conjugation.

Let E be an operator space. By the preceding definition of E the norm of Mn(E) = Mn⊗minEis characterized by the following identity

∀aj ∈Mn ∀xj ∈ E ‖∑n

1aj ⊗ xj‖Mn⊗minE

= ‖∑n

1aj ⊗ xj‖Mn⊗minE

.

In other words, the operator space structure of E is precisely defined so that Mn⊗minE is naturallyanti-isometric to Mn ⊗min E. For each matrix [aij ] in Mn(E), we simply have

‖[aij ]‖Mn(E) = ‖[aij ]‖Mn(E).

In sharp contrast to B(H), there are examples (see [60]) of von Neumann algebras E which failto be C∗-isomorphic to E.

Let H,K be Hilbert spaces. Assume H = `2(I1), K = `2(I2). Any ξ ∈ H ⊗2 K can berepresented by a kernel [ξ(i1, i2)] so that the series ξ =

∑I1×I2 ξ(i1, i2)ei1⊗ei2 converges in H⊗2K '

`2(I1×I2). To any pair ξ, η ∈ H⊗2K we associate a linear form on B(H)⊗minB(K) ⊂ B(H⊗2K)defined by

ϕξ,η(t) = 〈ξ, tη〉.

Let hξ : `2(I2)→ `2(I1) be the (Hilbert-Schmidt) linear operator associated to ξ in the usual way,so that if ξ = ei1 ⊗ ei2 then (with the usual matrix conventions) hξ = ei1i2 ((i1, i2) ∈ I1 × I2). Asusual we associate to an operator a ∈ B(H) the matrix [aij ] defined by aij = 〈ei, aej〉 for (i, j) ∈ I1

and similarly for b. We denote by tb ∈ B(K) the operator associated to the transposed matrix [bji].We observe that (note that h∗ξahη

tb ∈ S1(K,K))

(2.5) ∀a ∈ B(H), b ∈ B(K), ϕξ,η(a⊗ b) = tr(h∗ξahηtb),

Indeed, it suffices to check this when ξ = ei1 ⊗ ei2 and η = ej1 ⊗ ej2 and then both terms in (2.5)are easily seen to be equal to ai1,j1bi2,j2 .

We now wish to apply the same formula to B(H)⊗B(K). In that case, to any ξ ∈ H ⊗2 K weassociate the kernel [ξ(i1, i2)] defined by ξ =

∑I1×I2 ξ(i1, i2)ei1 ⊗ ei2 so that hξ : K → H remains

the same. The formula becomes

(2.6) ∀a ∈ B(H), b ∈ B(K), ϕξ,η(a⊗ b) = tr(h∗ξahηb∗).

33

Indeed, tb can be identified with the adjoint operator b∗ ∈ B(H), with associated matrix [bji].

Let t ∈ B(H)⊗B(K), be a finite sum t =∑aj ⊗ bj . Then ‖t‖

B(H)⊗minB(K)= sup|ϕξ,η(t)| where

the sup runs over ξ, η in the unit ball of H ⊗2 K. This gives us

(2.7) ‖∑

aj ⊗ bj‖B(H)⊗minB(K)= sup

y,x∈B2

∣∣∣∑j

tr(y∗ajxb∗j )∣∣∣ ,

where B2 denotes the unit ball in S2(K,H). Let us denote by ‖ ‖2 the norm in S2(K,H) orS2(H,K). Then (2.7) can be rewritten as

(2.8) ‖∑

aj ⊗ bj‖B(H)⊗minB(K)= sup

x∈B2

‖∑

ajxb∗j‖2 = sup

y∈B2

‖∑

b∗jy∗aj‖2.

We need to record all variants of this for future reference:

Proposition 2.11. Consider finite sequences (aj) in B(H) and (bj) in B(K). Then we have

(2.9)∥∥∥∑ aj ⊗ bj

∥∥∥B(H)⊗minB(K)

= supx∈B2

∥∥∥∑ bjx∗a∗j

∥∥∥2

= supy∈B2

∥∥∥∑ a∗jybj

∥∥∥2

(2.10) = supx∈B2

∥∥∥∑ ajxb∗j

∥∥∥2

= supy∈B2

∥∥∥∑ b∗jy∗aj

∥∥∥2.

Proof. Since ‖∑aj ⊗ bj‖B(H)⊗minB(K)

= ‖∑bj ⊗ aj‖B(K)⊗minB(H)

by Remark 1.9, it is easy to

derive (2.9) from (2.8), exchanging the roles of (aj), B(H) and (bj), B(K). Then (2.10) followsusing ‖T‖2 = ‖T ∗‖2 for all T in S2(H,K) or S2(K,H).

When H = K and aj = bj we can make these formulae more precise:

Proposition 2.12. Consider a finite sequence (aj) in B(H). Then we have

(2.11)∥∥∥∑ aj ⊗ aj

∥∥∥B(H)⊗minB(H)

= sup∣∣∣∑

jtr(xa∗jy

∗bj)∣∣∣ | x, y ∈ BS2(H), x, y ≥ 0

.

Moreover, if∑aj ⊗ aj is self-adjoint, then the preceding supremum is unchanged if we restrict it

to x = y ≥ 0 in BS2(H).

Proof. Every x in S2 can be written as x1 − x2 + i(x3 − x4) with x1, . . . , x4 all ≥ 0 such that‖x1‖22 + ‖x2‖22 + ‖x3‖22 + ‖x4‖22 = ‖x‖22. From this fact (applied to both x and y) it is easy todeduce (2.11) from (2.7). Moreover, when

∑aj ⊗ aj is self-adjoint, the sesquilinear form (y, x) 7→∑

j tr(xa∗jy∗aj) is symmetric, and hence (by polarization) the supremum in (2.11) remains the same

if we restrict it to x = y.

Remark 2.13 (Opposite C∗-algebra). Let A be a C∗-algebra. We define the opposite C∗-algebra,that we denote by Aop, as the same C∗-algebra but with the reverse product, so that the productof a, b in Aop is defined as ba. The involution remains unchanged. It turns out that this is nothingbut another way to consider A. Indeed, as is easy to check, the mapping a 7→ a∗ is a (C-linear)isomorphism from A to Aop. Therefore for any C∗-algebra B and any aj ∈ A, bj ∈ B (1 ≤ j ≤ n)we have

(2.12) ‖∑

aj ⊗ bj‖A⊗minB= ‖

∑a∗j ⊗ bj‖Aop⊗minB.

If H,K are Hilbert spaces, we have clearly H ⊗2 K ' H ⊗2 K, and hence B(H)⊗min B(K) 'B(H)⊗min B(K). Therefore:

(2.13) ‖∑

aj ⊗ bj‖A⊗minB= ‖∑

aj ⊗ bj‖A⊗minB= ‖

∑aj ⊗ bj‖A⊗minB

.

34

Remark 2.14 (Conjugate of group representation). We will also use complex conjugation for grouprepresentations. Given a group G and a unitary group representation π : G→ B(H), let π(t) = π(t)for any t ∈ G. This defines a unitary representation π : G→ B(H).

2.4 Operator space dual

The existence of the operator space dual is a consequence of Ruan’s fundamental theorem, describingfor any vector space E, the sequences of norms on the spaces Mn(E) | n ≥ 1 that come from anembedding of E in B(H), in other words that are associated to an operator space structure on E.But actually we prefer to give a direct proof avoiding the use of Ruan’s theorem.

Theorem 2.15. Let E be an operator space. There is a Hilbert space H and an isometric embedding

J : E∗ → B(H)

such that, for all n ≥ 1 and all ξ = [ξij ] ∈Mn(E∗), we have

‖[J(ξij)]‖Mn(B(H)) = ‖uξ‖cb

where uξ : E →Mn is the linear map naturally associated to ξ.

Proof. Consider the set D that is the disjoint union of the unit balls in Mn(E), i.e. we have

D =·⋃

n≥1

BMn(E).

Then for any t ∈ D, we have t ∈ BMn(E) for some n = n(t) and we denote by

vt : E∗ →Mn(t)

the linear map associated to t. We then define

∀ξ ∈ E∗ J(ξ) =⊕t∈D

vt(ξ) ∈ (⊕∑

Mn(t))∞.

Since, by (1.3), ‖ξ‖cb = ‖ξ‖ and vt(ξ) = (IdMn(t)⊗ ξ)(t) ∈Mn(t), we have

‖vt(ξ)‖Mn(t)≤ ‖ξ‖

and hence ‖J(ξ)‖ ≤ ‖ξ‖, but using only t ∈ BM1(E) = BE we find ‖J(ξ)‖ ≥ ‖ξ‖. Thus J isisometric.

Consider now ξ = [ξij ] ∈ Mn(E∗) with associated linear map uξ : E → Mn. We have then by(1.2)

‖uξ‖cb = supt∈D‖(IdMn(t)

⊗ uξ)(t)‖Mn(t)(Mn)

but since (modulo permutation of factors)

Mn(t)(Mn) 3 (IdMn(t)⊗ uξ)(t) ' [vt(ξij)] ∈Mn(Mn(t))

and since such a permutation clearly preserves the norms we have

‖uξ‖cb = supt∈D‖[vt(ξij)]‖Mn(Mn(t)) = ‖[J(ξij)]‖Mn(B(H)),

where at the last step we used (1.13).

35

The operator space dual E∗ is defined as the one obtained by equipping E∗ with the sequenceof the norms on Mn(E∗) derived from the embedding J in the preceding statement. Thus we haveisometrically

(2.14) Mn(E∗) = Mn ⊗min E∗ = CB(E,Mn).

More generally, we have for any operator space G an isometric inclusion

(2.15) G⊗min E∗ ⊂ CB(E,G) (or equivalently E∗ ⊗min G ⊂ CB(E,G)).

Indeed, this is easy to deduce from the case G = Mn using (1.9) or (1.11).The norm in Mn(E∗) is described in a more suggestive way by the formula

(2.16) ∀ξ ∈Mn(E∗) ‖ξ‖Mn(E∗) = sup‖ξ · x‖Mn(Mm) | x ∈ BMm(E), m ≥ 1

whereξ · x =

∑1≤ij≤n,1≤k,l≤m

eij ⊗ ekl ξij(xkl).

By (1.18) the equality (2.16) still holds if we restrict the sup to m = n.The resulting formula then appears as an extension of ‖ξ‖E∗ = sup|ξ(x)| | x ∈ BE (case n = 1).

We can reverse the roles of E and E∗, as follows. We will show that for any G we also haveisometric embeddings

(2.17) E ⊗min G ⊂ CB(E∗, G) (or equivalently G⊗min E ⊂ CB(E∗, G)).

Let t ∈ E ⊗ G and let t : E∗ → G be the associated linear map. For any v ∈ CB(E,Mn) lettv ∈ Mn(E∗) be the associated tensor. By (2.14), the correspondence v 7→ tv is a bijection fromBCB(E,Mn) to BMn(E∗). Observe that (v ⊗ IdG)(t) = (IdMn ⊗ t)(tv). By (1.9) (with G and Einterchanged) we have ‖t‖min = sup‖(v ⊗ IdG)(t)‖Mn(G) | n ≥ 1, v ∈ BCB(E,Mn), and hence

‖t‖min = sup‖(IdMn ⊗ t)(tv)‖Mn(G) | n ≥ 1, tv ∈ BMn(E∗) = ‖t‖cb,

which proves that (2.17) is isometric.Taking G = Mn in (2.17) and recalling (2.14) we find that the inclusion Mn(E) ⊂ Mn((E∗)∗)

is isometric. This shows that the inclusion E ⊂ (E∗)∗ = E∗∗ is completely isometric. In particularif E is reflexive as a Banach space then (E∗)∗ = E completely isometrically.

From these remarks, the following statement emerges naturally:

Lemma 2.16. Let E,F be operator spaces. For any u ∈ CB(E,F ) the adjoint u∗ : F ∗ → E∗ isc.b. and

‖u∗‖cb = ‖u‖cb.

Proof. Composition by u on the left gives us a contraction from CB(F,Mn) to CB(E,Mn) whichcan be restated as ‖(u∗)n : Mn(F ∗)→Mn(E∗)‖ ≤ ‖u‖cb. This implies ‖u∗‖cb ≤ ‖u‖cb. Iterating wefind ‖(u∗)∗‖cb ≤ ‖u∗‖cb, and by the preceding remark ‖u‖cb ≤ ‖(u∗)∗‖cb.

2.5 Bi-infinite matrices with operator entries

In the completely bounded context, it is natural to wonder whether one can replace Mn(E) bythe space M∞(E) of bi-infinite matrices with entries in E. Let us first clarify what we mean byM∞(E). Let E ⊂ B(H) be an operator space. Recall the notation `2(H) = H ⊕H ⊕ · · · . We may

36

clearly represent an operator a ∈ B(`2(H)) by a matrix [aij ] with entries in B(H). We denote byM∞(B(H)) the space of such matrices equipped with the norm transplanted from B(`2(H)) (sothat M∞(B(H)) ' B(`2(H)) isometrically).

Then we define M∞(E) ⊂ M∞(B(H)) as the subspace formed of those a ∈ M∞(B(H)) suchthat aij ∈ E for all i, j.

The norm of a ∈ M∞(E) is easy to compute in terms of the truncated matrices: we have(assuming the indices i, j run over 1, 2, · · · )

‖a‖M∞(E) = supn ‖[aij ]1≤i,j≤n‖Mn(E).

Moreover, if we are given a priori the entries aij ∈ E then there is a ∈M∞(E) admitting these asits entries if and only if

supn ‖[aij ]1≤i,j≤n‖Mn(E) <∞.

Therefore it is evident that for any operator spaces E,F and any u ∈ CB(E,F ) we have

‖u‖cb = ‖u∞ : M∞(E)→M∞(F )‖

where u∞ : M∞(E)→M∞(F ) is defined by

u∞([aij ]) = [u(aij)].

While we used `2 for simplicity, we could just as well use an arbitrary Hilbert space H instead:We define MH(B(H)) = B(H⊗2 H), and

MH(E) = x ∈MH(B(H)) | x(ξ, η) ∈ E ∀ξ, η ∈ H

where this time, for any ξ, η ∈ H, we have denoted by x(ξ, η) the element of B(H) obtained underthe natural action of the functional fξ,η ∈ B(H)∗ defined by fξ,η(y) = 〈ξ, yη〉. Equivalently, for anyη ∈ H, we let vη : H → H⊗2 H be defined by vη(h) = η ⊗ h for any h ∈ H, and we set

x(ξ, η) = v∗ξxvη.

The collection of all the finite dimensional subspaces of H directed by inclusion gives us a simpleformula for the norm of a ∈MH(E). We have (the easy verification is left to the reader)

(2.18) ‖a‖MH(E) = sup ‖(v∗ ⊗ I)a(v ⊗ I)‖Mn(E)

where the supremum runs over all n and all isometric embeddings v : `n2 → H.Again if H is infinite dimensional, we have

(2.19) ‖u‖cb = ‖uH : MH(E)→MH(F )‖

where uH : MH(E) → MH(F ) is defined as the only mapping such that [uH(a)](ξ, η) = u(a(ξ, η))for any ξ, η ∈ H. Moreover, if we are given a priori a sesquilinear map (ξ, η) 7→ a(ξ, η) (linear in η,antilinear in ξ) from H×H to E then this will come from an element of MH(E) if and only if thematrices [avij ] defined by [avij ] = a(vei, vej) satisfy

(2.20) sup ‖[avij ]‖Mn(E) <∞

with the sup as before.

37

Using the definition of the dual operator space E∗ given in §2.4, we find that we have anisometric identity

(2.21) CB(E,B(H)) = MH(E∗).

Indeed, if dim(H) < ∞ this is the very definition of MH(E∗). Now, if H is arbitrary, for anyu ∈ CB(E,B(H)) we have ‖u‖cb = sup‖x 7→ v∗u(x)v‖cb where the sup is as before, and moreoverif u : E → B(H) is any linear map then u ∈ CB(E,B(H)) iff sup‖x 7→ v∗u(x)v‖cb < ∞ wherethe sup is as before. Comparing this with (2.18) and (2.20), we obtain (2.21).

Note that the algebraic tensor product B(H)⊗B(H) is weak*-dense in B(H⊗2 H). When thesubspaces F ⊂ B(H) and E ⊂ B(H) are weak* closed in B(H), we denote by F ⊗E ⊂ B(H⊗2 H)the weak* closure of E ⊗ F . With this notation we have B(H⊗2 H) = B(H)⊗B(H). We observefor later use:

Proposition 2.17. If E ⊂ B(H) is weak* closed in B(H), then for any H we have

(2.22) MH(E) = B(H)⊗E.

Proof. It is easy to check that if E is weak* closed in B(H) then MH(E) is weak* closed inB(H ⊗2 H), whence the inclusion B(H)⊗E ⊂ MH(E). To show the reverse assume H = `2 fornotational simplicity. Then for any a ∈ M∞(E) it is easy to see that the truncated matrices[aij ]1≤i,j≤n (which are obviously in B(H) ⊗ E) tend weak* to a in M∞(B(H)) = B(H ⊗2 H)when n → ∞. This proves the reverse inclusion. The case of a general H is similar. We skip thedetails.

Since we use the spaces Mn(E) throughout these notes it is worthwhile to observe that therepresentations of Mn are very special:

Proposition 2.18. Let π : Mn → A ⊂ B(H) be an isometric unital ∗-homomorphism embeddingMn in a C∗-algebra A. Let A1 = A∩ π(Mn)′ be the relative commutant of π(Mn) in A. Then A isisomorphic to Mn(A1), and with this isomorphism π can be identified with x 7→ x⊗ 1.

Remark 2.19. The case A = MN shows that the existence of a unital embedding Mn ⊂MN requiresthat n divides N , while for a non-unital one n ≤ N obviously suffices (we may just add zero entries).

We leave the proof of Proposition 2.18 to the reader, but the proof is an easy modification ofthe following one, which is a von Neumann variant of it, to be used toward the end of these notes.

Proposition 2.20. Let H be any Hilbert space. Let π : B(H) → M ⊂ B(H) be a normal(isometric and unital) ∗-homomorphism embedding B(H) in a von Neumann algebraM. LetM1 =M∩ π(B(H))′. Then M is isomorphic as a von Neumann algebra to B(H)⊗M1, and modulo thisisomorphism π can be identified with B(H) 3 b 7→ b⊗ 1.

Proof. We may assume H = `2(I) for some set I. For notational simplicity, we assume I =1, 2, · · · . Let Eij = π(eij) ∈M. We will use the standard identities

(2.23) EikEk′j = Eij if k = k′ and EikEk′j = 0 otherwise.

Note that M1 = M ∩ Eij | i, j ≥ 1′. For any x ∈ B(H) let x(n) = PnxPn where Pn is thecanonical projection onto the span of the first n basis vectors in H. Let Qn = π(Pn) =

∑n1 Eii.

Then x(n) → x, in particular Pn → I and hence π(x(n)) → π(x) and Qn → I; these limitsare meant when n → ∞ and with respect to the weak* topology, as all the limits in the rest

38

of the proof. For any a ∈ M we have QnaQn → a. Let aij = limn∑n

k=1EkiaEjk. We have∑1≤i,j≤nEijaij =

∑1≤i,j≤nEiiaEjj = QnaQn. Using (2.23) one checks easily that aij ∈ M1. We

define a mapping σ : B(H)⊗M1 →M by setting

σ(lim∑n

i,j=1eij ⊗ yij) = lim

∑n

i,j=1Eijyij .

It is easy to check with (2.23) that σ is a ∗-homomorphism, and since σ([aij ]) = a for any a ∈M, σ isonto M. Moreover, we have ‖a‖ = lim ‖QnaQn‖ = lim ‖

∑1≤i,j≤nEijaij‖ = ‖σ([aij ])‖. Therefore,

σ is an isometric ∗-isomorphism from B(H)⊗M1 to M. Since the latter are both von Neumannalgebras, we know (see Remark 26.38) that σ and its inverse are automatically normal (actually itis easy to see this directly in the present situation). Lastly, we have σ(b⊗ 1) = lim

∑ni,j=1Eijbij =

limπ(∑n

i,j=1 eijbij), and hence (since π is normal) σ(b⊗ 1) = π(b) for any b ∈ B(H).

For more information on all questions involving operator spaces or algebras and dual topologies,we refer the reader to [27].

2.6 Free products of C∗-algebras

We start by recalling the definition of the free product of algebras: let (Ai)i∈I be a family ofalgebras (resp. unital algebras). We will denote by A (resp. A) their free product in the category ofalgebras (unital algebras). This object is characterized as the unique algebra (resp. unital algebra) Acontaining each Ai as a (resp. unital) subalgebra and such that if we are given another object B andmorphisms ϕi : Ai → B (i ∈ I), there is a unique morphism ϕ : A→ B such that ϕ|Ai = ϕi for alli. Moreover A is generated by the union of the Ai’s viewed as subalgebras (resp. unital subalgebras)of A. Similarly, if all the Ai’s are ∗-algebras, we can equip A with a ∗-algebra structure (meaningwith an involution) so that for any ∗-algebra B and ∗-morphisms ϕi : Ai → B, there is a unique∗-morphism such that ϕ|Ai = ϕi for all i, and the Ai’s can be viewed as ∗-subalgebras of A.

One can think of a typical element a of A (resp. A) as a sum of products of elements of theunion of the Ai’s. More precisely, a =

∑ak with each ak of the form ak = ak(i1) · · · ak(i`(k))

with ak(ij) ∈ Aij (1 ≤ j ≤ `(k)); when two consecutive terms are in the same algebra, saywhen ij = ij+1 we may replace ak(ij)ak(ij+1) by the single term (ak(ij)ak(ij+1)) ∈ Aij , thusreducing the product. When all such reductions have been done, we obtain a product of thesame form but for which i1 6= i2 6= · · · 6= i`(k), the product is then called reduced. For any

scalar λ ∈ C we have λa =∑

(λak(i1)) · · · ak(i`(k)). The product in A (resp. A) is defined by(∑

k ak)(∑

m bm) =∑

k,m akbm and by concatenation of the product terms akbm i.e.

[ak(i1) · · · ak(i`(k))][bm(j1) · · · bm(j`(m))] = ak(i1) · · · ak(i`(k))bm(j1) · · · bm(j`(m)).

If we now assume that (Ai)i∈I is a family of C∗-algebras (resp. unital ones), then we can equipA (resp. A) with a (resp. unital) C∗-algebra structure in the following way.

Let F be either A or A. Let C be the collection of all ∗-homomorphisms π : F → B(Hπ)(automatically such that ‖π|Ai‖ ≤ 1 for all i in I). Let j : F →

⊕π∈C B(Hπ) be the embedding

defined by j(x) =⊕

π∈C π(x) for all x in F . Clearly j is a ∗-homomorphism and (by standardalgebraic facts) it is injective. This allows us to equip F with the noncomplete C∗-algebra structureassociated to j and, after completion with respect to the norm

(2.24) ∀x ∈ F ‖x‖ = supπ∈C ‖π(x)‖

we obtain a C∗-algebra (resp. a unital one), admitting F as a dense subalgebra. We will denoteby ∗i∈I Ai (resp. ∗i∈I Ai) the resulting (resp. unital) C∗-algebra, which we call the free product ofthe family of (resp. unital) C∗-algebras (Ai)i∈I . See [15] and [255] for basic facts on free products.

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Let A be any of these two free products. Let σi : Ai → A be the natural embedding. Then forany family of morphisms πi : Ai → B(H) there is a unique morphism π : A → B(H) such thatπσi = πi for all i.

Remark 2.21. The constructions of ∗i∈I Ai or ∗i∈I Ai that we just sketched deliberately avoids afew points. For instance it is true but not obvious that the canonical morphism from A to ∗i∈I Aiis injective. This point is addressed in [23] and [171, p. 37]. See also [255, p.4].Using this it is easy to check that the norm we just defined in (2.24) on either A or A is the maximalC∗-norm.

Let A1, A2 be unital C∗-algebras with subspaces Ej ⊂ Aj (j = 1, 2). By suitably restricting theproduct map, we have a natural embedding E1⊗E2 ⊂ A1 ∗A2 with range equal to the linear spanof the set E1E2 ⊂ A1 ∗A2. The induced operator space structure on E1⊗E2 is that of the so-calledHaagerup tensor product E1⊗hE2, which, despite its importance, we chose not to present in detailin the present volume. The reader will find a full treatment in our previous book [208, Ch. 5] (orin [80, 196]). In the present one, the main result we need is stated as the next lemma.

Let A1, A2 be unital C∗-algebras with subspaces Ej ⊂ Aj (j = 1, 2). Let E ⊂ A1 ∗ A2 be thelinear span of the set E1E2 ⊂ A1 ∗ A2. We wish to describe the operator space structure on Einduced by this embedding, i.e. the norm in Mn(E) for all n. Actually we can do this directly forn =∞:

Lemma 2.22. Assume dim(H) =∞. Let t ∈ B(H)⊗E. Then ‖t‖min ≤ 1 if and only if there aretj ∈ B(H)⊗ Ej with ‖tj‖min ≤ 1 such that

(2.25) t = t1 t2,

where the sign means the bilinear mapping defined from (B(H)⊗E1)×(B(H)⊗E2)→ B(H)⊗Eby (b1 ⊗ e1) (b2 ⊗ e2) = (b1b2 ⊗ e1e2) (bj ∈ B(H), ej ∈ Ej). Equivalently, is the restriction ofthe usual product on B(H)⊗ (A1 ∗A2).

About the proof. Since t1, t2 are tensors of finite rank, the statement reduces immediately to thecase when E1, E2 are finite dimensional. Using mainly results from [50] it was proved in [204] thatE can be identified completely isometrically with the Haagerup tensor product E1 ⊗h E2. Thenthe result follows from [208, Cor. 5.9 p. 95].

Actually Lemma 2.22 is valid for any number n of factors Ej ⊂ Aj (1 ≤ j ≤ n).More generally, one can describe very efficiently the (operator space) structure of a free product∗i∈IAi of C∗-algebras, as follows.

Theorem 2.23 (Blecher-Paulsen factorization [28]). Let (Ai)i∈I be a family of unital C∗-algebras,let A = ∗i∈IAi be the unital free product, and let A ⊂ A be the algebraic free product. Let n ≥ 1and consider x ∈ Mn(A). Then ‖x‖ < 1 if and only if there are m, i1 6= i2 6= · · · 6= im in I andrectangular matrices xk ∈Mpk×qk(Aik) with ‖xk‖ < 1 such that p1 = qm = n and

(2.26) x = x1 · · ·xm.

About the proof. This beautiful result is a consequence of the Blecher-Ruan-Sinclair (BRS in short)characterization of operator algebras from [29]. One equips Mn(A) with the norm

(2.27) ‖x‖n = inf∏‖xk‖

40

where the infimum runs over all factorizations x = x1 · · ·xm of the kind just described. By theBRS characterization there is for some H a unital homomorphism π : A → B(H) such that

(2.28) ‖x‖n = ‖(IdMn ⊗ π)(x)‖Mn(B(H)).

Let πi = π|Ai . Clearly ‖x‖n ≤ ‖x‖Mn(A) whenever x ∈ Mn(Ai). Therefore by (2.28) ‖πi‖cb ≤ 1.Since πi(1) = 1, πi is c.p. by Theorem 1.35 and a fortiori self-adjoint. Consequently πi must be a∗-homomorphism for any i. This shows that π = ∗πi on A. By definition of A = ∗i∈IAi, π extendsto a (contractive) ∗-homomorphism on A. But now by (2.27) we clearly have ‖x‖Mn(A) ≤ ‖x‖nfor any n, in particular ‖x‖A ≤ ‖π(x)‖ for any x ∈ A. By the maximality of ‖x‖A (as reflectedin (2.24)) equality must hold, so that π is the restriction of an isometric, and hence completelyisometric, ∗-homomorphism from A to B(H). Thus we conclude that ‖x‖Mn(A) = ‖x‖n for anyx ∈ A.

The following very useful result is due to Boca [30].

Theorem 2.24 (Boca’s theorem). Let (Ai)i∈I be a family of unital C∗-algebras and let A = ∗i∈IAibe the unital free product. For each i ∈ I let fi be a state on Ai. Let ui : Ai → B be unital c.p.maps with values in another unital C∗-algebra B. There is a unital c.p. map u : A→ B such thatfor any m, any i1 6= i2 6= · · · 6= im in I and any ak ∈ Aik such that fik(ak) = 0 for all k we have

(2.29) u(a1a2 · · · am) = ui1(a1)ui2(a2) · · ·uim(am).

See [68] for a recent proof of Boca’s theorem using the Stinespring dilation Theorem 1.22.

2.7 Universal C∗-algebra of an operator space

Let E be an operator space. The universal C∗-algebra generated by E will be denoted by C∗〈E〉.Its definition will be given after the next statement.

Theorem 2.25. Let E be an operator space. There is a (resp. unital) C∗-algebra A and a com-pletely isometric embedding j : E → A with the following properties:

(i) For any (resp. unital) C∗-algebra B and any completely contractive map u : E → B there is a(resp. unital) representation π : A→ B extending u, i.e. such that πj = u.

(ii) The (resp. unital) algebra generated by j(E) is dense in A.Moreover, (ii) ensures that the (resp. unital) representation π in (i) is unique.

Proof. The proof is immediate. Let I be the “collection” of all u as in (i) with range generating a(resp. unital) C∗-algebra denoted by Bu, so that u : E → Bu satisfies ‖u‖cb ≤ 1. Let

BE =(⊕∑

u∈IBu

)∞.

We define j : E → BE byj(x) = (u(x))u∈I .

By (1.13) it is easy to check that j is completely isometric. Then, if we define A to be the(resp. unital) C∗-algebra generated by j(E) in BE , the announced universal property of A isimmediate.

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Notation: We will denote by C∗〈E〉 (resp. C∗u〈E〉) the (resp. unital) C∗-algebra A appearing inthe preceding statement.

Note that C∗〈E〉 is essentially unique. Indeed, if j1 : E → A1 is another completely isometricembedding into a C∗-algebra A1 with the property in Theorem 2.25, then the universal propertyof A (resp. A1) implies the existence of a representation π : A→ A1 (resp. π1 : A1 → A) such thatπj = j1 (resp. π1j1 = j). Since C∗-representations are automatically contractive we have ‖π‖ ≤ 1,‖π1‖ ≤ 1 and π1 = π−1 on j1(E) hence on the ∗-algebra generated by j1(E), which is dense in A1

by assumption. This implies that π is an isometric isomorphism from A onto A1.Similarly, C∗u〈E〉 is characterized as the unique unital C∗-algebra C containing E completely

isometrically in such a way that, for any unital C∗-algebra B (actually we may restrict to B = B(H)with H arbitrary), any c.c. map u : E → B uniquely extends to a unital representation (i.e. ∗-homomorphism) from C to B.

It is easy to see that C∗〈E〉 can be identified to the C∗-algebra generated by E in C∗u〈E〉.Indeed, the latter has the property in Theorem 2.25.

Remark 2.26. If two operator spaces E,F are completely isometrically isomorphic, then E and Fcan be realized as “concrete” operator subspaces E ⊂ A and F ⊂ B of two isomorphic C∗-algebrasA and B, for which there is an isometric ∗-homomorphism π : A→ B such that π(E) = F .

Indeed, let A = C∗〈E〉 and B = C∗〈F 〉, let u : E → F be a completely isometric isomorphism,let π : C∗〈E〉 → C∗〈F 〉 be the (unique) extension of u (as in Theorem 2.25) and let σ : C∗〈F 〉 →C∗〈E〉 be the (unique) extension of u−1. Then clearly we must have (by unicity again) σπ = IdC∗〈E〉and πσ = IdC∗〈F 〉 so that σ = π−1.

Remark 2.27 (Universal C∗-algebra of a contraction). Consider for example the simplest choice ofE, namely E = C. Then C∗〈C〉 can be described more explicitly as the completion of the space ofpolynomials P in the formal variables X,X∗ equipped with the C∗-norm ‖P‖ = sup‖P (x, x∗)‖where the sup runs over all H and all x ∈ B(H) with ‖x‖ ≤ 1. Indeed, the linear mappingux : C → B(H) taking 1 to x satisfies trivially ‖ux‖cb = ‖x‖, so that ‖x‖ ≤ 1 if and only if uxextends to a ∗-homomorphism πx : C∗〈C〉 → B(H) with ‖πx‖ = 1.

The analogue for the unital case requires that we consider polynomials “with a constant term”of the form λ1 + P (X,X∗) with λ ∈ C. We then equip the resulting unital algebra of polynomialswith the norm sup‖λIdH + P (x, x∗)‖ over all H’s and all contractions x ∈ B(H) as before. Thecompletion can now be identified with C∗u〈C〉.

2.8 Completely positive perturbations of completely bounded maps

Warning: this section is devoted to a rather technical point. To avoid interrupting the flow of ourpresentation, we advise the reader to skip it until it becomes needed (in §9.4).Notation (Order defined by the cone of c.p. maps). Let u, v : E → B be linear maps from anoperator system E to a C∗-algebra. If v − u ∈ CP (E,B) we will write

(2.30) u 4 v.

This is the partial order on CB(E,B) for which CP (E,B) is the positive cone.As we saw previously (see Theorem 1.35), in analogy with a well known fact in measure theory,

if a complete contraction on an operator system is unital, then it is automatically c.p. In thissection, we will prove quantitative versions of this: if a unital map has a cb-norm close to 1, thenthe map is close to a c.p. map. In the first result the range is an arbitrary C∗-algebra, while in thesecond one it is B(H). The first case is restricted to a finite dimensional domain but the secondone is not.

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Theorem 2.28 ([77]). Let E be an n-dimensional operator system, B a unital C∗-algebra.

(i) For any ϕ ∈ CP (E,B) there is a linear form f ∈ E∗ such that f ≥ 0, ‖f‖ ≤ 2n‖ϕ‖ andthe mapping Ψ : E → B defined by Ψ(x) = f(x)1 − ϕ(x) is c.p. (equivalently we have0 4 ϕ 4 f(·)1).

(ii) For any self-adjoint unital linear map ϕ : E → B such that ‖ϕ‖cb ≤ 1+δ, where 0 < δ < 1/2n,there is a u.c.p. map ψ : E → B such that ‖ϕ− ψ‖cb ≤ 8nδ.

Proof. We reproduce the proofs from [77] with a cosmetic change.(i) We first consider the case of a self-adjoint (not necessarily c.p.) linear map ϕ : E → B of rank1 with ‖ϕ‖ ≤ 1 of the form ϕ(x) = f(x)b for some f ∈ BE∗ and b ∈ BB both self-adjoint. We willdenote this map by f ⊗ b.We claim that there is F ∈ BE∗ with F ≥ 0 and ‖F‖ ≤ 1 such that, with respect to the naturalordering (2.30) of CB(E,B), we have

−F ⊗ 1 4 f ⊗ b 4 F ⊗ 1.

Assuming E ⊂ B(H), f can be extended to a self-adjoint form f ′ ∈ B(H)∗ with ‖f ′‖ ≤ ‖f‖(indeed, we may take for f ′ the real part of the Hahn-Banach extension of f). By the classicalHahn decomposition (see Remark 1.54), we have f ′ = f ′+ − f ′− with f ′± ≥ 0 such that‖f ′+‖+ ‖f ′−‖ ≤ ‖f ′‖. By restricting this to E we find a decomposition f = f+ − f− with f± ∈ E∗positive and such that ‖f+‖+ ‖f−‖ ≤ ‖f‖.In parallel, any b ∈ BB with b = b∗ can be decomposed as b = b+ − b− with b± ≥ 0 in B such that‖b+ + b−‖ ≤ 1. But it is easy to check that for the c.p. order for any 0 ≤ f1 ≤ f2 (f1, f2 ∈ E∗) and0 ≤ b1 ≤ b2 (b1, b2 ∈ B) we have 0 4 f1 ⊗ b1 4 f2 ⊗ b2. Therefore we have

−T 4 (f+ − f−)⊗ (b+ − b−) 4 T

with T = (f+ + f−)⊗ (b+ + b−) and a fortiori also with T = (f+ + f−)⊗ 1 since 0 ≤ b+ + b− ≤ 1.Thus our claim follows with F = f+ + f−.Now, since dim(E) = n, by Auerbach’s Lemma 1.55, there is a biorthogonal system (ξj , xj) inBE∗ × BE such that IdE =

∑n1 ξj ⊗ xj . Note that for any x ∈ E we have x∗ = (

∑ξj(x)xj)

∗ and

also (since x∗ ∈ E) x∗ =∑n

1 ξj(x∗)xj . Thus (setting ξj∗ = ξj(x∗)) we have

∑ξj ⊗xj =

∑ξj∗⊗x

∗j ,

and hence∑ξj ⊗ xj =

∑(ξj + ξj∗)/2⊗ (xj + x∗j )/2−

∑(ξj − ξj∗)/(2i)⊗ (xj − x∗j )/(2i) =

∑2n

1fj ⊗ yj ,

with fj , yj self-adjoint in BE∗ ×BE . This gives us

(2.31) ∀x ∈ E x =∑2n

1fj(x)yj .

Now let ϕ ∈ CP (E,B). Using (2.31), we may write ϕ as ϕ = Σfj ⊗ bj with bj = ϕ(yj). We mayassume ‖ϕ‖ = 1 (by homogeneity). Note bj = b∗j and ‖bj‖ ≤ ‖ϕ‖‖yj‖ ≤ 1. Therefore, using thefirst part of the proof we obtain Fj ≥ 0 with ‖Fj‖ ≤ 1, and hence ‖ΣFj‖ ≤ 2n, such that

−(ΣFj)⊗ 1 4 ϕ 4 (ΣFj)⊗ 1.

In particular, 0 4 ϕ 4 f(·)1 with f = ΣFj ≥ 0 and ‖f‖ ≤ 2n. In other words x 7→ Ψ(x) =f(x)1− ϕ(x) is in CP (E,B).

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(ii) Assume B ⊂ B(H). By Corollary 1.53 we can write ϕ = ϕ1−ϕ2 with ϕ1, ϕ2 ∈ CP (E,B(H))such that ‖ϕ1 +ϕ2‖ ≤ ‖ϕ‖cb. Let aj = ϕj(1). Then a1, a2 ≥ 0, a1−a2 = 1 and ‖a1‖ ≤ ‖a1 +a2‖ ≤1 + δ. Thus, for any unit vector ξ in H we have

〈ξ, a2ξ〉 = 〈ξ, (a1 − 1)ξ〉 = 〈ξ, a1ξ〉 − 1 ≤ δ,

and hence by (1.27)‖ϕ2‖ = ‖a2‖ ≤ δ.

By part (i) there is f ∈ E∗ with f ≥ 0 and ‖f‖ ≤ 2nδ such that the mapping ψ2 : E → B(H)defined by ψ2(x) = f(x)1− ϕ2(x) is c.p. We then set

ψ0 = ϕ1 + ψ2.

Note that, since ψ0(x)− ϕ(x) = f(x)1 ∈ C1, we have ψ0(E) ⊂ B, ψ0 ∈ CP (E,B) and

(2.32) ‖ψ0 − ϕ‖cb ≤ 2nδ.

To obtain a unital ψ we need to work a bit more. Let b = ψ0(1). Then b ≥ 0 and ‖b − 1‖ =‖ψ0(1) − ϕ(1)‖ ≤ 2nδ < 1, and hence b is invertible. A fortiori, by functional calculus (since|√b− 1| ≤ |b− 1| ∀b ∈ [0, 2]), we have ‖

√b− 1‖ ≤ ‖b− 1‖ ≤ 2nδ. For any y ∈ B we have

‖b12 yb

12 − y‖ = ‖(b

12 − 1)yb

12 + y(b

12 − 1)‖ ≤ (‖b

12 − 1‖ ‖b

12 ‖+ ‖b

12 − 1‖)‖y‖ ≤ 6nδ‖y‖.

A similar estimate holds for any k and any y = [yij ] ∈Mk(B). Thus if we now set

ψ(x) = b−12ψ0(x)b−

12 ,

so that ψ0(·) = b12ψ(·)b

12 , we find ‖ψ0 − ψ‖cb = ‖b

12ψb

12 − ψ‖cb ≤ 6nδ‖ψ‖cb = 6nδ. Thus we obtain

by (2.32)‖ϕ− ψ‖cb ≤ ‖ϕ− ψ0‖cb + ‖ψ0 − ψ‖cb ≤ 8nδ,

and ψ ∈ CP (E,B) is unital with ‖ψ‖cb = ‖ψ(1)‖ = 1 (by (1.27)).

The case when B = B(H) is much simpler:

Lemma 2.29 (Kirchberg). Let A be a unital C∗-algebra and let ϕ : A → B(H) be a completelybounded self-adjoint unital map. Then there is a u.c.p. map ψ : A→ B(H) such that ‖ϕ− ψ‖cb ≤‖ϕ‖cb − 1.

Proof. Let ε = ‖ϕ‖cb − 1. By the factorization Theorem 1.50 there are H, π : A → B(H) andV,W ∈ B(H, H) such that ‖V ‖‖W‖ = 1 + ε and ϕ(a) = V ∗π(a)W for all a in A. By homogeneity

we may assume ‖V ‖ = ‖W‖ = (1 + ε)12 . Since ϕ is unital we have V ∗W = IdH , and since it is

self-adjoint V ∗π(a)W = W ∗π(a)V . Note W ∗V = IdH . Let ψ(a) = (V ∗π(a)V +W ∗π(a)W )/2. Wehave then

ψ(a)− ϕ(a) =1

2(V ∗π(a)V +W ∗π(a)W − V ∗π(a)W −W ∗π(a)V ) =

1

2(V −W )∗π(a)(V −W ).

Note that ψ−ϕ (as well as ψ) is c.p. Moreover (V −W )∗(V −W ) = V ∗V +W ∗W −2IdH ≤ 2εIdH .Therefore, ‖ψ − ϕ‖ = ‖ψ − ϕ‖cb ≤ ε.

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2.9 Notes and Remarks

For Schur multipliers as in Remark 1.58 the equality ‖u‖ = ‖u‖cb is due to Haagerup, and for“module maps” as in Theorem 2.6 it was proved by Roger Smith in [233]. The theory of freeproducts of C∗-algebras can be traced back to Avitsur’s seminal paper [15], but it really took offwith Voiculescu’s “free probability” (see [255]). Note however that in the latter theory, it is thereduced free product of two C∗-algebras equipped with states that plays the central role, while forthe present volume we are mostly concerned with the maximal free product of unital C∗-algebras(i.e. amalgamated over C1). For this kind of free product it is rather striking that unital c.p. mapsare “admissible” morphisms as described by Boca’s theorem from [30]. As already mentioned theduality theory for operator spaces goes back to Effros-Ruan and Blecher-Paulsen. The results in§2.4 and §2.5 are due to them, except for Propositions 2.20 and 2.18, both well known facts. Theremarkable factorization in (2.26) (due to Blecher and Paulsen [28]) is a consequence of the no-less remarkable Blecher-Ruan-Sinclair characterization of operator algebras; the central underlyingconcept is the Haagerup tensor product of operator spaces. These important topics are treated indetail in our previous book [208] which explains our reluctance to expand on that same theme inthe present one. We describe these results in Theorem 2.23 and, making here an exception, referthe reader to [208] or [196] for detailed proofs.See Loring’s book [171] for a discussion of amalgamated full free products as solutions of a universalproblem. Loring also considers in [171] free products of unital algebras amalgamated over C1 butrelative to non-unital embeddings into the free product, which can be a quite different object.Concerning §2.8 and the completely positive ordering, we should mention the following Radon-Nikodym theorem due to Arveson [12]. Let A be a C∗-algebra, let u, v ∈ CP (A,B(H)). Letv(·) = V ∗π(·)V be the minimal Stinespring factorization of v (which means that V ∈ B(H, H) withV π(A)(H) = H). Then 0 4 u 4 v if and only if there is T ∈ π(A)′ ⊂ B(H) with 0 ≤ T ≤ 1 suchthat u(·) = V ∗Tπ(·)V . Moreover such a T is unique. See [197] for a sort of completely boundedanalogue.

3 C∗-algebras of discrete groups

Group representations are one of the main sources of examples of C∗-algebras. The universalrepresentation of a group G gives rise to the full or maximal C∗-algebra C∗(G), while the leftregular representation leads to the reduced C∗-algebra C∗λ(G). In this chapter we review some oftheir main properties when G is a discrete group.

3.1 Full (=Maximal) group C∗-algebras

We first recall some classical notation from non-commutative Abstract Harmonic Analysis on anarbitrary discrete group G.We denote by e (and sometimes by eG) the unit element. Let π : G → B(H) be a unitaryrepresentation of G. We denote by C∗π(G) the C∗-algebra generated by the range of π.Equivalently, C∗π(G) is the closed linear span of π(G).In particular, this applies to the so-called universal representation of G, a notion that we now recall.Let (πj)j∈I be a family of unitary representations of G, say

πj : G→ B(Hj)

in which every equivalence class of a cyclic unitary representation of G has an equivalent copy. Now

45

one can define the “universal” representation UG : G→ B(H) of G by setting

UG = ⊕j∈Iπj on H = ⊕j∈IHj .

Then the associated C∗-algebra C∗UG(G) is simply denoted by C∗(G) and is called the “full” (or the“maximal” ) C∗-algebra of the group G, to distinguish it from the “reduced” one that is describedin the sequel. Note that

C∗(G) = spanUG(t) | t ∈ G.

Let π be any unitary representation of G. By a classical argument, π is unitarily equivalent to adirect sum of cyclic representations, hence for any finitely supported function x : G→ C we have

(3.1)∥∥∥∑x(t)π(t)

∥∥∥ ≤ ∥∥∥∑x(t)UG(t)∥∥∥ .

In particular, if π is the trivial representation

(3.2) |∑

x(t)| ≤∥∥∥∑x(t)UG(t)

∥∥∥ .Equivalently (3.1) means∥∥∥∑x(t)UG(t)

∥∥∥B(H)

= sup

∥∥∥∑x(t)π(t)∥∥∥B(Hπ)

where the supremum runs over all possible unitary representations π : G→ B(Hπ) on an arbitraryHilbert space Hπ. More generally, for any Hilbert space K and any finitely supported functionx : G→ B(K) we have∥∥∥∑x(t)⊗ UG(t)

∥∥∥B(K⊗2H)

= sup

∥∥∥∑x(t)⊗ π(t)∥∥∥B(K⊗2Hπ)

where the sup is the same as before.

There is an equivalent description in terms of the group algebra C[G], the elements of whichare simply the formal linear combinations of the elements of G, equipped with the obvious natural∗-algebra structure. One equips C[G] with the norm (actually a C∗-norm)∑

t∈Gx(t)t 7→ sup

∥∥∥∑x(t)π(t)∥∥∥

where the supremum runs over all possible unitary representations π of G. One can then defineC∗(G) as the completion of C[G] with respect to the latter norm.

These formulae show that the norm of C∗(G) is the largest possible C∗-norm on C[G]. Whencethe term “maximal” C∗-algebra of G.

Remark 3.1 (A recapitulation). By (3.1) there is a 1 − 1 correspondence between the unitaryrepresentations π : G → B(H) and the ∗-homomorphisms ψ : C∗(G) → B(H). More precisely,for any π there is a unique ψ : C∗(G)→ B(H) such that ∀g ∈ G ψ(UG(g)) = π(g), or if we viewG as a subset of C[G] ⊂ C∗(G) (which means we identify g and UG(g)), we have

∀g ∈ G π(g) = ψ(g).

46

Remark 3.2 (c.b. and c.p. maps on C∗(G)). A linear map u : C∗(G) → B(K) is c.b. if and onlyif there exists a unitary group representation π : G → B(Hπ) and operators V,W : K → Hπ suchthat

∀t ∈ G u(UG(t)) = W ∗π(t)V.

Moreover, we have ‖u‖cb = inf‖W‖‖V ‖ and the infimum is attained. Indeed, in view of thepreceding remark this follows immediately from Theorem 1.50. The c.p. case is characterizedsimilarly but with V = W . When K = C and hence B(K) = C, this gives us a description of thedual of C∗(G), as well as a characterization of states on C∗(G).

The next result (in which we illustrate the preceding remark in the case of multipliers) isclassical, and fairly easy to check.

Proposition 3.3 (Multipliers on C∗(G)). Let ϕ : G→ C. Consider the associated linear operatorMϕ (a so-called multiplier, see §3.4) defined on spanUG(t) | t ∈ G by Mϕ(

∑x(t)UG(t)) =∑

x(t)ϕ(t)UG(t). Then Mϕ extends to a bounded operator on C∗(G) if and only if there are aunitary representation π : G→ B(Hπ) and ξ, η in Hπ such that

(3.3) ∀ t ∈ G ϕ(t) = 〈η, π(t)ξ〉.

Moreover we have for the resulting bounded operator (still denoted by Mϕ)

(3.4) ‖Mϕ‖ = ‖Mϕ‖cb = inf‖ξ‖‖η‖

where the infimum (which is attained) runs over all possible π, ξ, η for which this holds. Lastly, ifMϕ is positive (3.3) holds with ξ = η, and then Mϕ is completely positive on C∗(G).

Proof. If ‖Mϕ : C∗(G)→ C∗(G)‖ ≤ 1, let f(x) =∑

t∈G ϕ(t)x(t). Then by (3.2) f ∈ C∗(G)∗ with‖f‖ ≤ 1. Note f(UG(t)) = ϕ(t). By Remark 1.54 there are π, ξ, and η with ‖ξ‖‖η‖ ≤ ‖f‖ ≤ 1 suchthat (3.3) holds. If Mϕ (and hence f) is positive we find this with ξ = η. For the converse, since(like any unitary group representation) the mapping UG(t) 7→ UG(t)⊗π(t) extends to a continuous∗-homomorphism σ : C∗(G) → B(H ⊗2 Hπ), we have Mϕ(·) = V ∗2 σ(·)V1, with V1h = h ⊗ ξ andV2h = h⊗ η (h ∈ H) from which we deduce by (1.30) ‖Mϕ‖cb ≤ ‖ξ‖‖η‖. If ξ = η then V1 = V2 andhence Mϕ is c.p. on C∗(G).

Remark 3.4. By Remark 3.2 and (3.4) the space of bounded multipliers on C∗(G) can be identifiedisometrically with C∗(G)∗. If fϕ is the linear form on C∗(G) taking UG(t) to ϕ(t) (t ∈ G) we have

‖Mϕ‖ = ‖fϕ‖C∗(G)∗ .

Proposition 3.5. Let G be a discrete group and let Γ ⊂ G be a subgroup. Then the correspondenceUΓ(t) → UG(t), (t ∈ Γ) extends to an isometric (C∗-algebraic) embedding J of C∗(Γ) into C∗(G).Moreover, there is a completely contractive and completely positive projection P from C∗(G) ontothe range of this embedding, defined by P (UG(t)) = UG(t) for any t ∈ Γ and P (UG(t)) = 0 otherwise.

Proof. By the universal property of C∗(Γ) the unitary representation Γ ⊃ γ 7→ UG(γ) extends toa ∗-homomorphism J : C∗(Γ) → C∗(G) with ‖J‖ = 1. Let ϕ = 1Γ. The projection P describedin Proposition 3.5 coincides with the multiplier Mϕ acting on C∗(G). Thus, by Proposition 3.3 itsuffices to show that there is a unitary representation π : G→ B(Hπ) of G and a unit vector ξ ∈ Hπ

such that ϕ(t) = 〈ξ, π(t)ξ〉. Let G =⋃s∈G/Γ sΓ be the disjoint partition of G into left cosets. For

any t ∈ G the mapping sΓ 7→ tsΓ defines a permutation σ(t) of the set G/Γ, and t 7→ σ(t) is ahomomorphism. Let Hπ = `2(G/Γ) and let π : G → B(Hπ) be the unitary representation definedon the unit vector basis by π(t)(δs) = δσ(t)(s) for any s ∈ G/Γ. Let [[Γ]] ∈ G/Γ denote the coset Γ(i.e. sΓ for s = 1G) and let ξ = δ[[Γ]]. Then it is immediate that ϕ(t) = 〈ξ, π(t)ξ〉 for any t ∈ G.

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Remark 3.6. Let G be a discrete group and let E ⊂ C∗(G) be any separable subspace. We claimthat there is a countable subgroup Γ ⊂ G such that with the notation of Proposition 3.5 we haveE ⊂ J(C∗(Γ)). Indeed, since C∗(G) ⊂ span[UG(t) | t ∈ G] for any fixed x ∈ C∗(G) there is clearlya countable subgroup Γx ⊂ G and an analogous Jx such that x ⊂ Jx(C∗(Γx)). Arguing like thisfor each x in a dense countable sequence in E and taking the group generated by all the resultingΓx’s gives us the claim.By Proposition 3.5 this shows that there is a separable C∗-subalgebra C ⊂ C∗(G) with E ⊂ C forwhich there is a c.p. projection P : C∗(G)→ C.

Remark 3.7. Let G be any discrete group, let A = C∗(G). Then A ' A. Indeed, since for anyunitary representation π on G, the complex conjugate π (as in Remark 2.14) is also a unitaryrepresentation, the correspondence π 7→ π is a bijection on the set of unitary representations, fromwhich the C-linear isomorphism Φ : C∗(G) → C∗(G) follows immediately. Denoting by UG theuniversal representation of G, this isomorphism takes UG(t) to UG(t). Note that A ' A is in generalnot true (see [60]).

3.2 Full C∗-algebras for free groups

In this section, we start by comparing the C∗-algebras of free groups of different cardinals. Ourgoal is to make clear that we can restrict to C = C∗(F∞) (or if we wish to C∗(F2)) for the variousproperties of interest to us in the sequel. Then we describe the operator space structure of the spanof the free generators in C∗(F) when F is any free group. The following simple lemma will be ofteninvoked when we wish to replace C∗(F) by C∗(F∞).

Lemma 3.8. Let F be a free group with generators (gi)i∈I . Let E ⊂ C∗(F) be any separablesubspace. Then the inclusion E ⊂ C∗(F) admits an extension TE : C∗(F) → C∗(F) that can befactorized as

TE : C∗(F)w−→C∗(F∞)

v−→C∗(F)

where v, w are contractive c.p. maps.For any C∗-algebra D and any x ∈ D ⊗ E we have

(3.5) ‖x‖D⊗maxC∗(F) = ‖(IdD ⊗ w)(x)‖D⊗maxC∗(F∞).

In particular, E ⊂ C∗(F) is completely isometric to w(E) ⊂ C∗(F∞).

Proof. For any x ∈ C∗(F) there is clearly a countable subgroup Γx ⊂ F such that

x ∈ span[UF(t) | t ∈ Γx].

By the separability of E, we can find a countable subgroup Γ such that E ⊂ span[UF(t) | t ∈ Γ].Since any element of t ∈ Γ can be written using only finitely many “letters” in gi | i ∈ I, we mayassume that Γ is the free subgroup generated by (gi)i∈I′ for some countable subset I ′ ⊂ I. Then,identifying span[UF(t) | t ∈ Γ] with C∗(Γ), Proposition 3.5 yields a mapping T = JP : C∗(F) →C∗(F) with the required factorization through C∗(Γ) = C∗(FI′) that is the identity when restrictedto E. If I ′ is infinite the proof is complete: since C∗(FI′) = C∗(F∞) we may take TE = T .Otherwise, we note that FI′ ⊂ F∞ as a subgroup and hence by Proposition 3.5 again we have a

factorization of the same type C∗(FI′)J ′−→C∗(F∞)

P ′−→C∗(FI′) from which it is easy to conclude.Note

‖x‖D⊗maxC∗(F) = ‖(IdD ⊗ TE)(x)‖D⊗maxC∗(F) = ‖(IdD ⊗ vw)(x)‖D⊗maxC∗(F).

48

By Corollary 4.18 since v, w are c.p. contractions we have

‖x‖D⊗maxC∗(F) ≤ ‖(IdD ⊗ w)(x)‖D⊗maxC∗(F∞) and ‖(IdD ⊗ w)(x)‖D⊗maxC∗(F∞) ≤ ‖x‖D⊗maxC∗(F),

from which (3.5) follows.

Let F be a free group with generators (gi)i∈I . We start with a basic property of the span of thefree generators in C∗(F).

Lemma 3.9. Let F be a free group with generators (gi)i∈I . Let Ui = UF(gi) ∈ C∗(F). Let E =span[(Ui)i∈I , 1] ⊂ C∗(F) and EI = span[(Ui)i∈I ] ⊂ C∗(F). Then for any linear map u : E → B(H)and any v : EI → B(H) we have

(3.6) ‖u‖cb = ‖u‖ = maxsupi∈I ‖u(Ui)‖, ‖u(1)‖ and ‖v‖cb = ‖v‖ = maxsupi∈I ‖v(Ui)‖.

Proof. It clearly suffices to show that maxsupi∈I ‖u(Ui)‖, ‖u(1)‖ ≤ 1 implies ‖u‖cb ≤ 1. Whenu(1) = 1 and all u(Ui) are unitaries this is easy: indeed there is a (unique) group representationσ : F→ B(H) such that σ(gi) = u(Ui) and the associated linear extension uσ : C∗(F)→ B(H) is a∗-homomorphism automatically satisfying ‖uσ‖cb = 1, and hence ‖u‖cb = 1. This same argumentworks if we merely assume that u(1) is unitary. Indeed, we may replace u by x 7→ u(1)−1u, whichtakes us back to the previous easy case. Since the general case is easy to reduce to that of a finiteset, we assume that I is finite. Then the Russo-Dye Theorem 26.18 shows us that any u such thatmaxsupi∈I ‖u(Ui)‖, ‖u(1)‖ ≤ 1 lies in the closed convex hull of u’s for which u(1) and all theu(Ui)’s are unitaries, and hence ‖u‖cb ≤ 1 in that case also.

The first part of the next result is based on the classical observation that a unitary representationπ : F → B(H) is entirely determined by its values ui = π(gi) on the generators, and if we let πrun over all possible unitary representations, then we obtain all possible families (ui) of unitaryoperators. The second part is also well known.

Lemma 3.10. Let A ⊂ B(H) be a C∗-algebra. Let F be a free group with generators (gi)i∈I . LetUi = UF(gi) ∈ C∗(F). Let (xi)i∈I be a family in A with only finitely many non-zero terms. Considerthe linear map T : `∞(I)→ A defined by T ((αi)i∈I) =

∑i∈I αixi. Then we have

(3.7)∥∥∥∑

i∈IUi ⊗ xi

∥∥∥C∗(F)⊗minA

= ‖T‖cb = sup∥∥∑ui ⊗ xi

∥∥min

where the sup runs over all possible Hilbert spaces K and all families (ui) of unitaries on K.Actually, the latter supremum remains the same if we restrict it to finite dimensional Hilbert spacesK. Moreover, in the case when A = B(H) with dim(H) =∞, we have

(3.8)∥∥∥∑

i∈IUi ⊗ xi

∥∥∥C∗(F)⊗minB(H)

= inf

∥∥∥∑ yiy∗i

∥∥∥1/2 ∥∥∥∑ z∗i zi

∥∥∥1/2

where the infimum, which runs over all possible factorizations xi = yizi with yi, zi in B(H), isactually attained.Moreover, all this remains true if we enlarge the family (Ui)i∈I by including the unit element ofC∗(F).

Proof. It is easy to check going back to the definitions that on one hand∥∥∑Ui ⊗ xi∥∥

min= sup

∥∥∑ui ⊗ xi∥∥

min,

49

where the sup runs over all possible families of unitaries (ui), and on the other hand that

‖T‖cb = sup∥∥∑ ti ⊗ xi

∥∥min,

where the sup runs over all possible families of contractions (ti). By the Russo-Dye Theorem 26.18,any contraction is a norm limit of convex combinations of unitaries, so (3.7) follows by convexity.Actually, the preceding sup obviously remains unchanged if we let it run only over all possiblefamilies of contractions (ti) on a finite dimensional Hilbert space. Thus it remains unchanged whenrestricted to families of finite dimensional unitaries (ui).Now assume ‖T‖cb = 1. By the factorization of c.b. maps we can write T (α) = V ∗π(α)W whereπ : `∞(I) → B(H) is a representation and where V,W are in B(H, H) with ‖V ‖ ‖W‖ = ‖T‖cb.Since we assume dim(H) = ∞ and may assume I finite (because i 7→ xi is finitely supported), byRemark 1.51 we may as well take H = H. Let (ei)i∈I be the canonical basis of `∞(I), we set

yi = V ∗π(ei) and zi = π(ei)W.

It is then easy to check ‖∑yiy∗i ‖

1/2 ‖∑z∗i zi‖

1/2 ≤ ‖V ‖ ‖W‖ = ‖T‖cb. Thus we obtain one directionof (3.8). The converse follows from (2.2) (easy consequence of Cauchy-Schwarz) applied to ai =Ui ⊗ yi and bi = 1⊗ zi. Finally, the last assertion follows from the forthcoming Remark 3.12.

Remark 3.11 (Russo-Dye). The Russo-Dye Theorem 26.18 shows that the sup of any continuousconvex function on the unit ball of a unital C∗-algebra coincides with its sup over all its unitaryelements.

Remark 3.12. Let 0 be a singleton disjoint from the set I and let I = 0 ∪ I. Then for anyfinitely supported family xj | j ∈ I in B(H) (H arbitrary) we have

(3.9)∥∥∥I ⊗ x0 +

∑i∈I

Ui ⊗ xi∥∥∥

min= sup

∥∥∥∑j∈I

uj ⊗ xj∥∥∥

min

where the supremum runs over all possible families (uj)j∈I of unitaries.Indeed, since ∥∥∥∑

j∈Iuj ⊗ xj

∥∥∥min

=∥∥∥I ⊗ x0 +

∑i∈I

u−10 ui ⊗ xi

∥∥∥min

,

the right hand side of (3.9) is the same as the supremum of

(3.10)∥∥∥I ⊗ x0 +

∑i∈I

ui ⊗ xi∥∥∥

min

over all possible families of unitaries (ui)i∈I . Therefore (recalling U(gi) = Ui) we find∥∥∥I ⊗ x0 +∑

i∈IUi ⊗ xi

∥∥∥min

= sup∥∥∥I ⊗ x0 +

∑i∈I

ui ⊗ xi∥∥∥

min| ui unitary

,

where the sup runs over all Hilbert spaces H and all families (ui) of unitaries in B(H).Moreover, by the same argument we used for Lemma 3.10, we can restrict to finite dimensionalH’s:

(3.11)∥∥∥I ⊗ x0 +

∑i∈I

Ui ⊗ xi∥∥∥

min= supn≥1

∥∥∥I ⊗ x0 +∑

i∈Iui ⊗ xi

∥∥∥min| ui n× n unitaries

so that the supremum on the right-hand side is restricted to families of finite dimensional unitaries.Indeed, by Russo-Dye (Remark 3.11) the suprema of (3.10) taken over ui’s in the unit ball of B(H)and over unitary ui’s are the same. Replacing ui by PEui|E with E ⊂ H,dim(E) < ∞ showsthat the supremum of (3.10) is the same if we restrict it to ui’s in the unit ball of B(E) withdim(E) <∞. Then, invoking Russo-Dye (Remark 3.11) again, we obtain (3.11).

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Remark 3.13. Using (3.11) when I is a singleton and the fact that a single unitary generates acommutative unital C∗-algebra, it is easy to check that ‖T‖ = ‖T‖cb for any T : `2∞ → B(H).

Remark 3.14 (`1(I) as operator space). In the particular case A = C, (3.7) becomes

(3.12)∥∥∥∑

i∈IUixi

∥∥∥C∗(F)

=∑

i∈I|xi|,

which shows that EI = span[Ui, i ∈ I] ' `1(I) isometrically.Note that (3.8) generalizes the classical fact that B`1 = B`2B`2 for the pointwise product.More generally, Lemma 3.9 shows that the dual operator space E∗I can be identified with the vonNeumann algebra `∞(I) equipped with its natural operator space structure as a C∗-algebra, i.e.the one such that we have Mn(`∞(I)) = `∞(I;Mn) isometrically for all n. Lemma 3.10 describesthe dual operator space of the operator space (actually a C∗-subalgebra) c0(I) ⊂ `∞(I) that isthe closed span of the canonical basis in `∞(I). We obtain c0(I)∗ = EI completely isometrically,which is the operator space analogue of the isometric identity c0(I)∗ = `1(I). Indeed, together withLemma 3.9, (3.7) tells us that CB(c0(I),Mn) = Mn(EI) isometrically for all n.

3.3 Reduced group C∗-algebras. Fell’s absorption principle

We denote by C∗λ(G) (resp. C∗ρ(G)) the so-called “reduced” C∗-algebra generated in B(`2(G)) byλG (resp. ρG). Equivalently, C∗λ(G) = spanλG(t) | t ∈ G and C∗ρ(G) = spanρG(t) | t ∈ G.Note that λG(t) and ρG(s) commute for all t, s in G.

We denote λG and ρG simply by λ and ρ (and UG by U) when there is no ambiguity.The following very useful result is known as Fell’s “absorption principle”.

Proposition 3.15. For any unitary representation π : G→ B(H), we have

λG ⊗ π ' λG ⊗ I (unitary equivalence).

Here I stands for the trivial representation of G in B(H) (i.e. I(t) = IdH ∀t ∈ G). In particular,for any finitely supported functions a : G→ C and b : G→ B(`2), we have

(3.13) ‖∑

a(t)λG(t)⊗ π(t)‖C∗λ(G)⊗minB(H) = ‖∑

a(t)λG(t)‖,

‖∑

b(t)⊗ λG(t)⊗ π(t)‖B(`2)⊗minC∗λ(G)⊗minB(H) = ‖

∑b(t)⊗ λG(t)‖B(`2)⊗minC

∗λ(G).

Proof. Note that λG ⊗ π acts on the Hilbert space K = `2(G)⊗2 H ' `2(G;H). Let V : K → Kbe the unitary operator taking x = (x(t))t∈G to (π(t−1)x(t))t∈G. A simple calculation shows that

V −1(λG(t)⊗ IdH)V = λG(t)⊗ π(t).

We will often use the following immediate consequence:

Corollary 3.16. For any unitary representation π : G→ B(H), the linear map

σπ : span[λG(G)]→ B(`2(G)⊗2 H) defined by σπ(λG(g)) = λG(t)⊗ π(t) (∀t ∈ G)

extends to a (contractive) ∗-homomorphism from C∗λ(G) to B(`2(G)⊗2 H).

51

Remark 3.17. Let F be a free group with free generators (gj). Then for any finitely supportedsequence of scalars (aj), for any H and for any family (uj) of unitary operators in B(H) we have

‖∑

ajλ(gj)⊗ uj‖min = ‖∑

ajλ(gj)‖.

Indeed, this follows from (3.13) applied to the function a defined by a(gj) = aj and = 0 elsewhere,and to the unique unitary representation π of F such that π(gj) = uj .

Proposition 3.18. Let G be a discrete group and let Γ ⊂ G be a subgroup. Then the correspondenceλΓ(t) → λG(t), (t ∈ Γ) extends to an isometric (C∗-algebraic) embedding Jλ : C∗λ(Γ) → C∗λ(G).Moreover there is a completely contractive and completely positive projection Pλ from C∗λ(G) ontothe range of this embedding, taking λG(t) to 0 for any t 6∈ Γ.

Proof. Let Q = G/Γ and let G =⋃q∈Q

Γgq be the partition of G into (disjoint) right cosets. For

convenience, let us denote by 1 the equivalence class of the unit element of G. Since G ' Γ × Q,we have an identification

`2(G) ' `2(Γ)⊗2 `2(Q)

such that∀ t ∈ Γ λG(t) = λΓ(t)⊗ I.

This shows of course that Jλ is an isometric embedding. Moreover, we have a natural (linear)isometric embedding V : `2(Γ) → `2(G) (note that the range of V coincides with `2(Γ) ⊗ δ1 inthe preceding identification), such that λΓ(t) = V ∗λG(t)V for all t ∈ Γ. Let u(x) = V ∗xV .Clearly for any t ∈ G we have u(λG(t)) = λΓ(t) if t ∈ Γ and u(λG(t)) = 0 if t /∈ Γ. ThereforePλ = Jλu is the announced completely positive and completely contractive projection from C∗λ(G)onto JλC

∗λ(Γ).

As an immediate application, we state for further use the following particular case:

Corollary 3.19 (The diagonal subgroup in G × G). Let ∆ = (g, g) | g ∈ G ⊂ G × G be thediagonal subgroup. There are:− a complete isometry J∆ : C∗λ(G)→ C∗λ(G)⊗minC

∗λ(G) such that J∆(λG(t)) = λG(t)⊗λG(t), and

− a c.p. map Q∆ : C∗λ(G)⊗min C∗λ(G)→ C∗λ(G) with ‖Q∆‖ = 1 such that Q∆(λG(t)⊗ λG(s)) = 0

whenever s 6= t and Q∆(λG(t)⊗ λG(t)) = λG(t).

Proof. We apply Proposition 3.18 to the subgroup ∆ and we use the identification

C∗λ(G)⊗min C∗λ(G) ' C∗λ(G×G)

which follows easily from the definition of both sides (see §4.3 for more such identifications).

The projection Pλ in the preceding proposition is an example of mapping associated to a “mul-tiplier”.

3.4 Multipliers

Let ϕ : G → C be a (bounded) function. Let π be a unitary representation of G. Let Mϕ be thelinear mapping defined on the linear span of π(t) | t ∈ G by

∀t ∈ G Mϕ(π(t)) = ϕ(t)π(t).

52

As anticipated in Proposition 3.3, we say that ϕ is a bounded (resp. c.b. rresp. c.p.) multiplier onC∗π(G) if Mϕ extends to a bounded (resp. c.b. rresp. c.p.) linear map on C∗π(G).

We will be mainly interested in the cases when π = λG or π = UG.In the commutative case (or when G is amenable) the bounded or c.b. multipliers of C∗λ(G) coincidewith the linear combinations of positive definite functions, and the latter, as we explain next, arethe c.p. multipliers. However, in general the situation is more complicated. The next statementcharacterizes the c.b. case. We may even include B(H)-valued multipliers.

Theorem 3.20 ([136, 35]). Let G be a discrete group, H a Hilbert space. The following propertiesof a function ϕ : G→ B(H) are equivalent:

(i) The linear mapping defined on span[λ(t) | t ∈ G] by

Mϕ(λ(t)) = λ(t)⊗ ϕ(t)

extends to a c.b. map Mϕ : C∗λ(G)→ C∗λ(G)⊗min B(H) ⊂ B(`2(G)⊗2 H) with ‖Mϕ‖cb ≤ 1.

(ii) There is a Hilbert space H and bounded functions x : G → B(H, H) and y : G → B(H, H)with supt∈G ‖x(t)‖ ≤ 1 and sups∈G ‖y(s)‖ ≤ 1 such that

∀ s, t ∈ G ϕ(s−1t) = y(s)∗x(t).

Proof. Assume (i). Then by Theorem 1.50 there are a Hilbert space H, a representation π : C∗λ(G)→B(H) and operators Vj : `2(G)⊗2 H → H (j = 1, 2) with ‖V1‖‖V2‖ ≤ 1 such that

(3.14) ∀ θ ∈ G λ(θ)⊗ ϕ(θ) = Mϕ(λ(θ)) = V ∗2 π(λ(θ))V1.

We will use this for θ = s−1t, in which case we have 〈δs−1 , λ(θ)δt−1〉 = 1. We define x(t) ∈ B(H, H)and y(s) ∈ B(H, H) by x(t)h = π(λ(t)) V1(δt−1 ⊗ h) and y(s)k = π(λ(s))V2(δs−1 ⊗ k). Note thatwhen θ = s−1t

〈δs−1 ⊗ k, (λ(θ)⊗ ϕ(θ))(δt−1 ⊗ h)〉 = 〈k, ϕ(s−1t)h〉,

and hence (3.14) implies〈k, ϕ(s−1t)h〉 = 〈k, y(s)∗x(t)h〉,

and we obtain (ii).Conversely assume (ii). Define π : C∗λ(G)→ B(`2(G)⊗2 H) by π(x) = x⊗ Id

H. Let

Vj : `2(G)⊗2 H → `2(G)⊗2 H

be defined by V1(δt⊗h) = δt⊗x(t)h and V2(δs⊗k) = δs⊗y(s)k. Note that ‖V1‖ = supt∈G ‖x(t)‖ ≤ 1and ‖V2‖ = sups∈G ‖y(s)‖ ≤ 1. Then for any θ, t, s, h, k we have

〈δs ⊗ k, V ∗2 π(λ(θ))V1(δt ⊗ h)〉 = 〈δs, λ(θ)δt〉〈k, y(s)∗x(t)h〉 = 〈δs ⊗ k, (λ(θ)⊗ ϕ(θ))(δt ⊗ h)〉,

equivalently V ∗2 π(λ(θ))V1 = Mϕ(λ(θ)), so the converse part of Theorem 1.50 yields (ii) ⇒ (i).

In the particular case C = B(H) the preceding result yields:

Corollary 3.21 (Characterization of c.b. multipliers on C∗λ(G)). Consider a complex-valued func-tion ϕ : G→ C. Then ‖Mϕ : C∗λ(G)→ C∗λ(G)‖cb ≤ 1 if and only if there are Hilbert space valuedfunctions x, y with supt ‖x(t)‖ ≤ 1 and sups ‖y(s)‖ ≤ 1 such that

∀ s, t ∈ G ϕ(s−1t) = 〈y(s), x(t)〉.

53

Remark 3.22 (On positive definiteness). A function ϕ : G→ C is called positive definite if for anyn and any t1, · · · , tn ∈ G the n× n-matrix [ϕ(t−1

i tj)] is positive (semi)definite, i.e. we have

∀x ∈ Cn∑

xixjϕ(t−1i tj) ≥ 0.

Equivalently

∀x ∈ C[G]∑

x(s)x(t)ϕ(s−1t) ≥ 0.

Using the scalar product defined by the latter condition, we find, after passing to the quotient andcompleting in the usual way, a Hilbert space Hϕ and a mapping C[G]→ Hϕ denoted by x 7→ x with

dense range (so that ‖x‖2Hϕ =∑x(s)x(t)ϕ(s−1t) for all x ∈ C[G]) and a unitary representation πϕ

of G extending left translation on C[G]. Let δe ∈ C[G] denote the indicator function of the unitelement of G. We have

(3.15) 〈δe, πϕ(g)δe〉Hϕ = 〈δe, δg〉Hϕ = ϕ(g).

Thus ϕ is a (diagonal) matrix coefficient of π.Conversely, any ϕ of the form ϕ(g) = 〈ξ, π(g)ξ〉 (with π unitary and ξ ∈ Hπ) is positive definite.

Proposition 3.23. Let ϕ : G→ C. The following are equivalent:

(i) ϕ is a c.p. multiplier of C∗λ(G).

(ii) ϕ is positive definite.

Moreover, in that case we have ‖Mϕ‖ = ‖Mϕ‖cb = ϕ(e) where e is the unit of G.

Proof. Assume (i). Let t1, · · · , tn ∈ G. Consider the matrix a defined by aij = λG(ti)−1λG(tj).

Clearly a ∈ Mn(C∗λ(G))+. Then (IdMn ⊗Mϕ)(a) = [ϕ(t−1i tj)aij ] ∈ Mn(C∗λ(G))+. Therefore, for

any x1, · · · , xn ∈ `2(G) we have∑ϕ(t−1

i tj)〈xi, aijxj〉 ≥ 0. Choosing xj = λjδt−1j

(λj ∈ C) we find

〈xi, aijxj〉 = λiλj for all i, j, and we conclude that ϕ is positive definite.Assume (ii). By (3.15) we have for any g ∈ G

Mϕ(λG(g)) = ϕ(g)λG(g) = V ∗([λG ⊗ πϕ](g))V

where V : `2(G) → `2(G)⊗2 Hϕ is defined by V (h) = h⊗ δe. By Corollary 3.16 we have Mϕ(·) =V ∗(σπ(·))V , and hence Mϕ is c.p. on C∗λ(G). Moreover Mϕ(1) = ϕ(e)1, so ‖Mϕ(1)‖ = ϕ(e).

Remark 3.24. The reader can easily check that the preceding statement remains valid for B(H)-valued functions, in analogy with Theorem 3.20, for the natural extension of positive definiteness,defined by requesting that [ϕ(t−1

i tj)] ∈Mn(B(H))+ for all n. Such functions are sometimes calledcompletely positive definite.

In the preceding construction, we associated a linear mapping Mϕ to a function ϕ. We now goconversely. We will associate to a c.b. mapping a multiplier. In other words, we will describe alinear projection from the set of c.b. maps to the subspace formed by those associated to multipliers.

Proposition 3.25 (Haagerup). Let u : C∗λ(G) → C∗λ(G) be a c.b. map. Then the function ϕudefined by (recall e is the unit of G)

ϕu(t) = 〈δt, u(λG(t))δe〉

is a c.b. multiplier on C∗λ(G) with ‖Mϕu‖cb ≤ ‖u‖cb. If u is c.p. then the multiplier is also c.p.If u has finite rank then ϕu ∈ `2(G).Moreover, if u = Mϕ then ϕu = ϕ.

54

Proof. We have

‖IdC∗λ(G) ⊗ u : C∗λ(G)⊗min C∗λ(G)→ C∗λ(G)⊗min C

∗λ(G)‖ ≤ ‖u‖cb.

It is easy to see that C∗λ(G)⊗minC∗λ(G) can be identified with C∗λ(G×G). With this identification,

we have, for the mappings J∆, Q∆ in Corollary 3.19 , for any t ∈ G

ϕu(t)λG(t) = Q∆[IdC∗λ(G) ⊗ u]J∆(λG(t)).

In other words, Mϕu = Q∆[IdC∗λ(G) ⊗ u]J∆. All the assertions are now evident. We just note thatif u has rank 1, say u(x) = f(x)y with f ∈ C∗λ(G)∗ and y ∈ C∗λ(G), then ϕu(t) = f(λG(t))y(t), andt 7→ f(λG(t)) is bounded while y(t) = 〈δt, yδe〉 is in `2(G); this shows ϕu ∈ `2(G).

Remark 3.26. With the notation of the next section we have

ϕu(t) = τG(λG(t)∗u(λG(t))),

while with that of §11.2 it becomes ϕu(t) = 〈λG(t), u(λG(t))〉L2(τG).

The preceding two statements combined show that if u is decomposable as a linear combinationof c.p. maps on C∗λ(G) (as in chapter 6) then ϕu is a linear combination of positive definite functions.In particular:

Corollary 3.27. Let ϕ : G → C. The associated mapping Mϕ is decomposable on C∗λ(G) if andonly if ϕ is a linear combination of positive definite functions.

We will now complete the description started in Proposition 3.3 of multipliers on the full algebraC∗(G). In this case the picture is simpler.

Proposition 3.28. Let ϕ : G→ C. The following are equivalent:

(i) ϕ is a bounded multiplier on C∗(G).

(ii) ϕ is a linear combination of positive definite functions.

(iii) ϕ is c.b. multiplier on C∗(G).

Moreover, ϕ is positive definite if and only if Mϕ is c.p. on C∗(G).

Proof. We already know (i) ⇔ (iii) from Proposition 3.3. Assume (i). Then by Proposition 3.3 ϕsatisfies (3.3) for some π, η, ξ. By the polarization formula, we can rewrite ϕ as a linear combinationof four functions of the form t 7→ 〈ξ, π(t)ξ〉 with η = ξ. But the latter are clearly positive definite.This shows (i) ⇒ (ii). Assume ϕ positive definite. By (3.15) and by the case ξ = η in Proposition3.3 Mϕ is c.p. and hence a fortiori c.b. Now (ii) ⇒ (iii) is clear.

3.5 Group von Neumann algebra

We denote by MG ⊂ B(`2(G)) the von Neumann algebra generated by λG. This means thatMG = λG(G)′′. Equivalently MG is the weak* closure of the linear span of λG(G), and also theweak* closure of Cλ(G). See §26.16 for some background on von Neumann algebras (in particularon the bicommutant Theorem 26.46).Let f ∈ `2(G). Note that a priori, the operator of left convolution by f , Tf : x 7→ f ∗ x is onlybounded from `2(G) to `∞(G). An operator T ∈ B(`2(G)) belongs to MG if and only if there is

55

a (uniquely determined by f = Tf (δe)) function f ∈ `2(G) such that x 7→ f ∗ x defines a boundedoperator on `2(G) such that T = Tf .We have

M ′G = λG(G)′ = ρG(G)′′ and ρG(G)′ = MG.

Let Γ ⊂ G be a subgroup. Since the embedding Jλ : C∗λ(Γ) → C∗λ(G) in Proposition 3.18 isclearly bicontinuous with respect to the weak* topologies of B(`2(Γ)) and of B(`2(G)), it extendsto an embedding

MΓ ⊂MG,

with which we may identify MΓ to a von Neumann subalgebra of MG.Let δt | t ∈ G denote the canonical basis of `2(G). There is a distinguished tracial state τG

defined on MG byτG(T ) = 〈δe, T (δe)〉.

Of course this makes sense on the whole of B(`2(G)), but it is tracial only if we restrict to MG:

∀S, T ∈MG τG(TS) = τG(ST ).

Clearly τG is “normal” (meaning continuous for the weak* topology of B(`2(G)) ) and faithful(meaning τG(T ∗T ) = 0⇒ T = 0) and τG(1) = 1. Thus (MG, τG) is the basic example of a “tracial(or non-commutative) probability space” that we will consider in §12 when we discuss the Connesembedding problem.

Remark 3.29. Let ϕ be as in Corollary 3.21. Let Φ(s, t) = ϕ(s−1t). Then the Schur multiplier uΦ :B(`2(G))→ B(`2(G)) associated to Φ according to (iii) in Theorem 1.57 is completely contractiveon B(`2(G)) if and only if ϕ satisfies the equivalent conditions in Corollary 3.21. Moreover, thelatter Schur multiplier is weak* continuous, meaning continuous from B(`2(G)) to B(`2(G)) whenboth spaces are equipped with the weak* topology. Therefore, if we restrict to MG we obtaina weak* continuous (also called normal) complete contraction from MG to MG that extends themultiplier Mϕ : C∗λ(G)→ C∗λ(G). We will call the resulting maps weak* continuous multipliers onMG.A similar argument, based on Proposition 3.23, shows that ϕ is positive definite if and only if Mϕ

extends to a weak* continuous c.p. multiplier on MG.Lastly, the conclusion of Proposition 3.25 holds with the same proof for any c.b. map u : MG →MG.The resulting multiplier Mϕu is weak* continuous on MG, with ‖Mϕu‖cb ≤ ‖u‖cb. Moreover, if u isc.p. on MG, so is Mϕu .

3.6 Amenable groups

We review some basic facts on amenability.A discrete group G is called amenable if it admits an invariant mean, i.e. a functional ϕ in

`∞(G)∗+ with ϕ(1) = 1 such that ϕ(δt ∗ f) = ϕ(f) for any f in `∞(G) and any t in G.

Theorem 3.30. The following are equivalent:

(i) G is amenable.

(i)’ There is a net (hi) in the unit sphere of `2(G) that is approximately translation invariant,i.e. such that ‖λG(t)hi − hi‖2 → 0 for any t ∈ G.

(ii) C∗(G) = C∗λ(G).

56

(iii) For any finitely supported function f : G→ C we have |∑f(t)| ≤ ‖

∑f(t)λG(t)‖.

(iii)’ For any finite subset E ⊂ G, we have |E| =∥∥∑

t∈E λG(t)∥∥ .

(iv) There is a generating subset S ⊂ G with e ∈ S such that, for any finite subset E ⊂ S, wehave |E| =

∥∥∑t∈E λG(t)

∥∥ .(v) MG is injective.

Proof. Assume (i). Let ϕ be the invariant mean. Note that ϕ is in the unit ball of `1(G)∗∗+ .Therefore, there is a net (ϕi) in the unit ball of `1(G)+ tending in the sense of σ(`1(G)∗∗, `1(G)∗)to ϕ. Let 1 be the constant function equal to 1 on G. Since ϕi(1) → 1, we may assume afterrenormalization that ϕi(1) = ‖ϕi‖`1(G) = 1. Fix t ∈ G. Since δt ∗ ϕ = ϕ, we have δt ∗ ϕi − ϕi → 0when i→∞. But since δt ∗ ϕi − ϕi lies in `1(G) this means that limi→∞(δt ∗ ϕi − ϕi) = 0 for theweak topology of `1(G). By (Mazur’s) Theorem 26.9, passing to convex combinations of elementsof a subnet (here we leave some details to the reader, see Remark 26.10) we may assume thatlimi→∞ ‖δt ∗ϕi−ϕi‖`1(G) = 0. A priori, this was obtained for each fixed t, but, by suitably refiningthe argument (here again we skip some details), we can obtain the same for each finite subsetT ⊂ G. Let hi =

√ϕi. We claim that ‖δt ∗ hi − hi‖2 → 0 for any t ∈ T . This claim clearly

implies (i)’. To check the claim, using |x1/2 − y1/2| ≤ |x− y|1/2 for any x, y ∈ R+, we observe that|δt ∗ hi(s)− hi(s)| ≤ |δt ∗ ϕi(s)− ϕi(s)|1/2 and hence ‖δt ∗ hi − hi‖2 → 0 for any t ∈ T . This shows(i) ⇒ (i)’.Assume (i)’. Let x =

∑x(t)λG(t) ∈ span[λG(t) | t ∈ G]. Let π : G → B(H) be any unitary

representation. By the absorption principle (3.13) ‖∑x(t)λG(t)‖ = ‖

∑x(t)π(t) ⊗ λG(t)‖. We

claim that ‖∑x(t)π(t) ⊗ λG(t)‖ ≥ ‖

∑x(t)π(t)‖. Indeed, let fi be the state on B(`2(G)) defined

by fi(T ) = 〈hi, Thi〉. Then we have clearly

‖[Id⊗ fi](∑

x(t)π(t)⊗ λG(t))‖ ≤ ‖∑

x(t)π(t)⊗ λG(t)‖

but[Id⊗ fi](

∑x(t)π(t)⊗ λG(t)) =

∑x(t)π(t)fi(λG(t))→

∑x(t)π(t),

where at the last step we use fi(λG(t)) = 〈hi, δt ∗ hi〉 → 1. This implies the claim and hence‖∑x(t)λG(t)‖ ≥ ‖

∑x(t)π(t)‖. Taking the sup over the π’s we obtain (by “maximality” of UG)

‖∑x(t)λG(t)‖ = ‖

∑x(t)UG(t)‖. This shows (i)’ ⇒ (ii).

Assume (ii). Then (iii) holds by (3.2), and (iii) ⇒ (iii)’ ⇒ (iv) are trivial.Assume (iv). We will show (i)’. Fix E as in (iv). Let ME = |E|−1

∑t∈E λG(t) so that ‖ME‖ = 1.

There is a net (xi) in the unit sphere of `2(G) such that ‖ME(xi)‖ → 1. By the uniform convexityof `2(G) (see §26.3), this implies δt ∗ xi− xi → 0 in `2(G) for any t ∈ E. Rearranging the net (hereagain we leave the details to the reader) we find a net (hi) in the unit sphere of `2(G) such thatthe same holds for any t ∈ S, and since S generates G, still the same for any t ∈ G. This shows(iv) ⇒ (i)’.Assume (i)’. We will show (i). Let ϕ ∈ `∞(G)∗ be defined by

∀x ∈ `∞(G) ϕ(x) = limU∑

x(t)|hi(t)|2,

where U is an ultrafilter refining the net (see Remark 26.6). Let Dx ∈ B(`2(G)) be the diagonaloperator associated to x. Note that ϕ(x) = limU 〈hi, Dxhi〉, and also

(3.16) λG(t)DxλG(t)−1 = Dδt∗x.

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Therefore

ϕ(δt ∗ x) = limU 〈hi, Dδt∗xhi〉 = limU 〈λG(t)hi, DxλG(t)hi〉 = limU 〈hi, Dxhi〉 = ϕ(x).

Thus ϕ is an invariant mean, so (i) holds. This proves the equivalence of (i)-(iv), (i)’ and (iii)’. Itremains to show that (i) and (v) are equivalent.Assume (i). We will show that there is a c.p. projection P : B(`2(G)) → MG with ‖P‖ = 1. LetT ∈ B(`2(G)). We define ΦT : G→ B(`2(G)) by ΦT (g) = ρG(g)TρG(g)−1. We will define P (T ) asthe “integral” with respect to ϕ of the function ΦT , but some care is needed since ϕ is not really ameasure on G. Let [T (s, t)] be the “matrix” associated to T defined by T (s, t) = 〈δs, T δt〉 (s, t ∈ G).Observe that g 7→ ΦT (g)(s, t) is in `∞(G). Then we set

P (T )(s, t) = ϕ(ΦT (·)(s, t)).

This defines a matrix and it is easy to see that the associated linear operator on span[δt | t ∈ G]extends to a bounded one (still denoted by P (T )) on `2(G) such that ‖P (T )‖ ≤ ‖T‖. We haveΦT (g)(s, t) = T (sg, tg) and hence, by the left invariance of ϕ, P (T )(s, t) = P (T )(st−1, e). Thisshows that P (T ) acts on `2(G) as a left convolution bounded operator, in other words P (T ) ∈MG. Moreover, if T ∈ MG then T commutes with ρG so we have P (T ) = T . This proves thatP : B(`2(G))→MG is a contractive projection. A simple verification left to the reader shows thatit is c.p. (but this is automatic by Tomiyama’s Theorem 1.45). This shows (i) ⇒ (v).Assume (v). Let P : B(`2(G))→MG be a projection with ‖P‖ = 1. Invoking Theorem 1.45 again,we know that P is a c.p. conditional expectation. We define

∀x ∈ `∞(G) ϕ(x) = τG(P (Dx)) = 〈δe, P (Dx)δe〉.

Clearly ϕ ∈ `∞(G)∗+, ϕ(1) = 1 and by (3.16), (1.28) and the trace property of τG

∀t ∈ G ϕ(δt ∗ x) = τG[P (λG(t)DxλG(t)−1)] = τG[λG(t)P (Dx)λG(t)−1] = τG[P (Dx)] = ϕ(x).

Thus ϕ is an invariant mean on G. This shows (v) ⇒ (i).

Remark 3.31. If the generating set S is finite, the condition (iv) obviously reduces (by the triangleinequality) to

|S| =∥∥∥∑

t∈SλG(t)

∥∥∥ .Remark 3.32. The net (hi) in (i)’ is sometimes called asymptotically left invariant. By density (andafter renormalization) when it exists, it can always be found in the group algebra C[G].

Remark 3.33 (On Følner sequences). It is well known (see e.g. [194]) that for any amenable discretegroup G the net (hi) appearing in (i)’ in Theorem 3.30 can be chosen of the form hi = 1Bi |Bi|−1/2

for some family (Bi) of finite subsets of G. For (hi) of the latter form, (i)’ boils down to theassertion that the symmetric differences (tBi)∆Bi satisfy

∀t ∈ G |tBi∆Bi||Bi|−1 → 0.

A net of finite subsets (Bi) satisfying this is called a Følner net, and a Følner sequence when theindex set is N. Thus a (resp. countable) group G is amenable if and only if it admits a Følner net(resp. sequence). For instance, for G = Zd (1 ≤ d < ∞), the sequence Bn = [−n, n]d is a Følnersequence.

This gives us the following special property of the reduced C∗-algebra, called the CPAP in thesequel (see Definition 4.8):

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Lemma 3.34. If G is amenable, there is a net of finite rank maps ui ∈ CP (C∗λ(G), C∗λ(G)) (resp.ui ∈ CP (C∗(G), C∗(G))) with ‖ui‖ = 1 that tends pointwise to the identity on C∗λ(G) (resp. C∗(G)).Moreover, in both cases the ui’s are multiplier operators.

Proof. By Remark 3.32, there is a net (hi) in C[G] in the unit sphere of `2(G) such that ‖λG(t)hi−hi‖2 → 0 for any t in G. Let h∗i (t) = hi(t−1) (t ∈ G). A simple verification show that ϕi = h∗i ∗ hiis a positive definite function on G such that ϕi(e) = ‖hi‖22 = 1. Moreover, ϕi is finitely supportedand tends pointwise to the constant function 1 on G. Let ui be the associated multiplier operatoron C∗λ(G) (resp. C∗(G)). Its rank being equal to the cardinality of the support of ϕi is finite.By Proposition 3.23 (resp. Proposition 3.28), ui is c.p. and since ui(1) = ϕi(e)1 = 1, we have‖ui‖ = 1 by (1.20). For any x =

∑x(t)λG(t) (resp. x =

∑x(t)UG(t)) with x finitely supported,

ui(x) obviously tends to x in the norm of C∗λ(G) (resp. C∗(G)). Since such finite sums are densein C∗λ(G) (resp. C∗(G)) and supi ‖ui‖ <∞, we conclude that ui(x)→ x for any x ∈ C∗λ(G) (resp.C∗(G)).

Remark 3.35 (Examples of amenable groups). All commutative groups are amenable. If G iscommutative (and discrete), its dual G is defined as the group formed of all homomorphismsγ : G → T, which is compact for the pointwise convergence topology. For any finitely supportedfunction f : G → C we define its “Fourier transform” by f(γ) =

∑f(g)γ(g). (This is the usual

convention but we could remove the bar from γ(g) if we wished). As is entirely classical f 7→ fextends to an isometric isomorphism from `2(G) to L2(G,m), where m is the normalized Haarmeasure on G, and convolution of two functions on G is transformed into the pointwise product oftheir Fourier transforms. Using the latter fact one shows that the correspondence f 7→ f extendsto an isometric isomorphism from C∗λ(G) to the C∗-algebra C(G) of all continuous functions on G.Thus in the commutative case we have

(3.17) C∗(G) = C∗λ(G) ' C(G).

All finitely generated groups of polynomial growth are amenable. The growth is defined usingthe length. If G is generated by a symmetric set S the smallest number of elements of S needed towrite an element g ∈ G (as a word in letters in S) is denoted by `S(g). The growth function is thefunction Φ(R) = |g ∈ G | `S(g) ≤ R. The group G is called of polynomial growth if Φ(R) growsless than a power of R when R→∞. For instance G = Zn is of polynomial growth (but Fn is notwhenever n ≥ 2).

Remark 3.36. By Kesten’s famous work on the spectral radius of random walks on the free groupFn with n generators, the set S1 ⊂ Fn formed of the 2n elements of length 1 (i.e. these are eithergenerators or their inverses), satisfies

(3.18)∥∥∥∑

s∈S1

λFn(s)∥∥∥ = 2

√2n− 1.

Kesten also observed that it is not difficult to deduce from this that for any group G and anysymmetric subset S ⊂ G with |S| = k we have

‖∑

s∈SλG(s)‖ ≥ 2

√k − 1.

Akemann and Ostrand [2] proved that any S ⊂ S1 in Fn with |S| = k satisfies

(3.19) ‖∑

s∈SλFn(s)‖ = 2

√k − 1.

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In particular

(3.20) ‖∑n

j=1λFn(gj)‖ = 2

√n− 1.

The subsets S of a discrete group for which (3.19) holds have been characterized by Franz Lehnerin [166], as the translates of the union of a free set and the unit.Let S ⊂ Fn be the set formed of the unit and the n free generators, so that |S| = n + 1. Then avariant of what precedes is that for G = Fn

(3.21) ‖∑

s∈SλG(s)‖ = 2

√n.

When n ≥ 2, this is < n + 1, and hence (iii) or (iv) in Theorem 3.30 fails. This shows that Fn isnot amenable for n ≥ 2.

Since amenability passes to subgroups (by Proposition 3.18 and (i)⇔ (iv) in Theorem 3.30), anygroup containing a copy of F2 as a subgroup is non-amenable. The converse, whether non-amenablegroups must contain F2, remained a major open question for a long time but was disproved by A.Olshanskii, see [126] for details. See Monod’s [178] for what seems to be currently the simplestconstruction of non-amenable groups not containing F2 as a subgroup.

3.7 Operator space spanned by the free generators in C∗λ(Fn)

The next statement gives us a description up to complete isomorphism of the span of the generatorsin C∗λ(Fn) (and also implicitly in C∗λ(F∞)). See [168] for a more precise (completely isometric)description.

Theorem 3.37. Let (gj)1≤j≤n be the generators in Fn (n ≥ 1). Then for any Hilbert space H andany aj ∈ B(H) (1 ≤ j ≤ n) we have(3.22)

max‖∑

a∗jaj‖1/2, ‖∑

aja∗j‖1/2 ≤ ‖

∑aj ⊗ λFn(gj)‖min ≤ ‖

∑a∗jaj‖1/2 + ‖

∑aja∗j‖1/2.

In particular for any αj ∈ C we have

(∑|αj |2)1/2 ≤ ‖

∑αjλFn(gj)‖ ≤ 2(

∑|αj |2)1/2.

Proof. We will first prove the upper bound in (3.22). Let C+i ⊂ Fn (resp. C−i ⊂ Fn) be the subset

formed by all the reduced words which start by gi (resp. g−1i ). Note: except for the empty word

e, every element of G can be written as a reduced word in the generators admitting a well defined“first” and “last” letter (where we read from left to right). Let P+

i (resp. P−i ) be the orthogonalprojection on `2(Fn) with range span[δt | t ∈ C+

i ] (resp. span[(δt | t ∈ C−i ]). The 2n projectionsP+

i , P−i | 1 ≤ i ≤ n are mutually orthogonal. Then it is easy to check that

λFn(gj) = λFn(gj)P−j +λFn(gj)(1−P−j ) = λFn(gj)P

−j +P+

j λFn(gj)(1−P−j ),= λFn(gj)P−j +P+

j λFn(gj),

so that setting λFn(gj) = xj + yj with xj = λFn(gj)P−j and yj = P+

j λFn(gj) we find∥∥∥∑x∗jxj

∥∥∥ =∥∥∥∑P−j

∥∥∥ ≤ 1 and∥∥∥∑ yjy

∗j

∥∥∥ =∥∥∥∑P+

j

∥∥∥ ≤ 1.

Therefore for any finite sequence (aj) in B(H) we have by (1.11) (note aj ⊗ xj = (aj ⊗ 1)(1⊗ xj)and similarly for aj ⊗ yj)∥∥∥∑ aj ⊗ λFn(gj)

∥∥∥ ≤ ∥∥∥∑ aj ⊗ xj∥∥∥+

∥∥∥∑ aj ⊗ yj∥∥∥ ≤ ∥∥∥∑ aja

∗j

∥∥∥1/2+∥∥∥∑ a∗jaj

∥∥∥1/2.

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The inverse inequality follows from a more general one valid for any discrete group G: for anyfinitely supported function a : G→ B(H) we have

(3.23) max

∥∥∥∑ a(t)∗a(t)∥∥∥1/2

,∥∥∥∑ a(t)a(t)∗

∥∥∥1/2≤∥∥∥∑ a(t)⊗ λG(t)

∥∥∥min

.

To check this, let T =∑a(t)⊗ λG(t). For any h in BH we have T (h⊗ δe) =

∑a(t)h⊗ δt so that

‖T (h⊗ δe)‖ =(∑

t ‖a(t)h‖2)1/2

and hence∥∥∥∑ a(t)∗a(t)∥∥∥1/2

= suph∈BH

(∑‖a(t)h‖2

)1/2≤ ‖T‖.

Similarly since T ∗ =∑a(t−1)∗ ⊗ λG(t) we find∥∥∥∑ a(t)a(t)∗

∥∥∥1/2≤ ‖T ∗‖ = ‖T‖

and we obtain (3.23). In the case G = Fn, (3.23) implies the left hand side of (3.22). The secondinequality follows by taking aj = αj1.

Corollary 3.38. For any n,N ≥ 1 and any unitaries a ∈ Un, x1, · · · , xn ∈ UN we have

‖∑n

i,j=1aijxi ⊗ λFn(gj)‖ ≤ 2

√n

Proof. Let aj =∑

i aijxi. Since a is unitary a simple verification (using (26.13)) shows that wehave ‖

∑a∗jaj‖1/2 ≤ ‖

∑x∗jxj‖1/2 =

√n and ‖

∑aja∗j‖1/2 ≤ ‖

∑xjx∗j‖1/2 =

√n.

3.8 Free products of groups

Let (Gi)i∈I be a family of groups. The free product G = ∗i∈IGi is a group containing each Gi asa subgroup and possessing the following universal property that characterizes it: for any group G′

and any family of homomorphisms fi : Gi → G′, there is a unique homomorphism f : G → G′

extending each fi.When I = 1, 2 we denote G1 ∗G2 the free product ∗i∈IGi.When I = 1, · · · , n and G1 = · · · = Gn = Z it is easy to see that G = ∗i∈IGi can be identified

with Fn.More generally, any free group F that is generated by a family of free elements (gi)i∈I can be

identified with the free product ∗i∈IGi relative to Gi = Z for all i ∈ I. We denote that group byFI .

It is well known that any group G is a quotient of some free group. Indeed, if G is generated bya family (ti)i∈I , let f : FI → G be the (unique) homomorphism such that f(gi) = ti for all i ∈ I.Then f is onto G. Thus G ' FI/ ker(f). The analogous fact for C∗-algebras is the next statement.

Proposition 3.39. Any unital C∗-algebra A is a quotient of C∗(FI) for some set I. If A isseparable (resp. is generated by n unitaries) then we can take I = N (resp. I = 1, · · · , n).

Proof. Let G be the unitary group of A. Let f : FI → G be a surjective homomorphism. Letπ : C∗(FI)→ A be the associated ∗-homomorphism, as in Remark 3.1. By the Russo-Dye Theorem26.18, the range of π is dense in A, but since it is closed (see §26.14), π must be surjective. Thus Ais a quotient of C∗(FI). If A is generated as a C∗-algebra by a family of unitaries (ui)i∈I , we canreplace G in the preceding argument by the group generated by (ui)i∈I . This settles the remainingassertions.

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Remark 3.40. As we saw in Remark 3.36, F2 = Z ∗ Z is not amenable. More generally it can beshown that Zn∗Zm is not amenable if n ≥ 2 and m ≥ 3, and in fact contains a subgroup isomorphicto F∞. The group Z2 ∗ Z2 is a slightly surprising exception, it is amenable because it happens tohave polynomial growth (an exercise left to the reader).

3.9 Notes and Remarks

The main results of this section are by now well known, and sometimes for general locally compactgroups (for instance Proposition 3.5 is proved in greater generality in [226]), but we choose to focuson the discrete ones. §3.2 on free groups is just a reformulation of operator space duality illustratedon the pair (`1, `∞). Lemmas 3.9 and 3.10 are elementary facts from operator space theory (see[80, 208]). The classical reference that exploited C∗-algebra theory in non-commutative harmonicanalysis is Eymard’s thesis [85]. The name of Fell is attached to the notions of weak containment andweak equivalence of group representations, which apparently led him to the principle enunciated inTheorem 3.15. Concerning multipliers, those considered in Theorem 3.20 are sometimes called Herz-Schur multipliers (in honor of Carl Herz). The characterization in Theorem 3.20 and its Corollaryis due to Jolissaint [136], but the simple proof we give is due to Bozejko and Fendler [35]. Ourtreatment is inspired by Haagerup’s unpublished (but widely circulated) notes on multipliers, wherein particular he proves Proposition 3.25. There are many known characterizations of amenability,the main one going back to Kesten, with variants due to Hulanicki and many authors. We refer thereader to [194] (or [199]) for details and references. Theorem 3.37 appears in [118]. In [168] Lehnergives an exact computation of the norm of

∑aj ⊗ λFn(gj) when the coefficients aj are matricial or

equivalently when dim(H) <∞.

4 C∗-tensor products

A norm on a ∗-algebra A is called a C∗-norm if it satisfies

(4.1) ‖x‖ = ‖x∗‖, ‖xy‖ ≤ ‖x‖ ‖y‖ and ‖x∗x‖ = ‖x‖2

for any x, y in A.The completion A of (A, ‖.‖) then becomes a C∗-algebra. It is useful to point out that, after

completion, the norm is unique: there is only one C∗-norm on a C∗-algebra. In particular if twoC∗-norms on A are distinct then they are not equivalent, since otherwise they would produce thesame completion, where the C∗-norm is unique.

In particular any ∗-isomorphism between C∗-algebras must be isometric.More generally, it is useful to record here that any injective ∗-homomorphism between C∗-

algebras is automatically isometric. Consequently, a ∗-homomorphism between C∗-algebras musthave a closed range, and the range is isometric to a quotient C∗-algebra of the source of the map.Indeed, the kernel of any ∗-homomorphism u : A → B is a (closed two-sided and self-adjoint)ideal I ⊂ A. Passing to the quotient gives us an injective (and hence isometric) ∗-homomorphismA/I → B, which must have a closed range. When u is surjective we have B ∼= A/I. See Proposition26.24 for more details.

4.1 C∗-norms on tensor products

Let A1, A2 be two C∗-algebras. Their algebraic tensor product A1⊗A2 is a ∗-algebra for the naturaloperations defined by

(a1 ⊗ a2) · (b1 ⊗ b2) = a1b1 ⊗ a2b2

62

and(a1 ⊗ a2)∗ = a∗1 ⊗ a∗2.

Thus a norm ‖ ‖ on A1 ⊗A2 is a C∗-norm if it satisfies (4.1) for any x, y in A1 ⊗A2.This subject was initiated in the 1950’s by Turumaru in Japan. Later work by Takesaki and

Guichardet leads to the following result.

Theorem 4.1. There is a minimal C∗-norm ‖ ‖min and a maximal one ‖ ‖max, so that anyC∗-norm ‖ · ‖ on A1 ⊗A2 must satisfy

(4.2) ‖x‖min ≤ ‖x‖ ≤ ‖x‖max ∀x ∈ A1 ⊗A2.

We denote by A1⊗minA2 (resp. A1⊗maxA2) the completion of A1⊗A2 for the norm ‖ ‖min (resp.‖ ‖max).

The maximal C∗-norm is easy to describe. We simply write

(4.3) ‖x‖max = sup ‖π(x)‖B(H)

where the supremum runs over all possible Hilbert spaces H and all possible ∗-homomorphismsπ : A1 ⊗A2 → B(H).

The minimal (or spatial) norm can be described as follows: embed A1 and A2 as C∗-subalgebrasof B(H1) and B(H2) respectively, then for any x =

∑a1i ⊗ a2

i in A1 ⊗ A2, ‖x‖min coincides withthe norm induced by the space B(H1 ⊗2 H2), i.e. we have an embedding (i.e. an isometric ∗-homomorphism) of the completion, denoted by A1 ⊗min A2, into B(H1 ⊗2 H2). The resultingC∗-algebra does not depend on the particular embeddings A1 ⊂ B(H1) and A2 ⊂ B(H2).More generally, even if we allow completely isometric linear embeddings A1 ⊂ B(H1) and A2 ⊂B(H2), we obtain the same norm (i.e. the min-norm) induced on A1 ⊗ A2. So that, actually, theminimal tensor product of operator spaces, introduced in §1.1, coincides with the minimal C∗-tensorproduct when restricted to two C∗-algebras. See §1.1 for more on this.

Proof of Theorem 4.1. Let x 7→ ‖x‖α be a C∗-norm on A1 ⊗ A2. After completion we find a C∗-algebra A1⊗αA2 and a (Gelfand-Naimark) embedding π : A1⊗αA2 ⊂ B(H). For any x ∈ A1⊗A2

we have ‖x‖α = ‖π(x)‖, which shows ‖x‖α ≤ ‖x‖max. This proves the second inequality in (4.2).In particular, the minimal norm must satisfy ‖ ‖min ≤ ‖ ‖max. This goes back to Guichardet. Thelower bound ‖ ‖min ≤ ‖ ‖α is due to Takesaki and is much more delicate. For a proof, see either[241] or [146].

It is easy to see (at least in the unital case) that for any ∗-homomorphism π : A1⊗A2 → B(H)there is a pair of (necessarily contractive) ∗-homomorphisms πi : Ai → B(H) (i = 1, 2) withcommuting ranges such that

(4.4) π(a1 ⊗ a2) = π1(a1)π2(a2) ∀a1 ∈ A1 ∀a2 ∈ A2.

Indeed, in the unital case we just set π1(a1) = π(a1 ⊗ 1) and π2(a2) = π(1 ⊗ a2). In the generalcase, the same idea works with approximate units (see Remark 4.2).Conversely any such pair πj : Aj → B(H) (j = 1, 2) of ∗-homomorphisms with commuting rangesdetermines uniquely a ∗-homomorphism π : A1⊗A2 → B(H) by setting π(a1⊗a2) = π1(a1)π2(a2).

For any finite sum x =∑a1k ⊗ a2

k in A1 ⊗A2 we will use the notation

(4.5) (π1 · π2)(x) =∑

π1(a1k)⊗ π2(a2

k),

63

with which π = π1 · π2. Then, we can rewrite (4.3) as:

(4.6) ‖x‖max = sup∥∥∥∑π1(a1

k)π2(a2k)∥∥∥ = sup ‖(π1 · π2)(x)‖ ,

where the supremum runs over all possible such pairs (π1, π2).Since ‖π1‖ ≤ 1 and ‖π2‖ ≤ 1 we have ‖

∑a1k ⊗ a2

k‖max ≤∑‖a1

k‖‖a2k‖ and hence (see §26.1)

(4.7) ‖x‖max ≤ ‖x‖∧ = inf∑‖a1

k‖‖a2k‖

where the infimum runs over all possible ways to write x as a finite sum of tensors of rank 1.Incidentally, this ensures that (4.6) or (4.3) is finite.

Remark 4.2 ((4.4) still holds in the nonunital case). In the general a priori nonunital case, weclaim that it is still true that any ∗-homomorphism π : A1 ⊗A2 → B(H) that is “nondegenerate”(meaning here such that V = π(A1 ⊗A2)(H) is dense in H) must be of the form (4.4). Lett =

∑aj1 ⊗ a

j2 ∈ A1 ⊗ A2. We denote for x1 ∈ A1 (resp. x2 ∈ A2) x1 · t =

∑x1a

j1 ⊗ a

j2 (resp.

x2 · ·t =∑aj1⊗x2a

j2). Let ξ =

∑n1 π(tk)hk ∈ V (tk ∈ A1⊗A2). We define π1(x1)ξ =

∑n1 π(x1 ·tk)hk

and similarly π2(x2)ξ =∑n

1 π(x2 · ·tk)hk. Then π(x1 ⊗ x2)ξ = (π1 · π2)(x1 ⊗ x2)(ξ) and henceπ(t)ξ = (π1 · π2)(t)(ξ) for any t ∈ A1 ⊗ A2. Moreover πj extends to a bounded ∗-homomorphismπj : Aj → B(H). Indeed, a simple verification shows that ‖π1(x1)ξ‖2 ≤ ‖x1‖2‖ξ‖2 and similarly‖π2(x2)ξ‖2 ≤ ‖x2‖2‖ξ‖2. This proves the claim.

Remark 4.3 ((4.6) still holds in the nonunital case). Now if V is not assumed dense in H, theexistence of approximate units shows that ‖π(t)‖ = ‖π(t)|V‖ so that since we can always replace H

by V we conclude that (4.6) still holds in the general a priori nonunital case.

Remark 4.4. The norm ‖x‖∧ appearing in (4.7) is called the projective norm. It is the largestamong the reasonable tensor norms on tensor products of Banach spaces in Grothendieck’s sense(see §26.1), but it is not adapted to our context because it is not a C∗-norm.

Remark 4.5. It follows that any C∗-norm ‖ · ‖ on A1 ⊗A2 automatically satisfies

‖a1 ⊗ a2‖ = ‖a1‖ ‖a2‖ ∀a1 ∈ A1,∀a2 ∈ A2.

Indeed, it is easy to show that ‖a1 ⊗ a2‖max ≤ ‖a1‖ ‖a2‖ and ‖a1 ⊗ a2‖min ≥ ‖a1‖ ‖a2‖.Note that if α is either min or max we have canonically

(4.8) A1 ⊗α A2 ' A2 ⊗α A1.

Indeed, in both cases the flip a1 ⊗ a2 7→ a2 ⊗ a1 extends to an isomorphism.The basic definitions extend to tensor products of n-tuples A1, A2, . . . , An of C∗-algebras, but

when α = either min or max the resulting tensor products are “associative” so that we may reduceconsideration if we wish to the case n = 2. Indeed, “associative” means here the identity

(4.9) (A1 ⊗α A2)⊗α A3 = A1 ⊗α A2 ⊗α A3 = A1 ⊗α (A2 ⊗α A3),

which is easy to check and shows that the theory of multiple products reduces, by iteration, to thatof products of pairs.

Remark 4.6. Let A = (⊕∑n

1 Aj)∞ be the direct sum of a finite family of C∗-algebras. It is easy to

check that for any representation π : A→ B(H) that is nondegenerate (that is such that π(A)H =H) there is an orthogonal decomposition H = ⊕

∑n1 Hj and nondegenerate representations πj :

64

Aj → B(Hj) such that π can be unitarily identified with π1 ⊕ · · · ⊕ πn. Using this it is easy tocheck that, when α is either min or max, for any C∗-algebra B we have

(4.10) (⊕∑n

1Aj)∞ ⊗α B ' (⊕

∑n

1Aj ⊗α B)∞.

For α = min, a more general identity holds, see (1.14).

Let (B1, B2) be another pair of C∗-algebras and let πi : Ai → Bi (i = 1, 2) be ∗-homomorphisms.Then it is immediate from the definition that ‖(π1 ⊗ π2)(t)‖max ≤ ‖t‖max for any t ∈ A1 ⊗A2 andhence π1 ⊗ π2 defines a ∗-homomorphism from A1 ⊗max A2 to B1 ⊗max B2. For the minimal tensorproduct, this is also true because ∗-homomorphisms are automatically complete contractions (seeRemark 1.6). Indeed, consider c.b. maps ui : Ai → Bi (i = 1, 2). Then (see §1.1) u1⊗ u2 defines ac.b. map from A1 ⊗min A2 to B1 ⊗min B2 with ‖u1 ⊗ u2‖cb = ‖u1‖cb‖u2‖cb.In sharp contrast, the analogous property does not hold for the max-tensor products. However, itdoes hold if we moreover assume that u1 and u2 are completely positive (resp. decomposable) andthen (see the forthcoming Corollary 6.12 and §7.1) the resulting map u1 ⊗ u2 also is completelypositive (resp. decomposable) from A1 ⊗max A2 to B1 ⊗max B2, and we have

∀x ∈ A1 ⊗A2 ‖(u1 ⊗ u2)(x)‖B1⊗maxB2 ≤ ‖u1‖‖u2‖ ‖x‖A1⊗maxA2 ,

(resp. ‖u1 ⊗ u2‖dec ≤ ‖u1‖dec‖u2‖dec).

As we will see in Theorem 7.6 decomposable maps are the “right” analogue of c.b. maps when onereplaces the minimal tensor products by the maximal ones.If B1 = B(H1) and B2 = B(H2) (or merely if both B1 and B2 are assumed injective) then‖u1‖dec = ‖u1‖cb and ‖u2‖dec = ‖u2‖cb (see Proposition 6.7), so in this particular case there isno problem, tensor products of c.b. maps are bounded both on the minimal and maximal tensorproducts.

We have obviously a bounded ∗-homomorphism q : A1 ⊗max A2 → A1 ⊗min A2, which (asall C∗-representations) has a closed range, hence A1 ⊗min A2 is C∗-isomorphic to the quotient(A1⊗maxA2)/ker(q). The observation that in general q is not injective is at the basis of the theoryof nuclear C∗-algebras.

4.2 Nuclear C∗-algebras (a brief preliminary introduction)

In these notes, we will emphasize the notion of nuclear pair rather than that of nuclear C∗-algebra(which is by now well known), i.e. we will focus attention to specific pairs (A,B) of C∗-algebrassuch that the min and max C∗-norms coincide on A⊗B. See chapter 9. Thus in our presentationthe theory of nuclear C∗-algebras becomes embedded in that of nuclear pairs. Nevertheless, itseems more convenient to give here first a brief overview of nuclear C∗-algebras.

Definition 4.7. A C∗-algebra A is called nuclear if for any C∗-algebra B we have ‖ ‖min = ‖ ‖max

on A⊗B or in short if A⊗min B = A⊗max B. In that case, there is only one C∗-norm on A⊗B.

This notion was introduced (under a different name) by Takesaki and was especially investigatedby Lance [165], who saw the connection with the following property:

Definition 4.8. A C∗-algebra A has the completely positive approximation property (in shortCPAP) if the identity on A is the pointwise limit of a net of finite rank c.p. maps.

We will see in Corollary 7.12 that the CPAP is actually equivalent to nuclearity. Since we placethis result in a much broader context (see Theorem 7.10), we delay the full details of its proof till§10.2.

65

Remark 4.9. If A and B are nuclear then A ⊗min B is also nuclear. This is easy to check using(4.9). By (4.10), A⊕B is nuclear, as well as the direct sum of finitely many nuclear C∗-algebras.

Remark 4.10 (Examples of nuclear C∗-algebras). For example, if dim(A) <∞, A is nuclear, becauseA⊗B (there is no need to complete it!) is already a C∗-algebra, hence it admits a unique C∗-norm.All commutative C∗-algebras are nuclear.Indeed, any such algebra A is isometric to the space C0(T ) of continuous functions vanishing at ∞on a locally compact space T (that can be taken compact in the unital case). It is a well known factfrom Banach space theory that all such spaces have the metric approximation property, meaningthat the identity is the pointwise limit of a net of finite rank maps of norm at most 1. Moreover,in the particular case of A = C0(T ) we can arrange the latter net to be formed of positive maps.Since the latter maps are automatically c.p. (see Remark 1.28) this shows that A has the CPAP,and hence by Corollary 7.12 that A is nuclear.It is an easy exercise to show that K(H) has the CPAP, and hence is nuclear. For the samereason (although it is not so immediate) the Cuntz algebras are nuclear. The Cuntz algebra Onfor n ∈ N ∪ ∞ is the C∗-subalgebra of B(`2) generated by an n-tuple (a sequence if n = ∞)of isometries (Sj) on `2 such that

∑SjS

∗j = 1. It can be shown that, given any fixed n, all such

algebras are isomorphic, regardless of the choice of (Sj).In sharp contrast, B(`2) is not nuclear and C∗(F2) does not embed in a nuclear C∗-algebra (due toSimon Wasserman [256, 258]); we will prove these facts in the sequel (see Th. 12.29 or Cor. 18.12and Prop. 7.34).Other examples or counterexamples can be given among group C∗-algebras. For any discrete groupG, the full C∗-algebra C∗(G) or the reduced one C∗λ(G) (as defined in §3.1 and §3.3) is nuclear ifand only if G is amenable. See the subsequent Corollary 7.13 for details. So for instance if G = FIwith |I| ≥ 2 then C∗(G) and C∗λ(G) are not nuclear. (Note that for continuous groups the situationis quite different: Connes [61] proved that, for any separable connected locally compact group G,C∗(G) and C∗λ(G) are nuclear.)

4.3 Tensor products of group C∗-algebras

The following results are easy exercises:Let G1, G2 be two discrete groups. Then

(4.11) C∗(G1)⊗max C∗(G2) ' C∗(G1 ×G2),

(4.12) C∗λ(G1)⊗min C∗λ(G2) ' C∗λ(G1 ×G2),

and similarly for the free product G1 ∗G2:

(4.13) C∗(G1) ∗ C∗(G2) ' C∗(G1 ∗G2).

These identities can be extended to arbitrary families (Gi)i∈I in place of the pair (G1, G2). Inparticular, we have

∗i∈I

C∗(Gi) ' C∗(∗i∈I

Gi

).

The next result (essentially from [208, p.150]) illustrates the usefulness of the Fell principle (seeProposition 3.15).

66

Theorem 4.11. Let ΨG : C∗λ(G) ⊗max C∗λ(G) → MG ⊗max MG be the extension of the natural

inclusion C∗λ(G)⊗ C∗λ(G) ⊂MG ⊗MG. We have an isometric ∗-homomorphism

JG : C∗(G)→ C∗λ(G)⊗max C∗λ(G)

taking UG(t) to λG(t)⊗ λG(t) (t ∈ G), and a completely contractive c.p. mapping

PG : MG ⊗max MG → C∗(G)

such that IdC∗(G) = PGΨGJG as in the diagram

IdC∗(G) : C∗(G)JG−−−→C∗λ(G)⊗max C

∗λ(G)

ΨG−−−→MG ⊗max MGPG−−−→C∗(G).

Moreover, for all a, b ∈MG, such that a(δe) =∑

t∈G a(t)δt, b(δe) =∑

t∈G b(t)δt, we have (absolutelyconvergent series)

PG(a⊗ b) =∑

t∈Ga(t)b(t)UG(t).

We can express the preceding as a commuting diagram:

C∗λ(G)⊗max C∗λ(G)

ΨG //MG ⊗max MG

PG

C∗(G)?

JG

OO

IdC∗(G) // C∗(G)

Proof. Let x ∈MG ⊗MG (algebraic tensor product). For s ∈ G let fs ∈M∗G be the natural linearform defined by fs(a) = 〈δs, aδe〉, so that a =

∑fs(a)λG(s) (convergence in L2(τG)) for any a ∈MG.

Clearly x(s, t) = (fs ⊗ ft)(x) is well defined. Note that (a ⊗ b)(s, t) = fs(a)ft(b) = a(s)b(t) and(∑

s |fs(a)|2)1/2 ≤ ‖a‖MG(a, b ∈ MG). Thus (Cauchy-Schwarz)

∑t |(a ⊗ b)(t, t)| ≤ ‖a‖MG

‖b‖MG.

This shows that∑

t |x(t, t)| <∞ for any x ∈MG ⊗MG.We will show the following claim:

(4.14) ∀x ∈MG ⊗MG

∥∥∥∑tx(t, t)UG(t)

∥∥∥C∗(G)

≤ ‖x‖MG⊗maxMG.

Then we set PG(x) =∑

t x(t, t)UG(t). This implies the result. Indeed, in the converse direction wehave obviously by maximality∥∥∥∑x(t, t)λG(t)⊗ λG(t)

∥∥∥max≤∥∥∥∑x(t, t)UG(t)⊗ UG(t)

∥∥∥max≤∥∥∥∑x(t, t)UG(t)

∥∥∥ .Therefore (4.14) implies at the same time that the map JG (and also ΨGJG) defines an isometric∗-homomorphism and that PGΨG is a contractive map onto C∗(G). The proof of the claim willactually show that PG is c.p. Incidentally, JGPG is a “conditional expectation” onto JG(C∗(G)) ⊂C∗λ(G)⊗max C

∗λ(G), in the sense of Theorem 1.45.

We now prove the claim. Let π : G→ B(H) be a unitary representation of G. We introduce a pairof commuting representations (π1, π2) as follows:

π1(λG(t)) = λG(t)⊗ π(t) and π2(λG(t)) = ρG(t)⊗ I.

Note that both π1 and π2 extend to normal isometric representations of MG. For π1 this followsfrom the Fell absorption principle. For π2, it follows from the fact that ρG ' λG (indeed ifW : `2(G)→ `2(G) is the unitary taking δt to δt−1 , then WλG(·)W ∗ = ρG(·)).

67

Since π1 and π2 have commuting ranges, we have (recall the notation (4.5))

(4.15) ‖(π1.π2)(x)‖B(`2(G)⊗2H) ≤ ‖x‖MG⊗maxMG,

and hence if we restrict the left-hand side to K = δe ⊗ H ⊂ `2(G) ⊗2 H, we obtain (note that〈δe, λG(s)ρG(t)δe〉 = 1 if s = t and zero otherwise) Id⊗

∑t x(t, t)π(t) = PK(π1.π2)(x)|K and hence

(4.16)∥∥∥∑

tx(t, t)π(t)

∥∥∥B(H)

≤ ‖x‖MG⊗maxMG.

Finally, taking the supremum over π, we obtain the announced claim (4.14). This argument showsthat PG is c.p. and ‖PG‖cb ≤ 1.

Remark 4.12. By (4.16) applied when π is the trivial representation, we have for any x ∈MG⊗MG

|∑

x(t, t)| ≤ ‖x‖MG⊗maxMG.

Note that since the matrix of λG(t) has real entries (equal to 0 or 1) λG(t) = λG(t). Therefore thecorrespondence

∑f(t)λG(t) 7→

∑f(t)λG(t) is an isomorphism from C∗λ(G) to C∗λ(G), that extends

to an isomorphism from MG to MG. Thus we also have (say assuming (s, t) 7→ x(s, t) finitelysupported)(4.17)

|∑

x(t, t)| ≤ ‖∑

s,tx(s, t)λG(s)⊗λG(t)‖MG⊗maxMG

≤ ‖∑

s,tx(s, t)λG(s)⊗λG(t)‖

C∗λ(G)⊗maxC∗λ(G).

Remark 4.13. It will be convenient to record here the following fact similar to (4.16) (when π isthe trivial representation). Consider x ∈ C[G]⊗C[G]. Then, via the maps C∗(G)→ C∗λ(G) ⊂MG,x determines an element xU ∈ C∗(G)⊗max C

∗(G), an element xλ ∈ C∗λ(G)⊗max C∗λ(G), and lastly

xm ∈ MG ⊗max MG (we will not use this notation later on). Since λG and ρG obviously extend torepresentations on MG with commuting ranges, we have

(4.18) ‖(λG.ρG)(xm)‖B(`2(G)) ≤ ‖xm‖MG⊗maxMG≤ ‖xλ‖C∗λ(G)⊗maxC∗λ(G) ≤ ‖xU‖C∗(G)⊗maxC∗(G).

As earlier, for any x ∈MG ⊗MG, let x(s, t) = 〈δs ⊗ δt, x(δe ⊗ δe)〉 (s, t ∈ G).Since

∑x(t, t) = 〈δe, (λG.ρG)(x)δe〉, we have

(4.19) |∑

t∈Gx(t, t)| ≤ ‖(λG.ρG)(x)‖B(`2(G)).

4.4 A brief repertoire of examples from group C∗-algebras

It is often hard to calculate norms of operators, and hence also of tensors in C∗-tensor products.The group case provides us with many instances where there are nice formulae. For conveniencewe recapitulate them here.

Proposition 4.14. Let G be a discrete group, and π : G → B(H) a unitary representation. Letf : G→ C be any finitely supported function, then(4.20)

‖∑

f(t)UG(t)⊗UG(t)‖C∗(G)⊗minC∗(G) = ‖∑

f(t)UG(t)⊗UG(t)‖C∗(G)⊗maxC∗(G) = ‖∑

f(t)UG(t)‖.

(4.21) ∀f ≥ 0 ‖∑

f(t)UG(t)‖ =∑

f(t).

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(4.22) ‖∑

f(t)UG(t)⊗ λG(t)‖C∗(G)⊗minC∗λ(G) = ‖

∑f(t)λG(t)‖C∗λ(G).

(4.23) ‖∑

f(t)π(t)⊗ λG(t)‖C∗π(G)⊗minC∗λ(G) = ‖

∑f(t)λG(t)‖C∗λ(G).

(4.24) ‖∑

f(t)λG(t)⊗ λG(t)‖C∗λ(G)⊗maxC∗λ(G) ≥ |∑

f(t)|.

(4.25) ‖∑

f(t)λG(t)⊗ λG(t)‖C∗λ(G)⊗maxMG≥ |∑

f(t)|.

(4.26)

∀f ≥ 0 ‖∑

f(t)λG(t)⊗ λG(t)‖C∗λ(G)⊗maxC∗λ(G) = ‖∑

f(t)λG(t)⊗ λG(t)‖MG⊗maxMG=∑

f(t).

(4.27)

∀f ≥ 0 ‖∑

f(t)UG(t)⊗λG(t)‖C∗(G)⊗maxC∗λ(G) = ‖∑

f(t)UG(t)⊗λG(t)‖C∗(G)⊗maxMG=∑

f(t).

Proof. (4.20): It is easy to show that UG ⊗ UG dominates UG since UG contains the trivial repre-sentation, and the converse is obvious by maximality.(4.21): Indeed, ≥

∑f(t) holds because of the presence of the trivial representation in UG, and

≤∑|f(t)| follows from the triangle inequality.

Both (4.22) and (4.23) follow from Fell’s principle (see (3.13)).Let ‖x‖ (resp. ‖y‖) be the (common) left hand side of (4.24) and (4.26) (resp. of (4.27)). Then

‖y‖ ≥ ‖x‖ ≥ ‖∑

f(t)λG(t)ρG(t)‖ ≥ ‖∑

f(t)λG(t)ρG(t)δe‖ = |∑

f(t)|

where at the last step we use λG(t)ρG(t)δe = δe. When f ≥ 0 the triangle inequality gives theconverse.The same argument is valid for (4.25) and for the terms involving MG in (4.26) and (4.27) sinceλG and ρG obviously extend to mutually commuting ∗-homomorphisms on MG.

4.5 States on the maximal tensor product

By definition, a state on a C∗-algebra A is a positive linear form ϕ of unit norm. If A is unital, afunctional ϕ ∈ A∗ is a state if and only if ‖ϕ‖ = ϕ(1) = 1 (see Remark 1.33). If A is not unital, apositive functional ϕ on A is a state if and only if ϕ(xi) → 1 when (xi) is any fixed approximateunit in A as in (1.22). Indeed, this follows from Remark 1.24 and (1.22). When this holds we saythat ϕ is approximately unital.

Remark 4.15. Let ‖ ‖α be any C∗-norm on the algebraic tensor product A1⊗A2 of two C∗-algebras.The set of elements of the form x∗x | x ∈ A1⊗A2 is clearly α-dense in the set x∗x | x ∈ A1⊗αA2,or equivalently α-dense in (A1 ⊗α A2)+. This shows that when ϕ ∈ (A1 ⊗α A2)∗, if ϕ is positiveon A1 ⊗ A2 then it is positive on A1 ⊗α A2. If both algebras are unital then 1 ⊗ 1 ∈ A1 ⊗ A2,and hence, by (1.22) and Remark 1.24, the set of states on A1 ⊗α A2 is simply formed of the setof positive unital functionals on A1 ⊗ A2 that are α-continuous. In the non unital case, if xi ≥ 0(resp. yj ≥ 0) is any approximate unit in BA1 (resp. BA2) as in (1.22), then xi ⊗ yj is also one inthe unit ball of A1 ⊗α A2. Thus, in any case, the set of states on A1 ⊗α A2 is simply formed of theset of approximately unital positive functionals on the algebraic tensor product that happen to becontinuous for the norm ‖ ‖α.

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The following statement describes the states on A1⊗maxA2. Not surprisingly this is the largestpossibility: the set of all normalized positive functionals on A1 ⊗A2.

Theorem 4.16. Let ϕ : A1⊗A2 → C be a linear form and let uϕ : A1 → A∗2 be the correspondinglinear map defined by uϕ(a1)(a2) = ϕ(a1 ⊗ a2). The following are equivalent:

(i) ϕ extends to a positive linear form in (A1 ⊗max A2)∗.

(ii) uϕ : A1 → A∗2 is completely positive in the following sense:

(4.28)∑

i,juϕ(xij)(yij) ≥ 0 ∀n ∀x ∈Mn(A1)+ ∀y ∈Mn(A2)+.

(iii) ϕ is a positive linear form on A1 ⊗A2, in the sense that ϕ(t∗t) ≥ 0 for any t ∈ A1 ⊗A2.

When this holds

(4.29) ‖ϕ‖(A1⊗maxA2)∗ = ‖uϕ‖.

Thus ‖ϕ‖(A1⊗maxA2)∗ = 1 (i.e. ϕ is a state on A1 ⊗max A2) if and only if ‖uϕ‖ = 1.

Proof. Assume (i) with ϕ of norm 1. By the GNS construction (see §26.13), there are a represen-tation π : A1 ⊗max A2 → B(H) and ξ in the unit ball of H such that ϕ(·) = 〈ξ, π(·)ξ〉. We mayassume that π = π1 ·π2 as in (4.4). Let x, y be as in (ii). Let z = y1/2 so that yij =

∑k z∗kizkj (note

zik = z∗ki) we claim that the matrix [π1(xij)π2(yij)] is positive. Indeed, for each fixed k the matrix[akij ] defined by

akij = π2(zki)∗π1(xij)π2(zkj)

can be rewritten as a product C∗k [π1(xij)]Ck showing that it is positive and since π1, π2 havecommuting ranges we have

π1(xij)π2(yij) =∑

kπ2(zki)

∗π1(xij)π2(zkj) =∑

kakij .

This proves our claim. Let ξ ∈ H ⊕ · · · ⊕H (n times) be defined by ξ = ξ ⊕ · · · ⊕ ξ. Then we have∑uϕ(xij)(yij) = 〈ξ, [π1(xij)π2(yij)]ξ〉 ≥ 0,

which (by homogeneity) shows that (i) ⇒ (ii).Assume (ii). Consider t =

∑aj ⊗ bj in A1 ⊗A2. Then t∗t =

∑i,j a

∗i aj ⊗ b∗i bj hence by (ii)

ϕ(t∗t) =∑i,j

uϕ(a∗i aj)(b∗i bj) ≥ 0,

which shows that (iii) holds.Assume (iii). By the GNS construction applied to A1 ⊗A2, there are a ∗-homomorphism π : A1 ⊗A2 → B(H) and ξ in H such that ϕ(·) = 〈ξ, π(·)ξ〉. But any ∗-homomorphism π : A1⊗A2 → B(H)extends to one on the whole of A1 ⊗max A2. Thus ϕ also extends to A1 ⊗max A2 and satisfies (i).Moreover, if we assume A1, A2 and π unital we have ‖ϕ‖∗max = ϕ(1⊗1) = uϕ(1)(1) ≤ ‖uϕ‖, but also|uϕ(a1)(a2)| ≤ ‖ϕ‖∗max‖a1⊗a2‖max = ‖ϕ‖∗max for any unit vectors a1, a2, and hence ‖ϕ‖∗max = ‖uϕ‖.In the non-unital case it is easy to modify this argument using an approximate unit (we leave thedetails to the reader).

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Remark 4.17. Let ϕ be as in (4.28). Consider C∗-algebras Bj and uj ∈ CP (Bj , Aj) (j = 1, 2).Then the composition (u2)∗uϕu1 is obviously c.p. from B1 to B∗2 in the sense of (4.28).

By definition of the maximal tensor product, the following inequality (4.30) is clear when u1, u2

are ∗-homomorphisms, but we will crucially use the following more general fact.

Corollary 4.18. Let ui : Ai → Bi (i = 1, 2) be c.p. maps between C∗-algebras. Then the linearmap u1 ⊗ u2 : A1 ⊗A2 → B1 ⊗B2 extends to a c.p. map from A1 ⊗max A2 to B1 ⊗max B2 and

(4.30) ∀x ∈ A1 ⊗A2 ‖(u1 ⊗ u2)(x)‖B1⊗maxB2 ≤ ‖u1‖‖u2‖ ‖x‖A1⊗maxA2 .

Proof. We may assume ‖ui‖ ≤ 1. Consider any ϕ in the unit ball of (B1 ⊗max B2)∗+, define ψ byψ(t) = ϕ((u1 ⊗ u2)(t)) (t ∈ A1 ⊗ A2). We claim that ψ ∈ (A1 ⊗max A2)∗+. Indeed, uψ : A1 → A∗2is given by uψ = (u2)∗uϕu1, and the latter map is c.p. by Remark 4.17. By Theorem 4.16ψ ∈ (A1 ⊗max A2)∗+ and ‖ψ‖(A1⊗maxA2)∗ = ‖uψ‖ ≤ ‖uϕ‖‖u1‖‖u2‖ ≤ 1. Note

‖(u1 ⊗ u2)(t)‖max = sup|ϕ((u1 ⊗ u2)(t))| | ‖ϕ‖(B1⊗maxB2)∗ ≤ 1.

Since any element ϕ in the unit ball of the dual of a C∗-algebra such as (B1⊗maxB2)∗ decomposesas a linear combination of positive elements ϕ = ϕ1−ϕ2 + i(ϕ3−ϕ4) all in the unit ball, it is clearthat u1 ⊗ u2 must be bounded (say by 4) from (A1 ⊗A2, ‖ ‖max) to (B1 ⊗B2, ‖ ‖max) and henceextends to a bounded map u : A1 ⊗max A2 → B1 ⊗max B2. To complete the proof we show thatthe latter extension u is positive, i.e. that t ∈ (A1 ⊗max A2)+ ⇒ u(t) ∈ (B1 ⊗max B2)+. Indeed,it suffices to check that ϕ(u(t)) ≥ 0 for all ϕ in (B1 ⊗max B2)∗+, but with the preceding notationwe have ϕ(u(t)) = ψ(t) with ψ ∈ (A1 ⊗max A2)∗+, so the positivity of u is clear. Replacing A1 byMn(A1), we obtain its complete positivity. By (1.20), if A1, A2 are unital we have

‖u1 ⊗ u2 : A1 ⊗max A2 → B1 ⊗max B2‖ ≤ ‖(u1 ⊗ u2)(1⊗ 1)‖ = ‖u1(1)‖ · ‖u2(1)‖ ≤ ‖u1‖ · ‖u2‖.

In the non-unital case, we obtain the same conclusion using approximate units.

Corollary 4.19. Let ui : Ai → B(H) (i = 1, 2) be c.p. maps with commuting ranges. Then thelinear map

u1.u2 : A1 ⊗A2 → B(H)

defined by (u1.u2)(x1 ⊗ x2) = u1(x1)u2(x2) extends to a c.p. map from A1 ⊗max A2 to B(H) withnorm ≤ ‖u1‖‖u2‖.

Proof. Let Bi ⊂ B(H) be the C∗-subalgebra generated by ui(Ai). Since the ui’s are self-adjoint soare their ranges. Therefore B1 and B2 mutually commute. Let π : B1 ⊗max B2 → B(H) be the∗-homomorphism defined by π(b1⊗b2) = b1b2. Since u1.u2 = π(u1⊗u2), by the preceding corollaryu1.u2 is a composition of c.p. maps with ‖u1.u2‖ ≤ ‖π‖‖u1 ⊗ u2‖ ≤ ‖u1‖‖u2‖.

4.6 States on the minimal tensor product

We clearly have a surjective ∗-homomorphism Q : A1 ⊗max A2 → A1 ⊗min A2. Let I = ker(Q) sothat

A1 ⊗min A2 = (A1 ⊗max A2)/I.

Therefore(A1 ⊗min A2)∗ = I⊥ ⊂ (A1 ⊗max A2)∗.

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Note that by (1.3) and (1.10) we have a natural canonical inclusion

A∗1 ⊗A∗2 ⊂ (A1 ⊗min A2)∗.

By Corollary 1.15, we have

(4.31) ker(Q) = t ∈ A1 ⊗max A2 | 〈t, ξ1 ⊗ ξ2〉 = 0 ∀ξ1 ∈ A∗1, ξ2 ∈ A∗2.

Lemma 4.20.(A1 ⊗min A2)∗ = A∗1 ⊗A∗2

where the closure is with respect to pointwise convergence on A1⊗minA2. Moreover, when viewed asa subset of (A1⊗max A2)∗, the set (A1⊗min A2)∗ is the closure of A∗1⊗A∗2 with respect to pointwiseconvergence on A1 ⊗max A2.

Proof. By Hahn-Banach, to check the second assertion it suffices to show that any t ∈ A1 ⊗max A2

that vanishes on A∗1 ⊗A∗2 belongs to I = ker(Q), i.e. satisfies Q(t) = 0. But by (4.31) this is clear.The first assertion is then clear.

Let Aj ⊂ B(Hj) be C∗-algebras. Then, by definition, A1⊗min A2 ⊂ B(H1⊗2H2) isometrically.Therefore any state on A1⊗minA2 is the restriction of a state on B(H1⊗2H2). Let H = H1⊗2H2.We should first recall how to tackle states on B(H). First note that the unit ball of B(H)∗ is theweak* closure of the convex hull of the elements which come from rank one operators on H (see§26.10), i.e. the functionals of the form

ϕξ,η(T ) = 〈ξ, Tη〉

for some ξ, η ∈ BH . Similarly the positive part of the unit ball of B(H)∗ is the weak* closureof the convex hull of the set of functionals of the form ϕξ,ξ for some ξ ∈ BH . Lastly, the set ofstates on B(H) is the weak* closure of the convex hull of ϕξ,ξ | ‖ξ‖ = 1. Moreover, if F ⊂ His a dense linear subspace, the latter set is the same as the weak* closure of the convex hull ofϕξ,ξ | ‖ξ‖ = 1, ξ ∈ F.

Let A ⊂ B(H) be a C∗-algebra. We will say that a map u ∈ CP (A,Mn) is obtained bycompression if there is an isometry V : `n2 → H such that u(x) = V ∗xV ∈ B(`n2 ) ' Mn for anyx ∈ A. We will say that a linear form ϕ on A1 ⊗ A2 comes from a matricial state (resp. obtainedby compression) if there are integers n(j), maps uj ∈ CP (Aj ,Mn(j)) with ‖uj‖ ≤ 1 (j = 1, 2) (resp.both obtained by compression) and a state ψ on Mn(1) ⊗min Mn(2) such that

(4.32) ∀x ∈ A1 ⊗A2 ϕ(x) = ψ((u1 ⊗ u2)(x)).

Note that the notion of map u ∈ CP (A,Mn) or of state “obtained by compression” depends on theembedding A ⊂ B(H), while that of state coming from a matricial state does not.

We can now refine the description of the states on A1 ⊗min A2 given in the preceding lemma.

Theorem 4.21. Let Aj ⊂ B(Hj) be C∗-algebras. Let ϕ : A1 ⊗ A2 → C be a linear form and letuϕ : A1 → A∗2 be the corresponding linear map. We first assume A1, A2 unital and ϕ(1 ⊗ 1) = 1.The following are equivalent:

(i) The functional ϕ extends to a state on A1 ⊗min A2.

(i)’ The functional ϕ satisfies |ϕ(x)| ≤ ‖x‖min for any x ∈ A1 ⊗A2.

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(ii) The functional ϕ is the pointwise limit on A1 ⊗ A2 of a net of functionals that come frommatricial states obtained by compression.

(iii) The functional ϕ is the pointwise limit on A1 ⊗ A2 of a net of functionals that come frommatricial states.

(iv) The map uϕ : A1 → A∗2 is the pointwise limit with respect to the weak* topology on A∗2 of anet of finite rank c.p. maps of unit norm from A1 to A∗2.

Moreover, in the non-unital case, if we replace our assumption on ϕ by supϕ(x ⊗ y) | 0 ≤ x, 0 ≤y, ‖x‖ < 1, ‖y‖ < 1 = 1, the equivalence still holds.

Proof. By our normalization assumption, (i) and (i)’ are equivalent (see the discussion at thebeginning of §4.5). Assume (i) and Aj ⊂ B(Hj), j = 1, 2. Let H = H1 ⊗2 H2. Then ϕ is therestriction of a state on B(H1 ⊗2 H2). By the remarks before Theorem 4.21, ϕ is the pointwiselimit of states on B(H) of the form

(4.33) T 7→∑N

1λkϕξk,ξk(T )

where λk > 0,∑N

1 λk = 1 and the ξk’s are unit vectors in H. We may assume (by density) that ineach case there are finite dimensional subspaces Kj ⊂ Hj such that ξk ∈ K1 ⊗2 K2 for all k ≤ N .Then the resulting states come from matricial states. Indeed, letting uj(x) = PKjx|Kj ∈ B(Kj),and denoting by ψk the state on B(K1)⊗minB(K2) = B(K1⊗2K2) defined by ϕξk,ξk , we may writeϕξk,ξk(T ) = ψk((u1 ⊗ u2)(T )). Then the state

∑λkϕξk,ξk satisfies (4.32) with n(j) = dim(Kj) and

ψ =∑λkψk. This proves (i) ⇒ (ii) and (ii) ⇒ (iii) is trivial.

Note that (4.32) implies uϕ = (u2)∗uψu1, and hence by Remark 4.17 uϕ is a c.p. map of finite rankwhen ϕ comes from matricial states. Moreover, ‖uϕ‖ = ϕ(1⊗ 1) by (4.29).Assume (iii). Let ϕi be the pointwise approximating functionals coming from matricial states.Since we can replace them by ϕi/ϕi(1⊗ 1) and ϕi(1⊗ 1)→ 1 we obtain (iv).Assume (iv). Then by Theorem 4.16, ϕ is the pointwise limit on the set a1 ⊗ a2 | aj ∈ Aj (orequivalently, by linearity, on A1 ⊗ A2) of a net ϕi ∈ A∗1 ⊗ A∗2, formed of states on A1 ⊗max A2.By density, since this net is equicontinuous on A1 ⊗max A2, we have pointwise convergence on thewhole of A1 ⊗max A2, and hence by Lemma 4.20 we obtain (i).

Remark 4.22. Consider a C∗-algebra A ⊂ B(H). LetH = H⊕H⊕· · · . Then π : a 7→ a⊕a⊕· · · is anembedding of A in B(H). We will say that any such embedding A ⊂ B(H) has infinite multiplicity.When A ⊂ B(H) has infinite multiplicity, any state ϕ on A of the form a 7→

∑λk〈ξk, aξk〉 (with

unit vectors ξk ∈ H) can be rewritten as a vector state on B(H). More precisely, if we define

ξ′ = (λ1/2k ξk) then ξ′ is a unit vector in H and we have ϕ(T ) = 〈ξ′, π(T )ξ′〉. Thus, any state on A

is a pointwise limit of vector states (relative to H).

We will need an obvious generalization of this trick for A = A1⊗minA2 with H = H1⊗2H2 andAj ⊂ B(Hj). LetHj = Hj⊕Hj⊕· · · and πj : B(Hj)→ B(Hj) be again such that πj(a) = a⊕a⊕· · · .For notational simplicity we give ourselves fixed orthonormal bases in H1 and H2 which allow usto define unambiguously the transpose ta of a ∈ B(Hj) simply as the operator associated to thetransposed matrix. We then define tπj(a) = πj(

ta) = ta⊕ ta⊕ · · · for any a ∈ B(Hj).Thus we obtain:

Proposition 4.23. Let ϕ be a state on A1 ⊗min A2. With the preceding notation, there is a net offinite rank operators zi : H2 → H1 with Hilbert-Schmidt norm 1, i.e. tr(z∗i zi) = 1 such that

∀(a, b) ∈ A1 ×A2 ϕ(a⊗ b) = lim tr(z∗i π1(a)zitπ2(b)).

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Proof. The state ϕ is the limit of states of the form (4.33). Each state ϕξk,ξk can be written asdescribed in (2.5) (with unit vectors ξ = η). Thus it is easy to complete the proof using the sameidea as in Remark 4.22 (with zi acting diagonally).

Remark 4.24. The preceding proposition shows that a state ϕ on A1⊗min A2 is the pointwise limitof states that come from matricial vector states (that is for which ψ in (4.32) is a vector state onMn ⊗min Mm ' B(`n2 ⊗2 `

m2 )).

Actually, we will more often use the following variant for A1⊗minA2 (see also Proposition 2.11):

Proposition 4.25. In the preceding situation, for any state ϕ on A1⊗minA2 there is a net of finiterank operators zi : H2 → H1 with tr(z∗i zi) = 1 such that

∀(a, b) ∈ A1 ×A2 ϕ(a⊗ b) = lim tr(z∗i π1(a)ziπ2(b)∗).

Similarly, for any state ϕ on A1 ⊗min A2 there is a net of finite rank operators hi : H2 → H1 withtr(h∗ihi) = 1 such that

ϕ(a⊗ b) = lim tr(h∗iπ1(a)∗hiπ2(b)).

Proof. The first part is just a rewriting of the preceding Proposition with (2.6) in place of (2.5).For the second part we observe that b⊗ a 7→ ϕ(a⊗ b) defines a state on A2 ⊗min A1.

4.7 Tensor product with a quotient C∗-algebra

We will need the following basic fact on the behavior of C∗-tensor products with respect to quo-tient C∗-algebras. We will return to this topic specifically for the minimal tensor product moreextensively in §7.5.

Lemma 4.26. Let A,B be C∗-algebras and let I ⊂ A be a closed ideal so that A/I is a C∗-algebra.Let ‖ ‖α be any C∗-norm on A ⊗ B. Let E ⊂ B be an arbitrary subspace. We denote by A⊗αE(resp. I⊗αE) the closure of A⊗ E (resp. I⊗E) in A⊗α B. Let

Qα[E] =A⊗αEI⊗αE

Then, if E ⊂ F ⊂ B are arbitrary subspaces, we have a natural isometric embedding

Qα[E] ⊂ Qα[F ].

Proof. The proof uses the classical fact that the ideal I has a 2-sided approximate unit formed ofelements ai with 0 ≤ ai and ‖ai‖ ≤ 1 (see §26.15).Let Ti : A⊗αF → I⊗αF be the operator defined by Ti(x⊗ y) = aix⊗ y. If B is unital, this is justthe left multiplication by ai ⊗ 1. Note that ‖Ti‖ ≤ 1 and ‖I − Ti‖ ≤ 1. Moreover, Ti(ϕ) → ϕ forany ϕ in I ⊗ F .Let us denote by d(·, ·) the distance in the norm of A⊗α B. Note that by density we have for anyx ∈ A⊗B

d(x, I⊗αF ) = d(x, I ⊗ F ),

and similarly for E. We claim that for any x ∈ A⊗ F

d(x, I⊗αF ) = lim supi ‖(1− Ti)(x)‖.

74

Let y ∈ I ⊗ F . Since ‖1− Ti‖ ≤ 1 we have

‖x− Tix‖α ≤ ‖(I − Ti)(x− y)‖α + ‖(I − Ti)(y)‖α ≤ ‖x− y‖α + ‖(I − Ti)(y)‖α,

and ‖(I−Ti)(y)‖α → 0 for any y ∈ I⊗F . Thus we obtain lim supi ‖x−Tix‖α ≤ ‖x−y‖α and hencelim supi ‖x− Tix‖α ≤ d(x, I⊗αF ). Since Ti(x) ∈ I ⊗ F , we have lim infi ‖x− Tix‖α ≥ d(x, I⊗αF )which proves the claim (and the convergence of ‖x− Tix‖α). Now to show that for any x in A⊗Ewe have

d(x, I ⊗ F ) = d(x, I ⊗ E),

it suffices to observe that Tix ∈ I ⊗ E for any x ∈ A⊗ E. This gives us

d(x, I ⊗ E) ≤ lim infi ‖x− Tix‖α = d(x, I ⊗ F )

and the converse is obvious since I ⊗ E ⊂ I ⊗ F .

The following simple fact will be invoked several times.

Lemma 4.27. Let A,B be C∗-algebras and let I ⊂ A be a closed (two sided, self-adjoint) ideal.Let ‖ ‖α be any C∗-norm on A⊗B. Then

(A⊗B) ∩ I⊗Bα= I ⊗B.

Moreover, if we denote by Q : A ⊗ B → (A ⊗ B)/(I ⊗ B) the quotient map, then for anyt ∈ (A⊗B)/(I ⊗B) we have

(4.34) ‖t‖(A⊗αB)/I⊗Bα = inf‖t‖A⊗αB | t ∈ A⊗B, Q(t) = t.

Proof. Let t =∑n

1 ak ⊗ bk ∈ A ⊗ B. Let (xi) be a (bounded) approximate unit of I and (yj) onefor B in the sense of §26.15. Clearly for any z ∈ I ⊗ B we have ‖z − z(xi ⊗ yj)‖α → 0, and byequicontinuity this remains true for any z ∈ I ⊗Bα

. Therefore, if t ∈ I ⊗Bαthen ‖

∑nk=1 ak⊗bk−∑n

1 akxi ⊗ bkyj‖α → 0 and hence also (since ‖bkyj − bk‖ → 0) ‖∑n

k=1(ak − akxi)⊗ bk‖α → 0. Wemay assume the bk’s linearly independent. Then it follows that akxi → ak and hence ak ∈ I = Ifor any k. This proves (A⊗B) ∩ (I⊗αB) ⊂ I ⊗B and the converse is trivial.Let s ∈ A⊗ B be a representative of t modulo I ⊗ B. Then ‖t‖(A⊗αB)/I⊗Bα = inf‖s+ η‖α | η ∈I⊗Bα and by density this is = inf‖s+ η‖α | η ∈ I⊗B, which is the same as (4.34).

4.8 Notes and Remarks

The study of tensor products of C∗-algebras was initiated in the 50’s by Turumaru in Japan, andcontinued by Takesaki [239] and Guichardet [100] (see also [101]). Much work was then done onnuclear C∗-algebras, (to which we return in §10.2). In the process, this clarified what we know onC∗-tensor products. For instance, Lance’s paper [165] contains a lot of information on the latter, inparticular (4.11) and (4.12), and the fact that C∗λ(G) is nuclear if and only if G is amenable. Lance[165] showed that the CPAP implies nuclearity. Choi-Effros and Kirchberg [154, 45] independentlyproved the converse (see §10.2). In some variant, Theorem 4.11 appears in [208, p.150]). As anexample of application this shows that exactness is not stable by the max-tensor product (seeRemark 10.5), which was left open in Kirchberg’s early works on exactness (see [156, p. 75, (P3)]). Lance [165] found the description of the states of the maximal tensor product that appears in§4.5. The corresponding result for states on the minimal one in §4.6 is used in Kirchberg’s work[155] but was probably long known to experts.§4.7 is probably well known, but our treatment is influenced by Arveson’s [13].

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5 Multiplicative domains of c.p. maps

When dealing with a contraction u : A→ B between C∗-algebras it is often interesting to identifythe largest C∗-subalgebra of A on which u behaves as a ∗-homomorphism. For c.p. maps there isa useful description of the largest such C∗-subalgebra, called the multiplicative domain of u.

5.1 Multiplicative domains

The unreasonable effectiveness of completely positive contractions in C∗-algebra theory is partiallyelucidated by the next statement.

Theorem 5.1. Let u : A→ B be a c.p. map between C∗-algebras with ‖u‖ ≤ 1.

(i) Then if a ∈ A satisfies u(a∗a) = u(a)∗u(a), we have necessarily

u(xa) = u(x)u(a), ∀x ∈ A

and the set of such a’s forms an algebra.

(ii) Let Du = a ∈ A | u(a∗a) = u(a)∗u(a) and u(aa∗) = u(a)u(a)∗. Then Du is a C∗-subalgebraof A (called the multiplicative domain of u) and u|Du is a ∗-homomorphism. Moreover, we have

(5.1) ∀a, b ∈ Du ∀x ∈ A u(ax) = u(a)u(x), u(xb) = u(x)u(b) and u(axb) = u(a)u(x)u(b).

Proof. First recall a classical inequality for x ∈ L2(m) when m is a probability∫|x|2dm ≥ |

∫xdm|2.

We will show that u satisfies a similar Cauchy-Schwarz inequality (first used by Choi for 2-positivemaps, but see also [143] for earlier similar results for positive ones), as follows.

(5.2) ∀x ∈ A u(x∗x) ≥ u(x)∗u(x).

This is easy for c.p. maps. Indeed, by Theorem 1.22, we can write u as u(·) = V ∗π(·)V for somerepresentation π : A → B(H) and V : H → H with B ⊂ B(H). Then we have for all T with0 ≤ T ≤ 1

u(x∗x) = V ∗π(x)∗π(x)V ≥ V ∗π(x)∗Tπ(x)V,

and hence choosing T = V V ∗ we obtain (5.2). This implies that the “defect”

ϕ(x, y) = u(x∗y)− u(x)∗u(y)

behaves like a B-valued scalar product. In particular, we clearly have (by Cauchy-Schwarz)

|〈ξ, ϕ(x, y)ξ〉| ≤ 〈ξ, ϕ(x, x)ξ〉1/2〈ξ, ϕ(y, y)ξ〉1/2 ∀ξ ∈ H ∀x, y ∈ A.

This shows (taking y = a) that if ϕ(a, a) = 0 we have 〈ξ, ϕ(x, a)ξ〉 = 0 for all ξ, hence ϕ(x, a) = 0for all x in A. Changing x to x∗ (and recalling that a c.p. map is self-adjoint) we obtain

∀x ∈ A u(xa) = u(x)u(a).

Thus we have proved

(5.3) a ∈ A | u(a∗a) = u(a)∗u(a) = a ∈ A | u(xa) = u(x)u(a) ∀x ∈ A.

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It is easy to see that the right-hand side of this equality is an algebra (using associativity). Thisproves (i). To check (ii), we note that reversing the roles of a and a∗ in (i) we have u(aa∗) =u(a)u(a)∗ if and only if u(ay) = u(a)u(y) for all y in A. Note that a ∈ Du if and only if both aand a∗ belong to the set (5.3). Therefore Du is a C∗-algebra, we have u(ab) = u(a)u(b) for any a, bin Du which proves (ii) and (5.1) holds.

Remark 5.2. In the situation of Theorem 5.1, let π = u|Du : Du → B. Then ker(π) is a hereditaryC∗-subalgebra of A (and of course an ideal of Du). Indeed, if 0 ≤ y ≤ x with y ∈ A and x ∈ Du

then π(x) = 0⇒ u(y) = 0 and 0 ≤ u(y2) ≤ ‖y‖u(y) = 0 so that y ∈ Du and hence y ∈ ker(π).

As a consequence, we have

Corollary 5.3 (On bimodular maps). Let C ⊂ B be a C∗-subalgebra of a C∗-algebra B. Letπ : C → π(C) ⊂ B(H) be a representation. Then any contractive c.p. map (in particular anyunital c.p. map) u : B → B(H) extending π must satisfy

u(c1xc2) = π(c1)u(x)π(c2),

for all x ∈ B and all c1, c2 ∈ C i.e. u must be a C-bimodule map (for the action defined by π).

In particular (taking for π the identity on C) :

Corollary 5.4 (On conditional expectations). Let C ⊂ B be a C∗-subalgebra of a C∗-algebra B.Then any contractive c.p. projection P : B → C is a conditional expectation, i.e.

P (c1xc2) = c1P (x)c2, ∀x ∈ B ∀c1, c2 ∈ C.

5.2 Jordan multiplicative domains

In some situations, we will have to deal with a contractive mapping u : A → B that is merelypositive (and hence preserving self-adjointness). In that case, there is an analogue of Theorem 5.1where the product in A and B is replaced by the Jordan product defined by x y = (xy + yx)/2.A linear subspace ∆ of a C∗-algebra is called a Jordan subalgebra if it is stable under the Jordanproduct. A linear map u : ∆→ B(H) is called a Jordan morphism if

∀a, b ∈ ∆ u(a b) = u(a) u(b).

If in addition ∆ and u are self-adjoint (meaning a∗ ∈ ∆ and u(a∗) = u(a)∗ for any a ∈ ∆), we willsay that u is a Jordan ∗-morphism.

Theorem 5.5. Let u : A→ B be a positive unital map between unital C∗-algebras with ‖u‖ ≤ 1.If (and only if) a ∈ A satisfies u(a∗ a) = u(a)∗ u(a), we have

∀x ∈ A u(x a) = u(x) u(a).

The set ∆u of such a’s forms a closed self-adjoint Jordan subalgebra (called the Jordan multiplicativedomain of u) and u|∆u

is a Jordan ∗-morphism.

Proof. Since u preserves positivity it also preserves self-adjointness. Note that the commutativityof dispenses us from distinguishing left and right products. In particular, ∆u is self-adjoint. Leta ∈ A with a = a∗. We claim that

(5.4) u(a2) ≥ u(a)2.

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Let Ca be the C∗-algebra generated by a. Since Ca is commutative, the restriction of u to Ca is c.p.by Remark 1.32. Therefore, the claim follows from (5.2) applied to the latter restriction. Note forlater use that if u(a2) = u(a)2, Theorem 5.1 implies that u is multiplicative on Ca, and in particular

(5.5) u(a4) = u(a2)2.

Now for x ∈ A of the form x = a + ib with a, b self-adjoint we have x∗ x = a2 + b2 andu(x) = u(a) + iu(b). Therefore, (5.4) implies

(5.6) u(x∗ x) ≥ u(x)∗ u(x).

From that point on, the proof can be completed like for Theorem 5.1. Note that

(5.7) ∆u = a ∈ A | u(x a) = u(x) u(a) ∀x ∈ A.

However, some extra care is needed to show that ∆u (which is clearly self-adjoint) is a Jordanalgebra, because the Jordan product is not associative. Since ∆u is a self-adjoint subspace, itsuffices to show that a b ∈ ∆u for any pair a, b of self-adjoint elements of ∆u. Since a b =((a+ b)2 − (a− b)2)/4, it suffices to show that a = a∗ and a ∈ ∆u implies a2 ∈ ∆u, or equivalentlythat a = a∗ and u(a2) = u(a)2 implies u(a4) = u(a2)2, but we already observed this in (5.5).

At some point in the sequel we will crucially need the following result due to Størmer (see [123]).

Theorem 5.6. Let A be a C∗-algebra. Let r : A → B(H) be a Jordan ∗-morphism. There is aprojection p in r(A)′′ ∩ r(A)′ such that the decomposition

∀a ∈ A r(a) = pr(a) + (1− p)r(a)

decomposes r as the sum of a ∗-homomorphism a 7→ pr(a) (= pr(a)p) and ∗-antihomomorphisma 7→ (1− p)r(a) (= (1− p)r(a)(1− p)).

We will use this (without proof) via the following consequence:

Theorem 5.7. Let M,N be von Neumann algebras. Let ϕ : N → M be a normal (surjective)linear map such that ϕ(BN ) = BM . Then there are mutually orthogonal projections p, q in Nsuch that M embeds in pNp ⊕ (qNq)op as a von Neumann subalgebra admitting a contractiveconditional expectation onto it. More precisely, there is an injective normal ∗-homomorphism r :M → pNp⊕ (qNq)op and a normal contractive (and c.p.) projection P : pNp⊕ (qNq)op → r(M).

To prepare for the proof we first need some background on support projections.

Remark 5.8. Let N ⊂ B(H) be a von Neumann algebra. Let PN denote the set of (self-adjoint)projections in N . Let (pi)i∈I be any family in PN . Then there is a unique element denoted by∨pi | i ∈ I in PN that is minimal among all projections q ∈ PN such that q ≥ pi for all i ∈ I.To verify this just observe that by the bicommutant Theorem 26.46 a (self-adjoint) projectionp ∈ B(H) is in PN if and only if p commutes with N ′ or equivalently if p(H) ⊂ H is an invariantsubspace for N ′. By this criterion if E = span[∪i∈Ipi(H)] the projection PE is in N and hence wemay define simply ∨pi | i ∈ I = PE .

Remark 5.9. Let ϕ : N → B(H) be a positive map that is weak* to weak* continuous (in otherwords ϕ is “normal”). The support projection sϕ of ϕ in N is defined as

sϕ = 1− ∨p | p ∈ PN , ϕ(p) = 0.

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We claim that ϕ(1 − sϕ) = 0. Let J = x ∈ N | ϕ(x∗x) = 0. Recall x∗y∗yx ≤ ‖y‖2x∗x for anyx, y ∈ N . Thus J is a weak* closed left ideal in N . By Remark 26.36 there is a projection P ∈ Jsuch that J = NP . Now for p ∈ PN , ϕ(p) = 0 implies p ∈ NP and hence ker(P ) ⊂ ker(p) orequivalently p(H) ⊂ P (H) therefore also span[p(H) | ϕ(p) = 0] ⊂ P (H). It follows that 1−sϕ ≤ Pand hence ϕ(1− sϕ) = 0, which proves our claim.This implies that ϕ(x) = ϕ(sϕxsϕ) for any x ∈ N . Indeed, we have x − sϕxsϕ = x(1 − sϕ) +(1 − sϕ)xsϕ and it is easy to check (hint: compose with a state and use Cauchy-Schwarz) thatϕ(x(1− sϕ)) = ϕ((1− sϕ)y) = 0 for any x, y ∈ N .Lastly, ϕ(x) 6= 0 for any nonzero x ∈ (sϕNsϕ)+. Otherwise in the commutative von Neumannalgebra generated by x in sϕNsϕ we would find a nonzero projection p such that εp ≤ x for someε > 0 and hence ϕ(p) = 0, so that p ≤ 1− sϕ which is absurd for 0 6= p ∈ sϕNsϕ. Moreover, sϕ isthe largest projection in PN with the latter property.

Proof of Theorem 5.7. We follow closely Ozawa’s presentation in [189].Our first goal is to show that our assumption implies the existence of a normal positive unitalsurjective linear map ϕ′ : N →M such that ϕ′(BN ) = BM . Let C = x ∈ N | ‖x‖ ≤ 1, ϕ(x) = 1.Observe that C is a (nonvoid) convex subset (actually, as will soon become apparent, a “face”) ofthe unit ball of N . Since ϕ is normal, C is σ(N,N∗) compact and hence has extreme points bythe Krein-Milman theorem. Let U be an extreme point of C. We claim that U is necessarily anextreme point of BN . Indeed, if U is the midpoint of a segment [a, b] in BN , then ϕ(U) = 1 is themidpoint of the segment [ϕ(a), ϕ(b)] in the unit ball of M . But it is easy to see (by the uniformconvexity of the Hilbert space on which M is realized, see §26.3) that this forces ϕ(a) = ϕ(b), andhence ϕ(a) = ϕ(b) = 1. The latter means that a, b ∈ C, and since U ∈ ext(C), we must havea = b = U . This proves the claim that U ∈ ext(BN ). By a well known characterization of ext(BN )(see e.g. [241, p. 48]) U is a partial isometry. Since ϕ(U) = 1 we have

(5.8) ∀x ∈ N ϕ(x) = ϕ(UU∗x).

Indeed, for any normal state g on M we have g(ϕ(U)) = 1 so that the functional f ∈ N∗ definedby f(x) = g(ϕ(x)) satisfies ‖f‖N∗ = 1 = f(U), so by Lemma 26.41 we have f(x) = f(UU∗x) org(ϕ(x)) = g(ϕ(UU∗x)) and since this holds for any g, we obtain (5.8).Let us define ϕ′ : N → M by ϕ′(x) = ϕ(Ux). Then ϕ′(1) = 1 = ‖ϕ′‖ and, by Remark 1.34, ϕ′ isautomatically positive. By (5.8) ϕ′ still takes the closed unit ball of N onto that of M . Thus wereach our first goal.

Thus, replacing ϕ by ϕ′ we may assume that ϕ is in addition positive and unital. Let P denotethe set of projections in N . Let e ∈ P denote the support projection sϕ of ϕ (see Remark 5.9).Then ϕ(a) 6= 0 for any nonzero a ∈ (eNe)+. Replacing N by eNe (which has e as its unit) and ϕby its restriction to eNe, we may assume that e = 1. Since ϕ(x) = ϕ(exe) for any x ∈ N , after thischange fromN to eNe our assumption ϕ(BN ) = BM still holds. Let ∆ϕ be the Jordan multiplicativedomain. By Theorem 5.5, the latter is a self-adjoint Jordan subalgebra, which is weak* closed ascan be easily deduced from (5.7), and ϕ|∆ϕ

: ∆ϕ → M is a normal Jordan ∗-morphism. We firstclaim that ϕ|∆ϕ

is injective. Indeed, for x ∈ ∆ϕ, if ϕ(x) = 0 then ϕ(x)∗ ϕ(x) = ϕ(x∗ x) = 0 andsince 1 = e is the support of ϕ we have x∗ x = 0 and hence x = 0.Secondly, we claim that ϕ|∆ϕ

is surjective. Let v ∈ U(M). We will show that there is w ∈ U(N)such that ϕ(w) = v. Since ϕ(BN ) = BM we know there is w ∈ BN satisfying this. But now thepositivity of ϕ implies ϕ(1− w∗ w) ≥ 0 and by (5.6) ϕ(1− w∗ w) ≤ 0. Therefore we must have1 − w∗ w = 0, so that w is necessarily unitary. Lastly, since w and ϕ(w) are both unitary andϕ(1) = 1, we must have w ∈ ∆ϕ. This shows that the range of ϕ|∆ϕ

contains U(M), and since Mis linearly spanned by U(M), this proves the surjectivity of ϕ|∆ϕ

.

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Thus ϕ|∆ϕis an invertible Jordan ∗-morphism from ∆ϕ onto M . We now apply Theorem 5.6

to the inverse Jordan ∗-morphism ψ : M → ∆ϕ ⊂ N : there are mutually orthogonal projectionsp, q ∈ P with p+ q = 1 in ∆′′ϕ∩∆′ϕ ⊂ N such that the mapping r : M → pNp⊕ (qNq)op defined by

r(x) = pψ(x)⊕ qψ(x)

is an injective ∗-homomorphism. We may write just as well r(x) = pψ(x)p ⊕ qψ(x)q since p, qcommute with the range of ψ. Moreover, since ψ(M) = ∆ϕ is weak* closed in N and commuteswith p (and q), r(M) is also weak* closed in pNp⊕ (qNq)op. Thus r(M) is a von Neumann algebra,r : M → r(M) is a ∗-isomorphism and hence (recall Remark 26.38) r is automatically normal.Consider y = y1 ⊕ y2 ∈ pNp ⊕ (qNq)op. Let t(y) = y1 + y2 ∈ N . Then t : pNp ⊕ (qNq)op → Nis isometric (positive but in general not c.p.). The mapping P : pNp ⊕ (qNq)op → r(M) definedby P (y) = (rϕt)(y) is a contractive (normal) projection onto r(M) (because tr = ψ and ϕψ is theidentity on M). By Tomiyama’s theorem 1.45, it is a c.p. projection.

Remark 5.10. Conversely, if r, p, q, P are as in the conclusion of Theorem 5.7, then the map ϕ of theform ϕ(x) = r−1P ((pxp, qxq)), is a normal (positive unital) map onto M such that ϕ(BN ) = BM .

5.3 Notes and Remarks

The theory of multiplicative domains for c.p. (or merely 2-positive) maps is due to Choi [44]. Itwas preceded by important work on positive maps by Kadison and Størmer; see [236] for referencesand information on the latter maps. See [123] for information on Jordan algebras. More recentresults on Jordan multiplicative domains appear in Størmer’s paper [235].

6 Decomposable maps

This chapter is devoted to linear maps that are decomposable as linear combinations of c.p. mapsand to the appropriate norm denoted by ‖·‖dec. As will soon be clear, these maps and the dec-normplay the same role for the max-tensor product as cb-maps and the cb-norm with respect to themin-tensor product.

6.1 The dec-norm

Let A ⊂ B(H) be a closed subspace forming an operator system and B a C∗-algebra. We will denoteby D(A,B) the set of all “decomposable” maps u : A→ B, i.e. the maps that are in the linear spanof CP (A,B). This means that u ∈ D(A,B) if and only if there are uj ∈ CP (A,B) (j = 1, 2, 3, 4)such that

u = u1 − u2 + i(u3 − u4).

A simple minded choice of norm would be to take ‖u‖ = inf∑4

1 ‖uj‖, but this is not the optimalchoice. In many respects, the “right” norm on D(A,B) is the following one, introduced by Haagerupin [104]. We denote

(6.1) ‖u‖dec = infmax‖S1‖, ‖S2‖

where the infimum runs over all maps S1, S2 ∈ CP (A,B) such that the map

(6.2) V : x→(S1(x) u(x)u(x∗)∗ S2(x)

)

80

is in CP (A,M2(B)).We will use the notation

u∗(x) = u(x∗)∗.

Note that u = u∗ if and only if u takes self-adjoint elements of A to self-adjoint elements of B.This holds in particular for any c.p. map u.

With this notation, we can write

V =

(S1 uu∗ S2

).

Then D(A,B) equipped with the norm ‖ ‖dec is a Banach space. To clarify this, let us denote byD′(A,B) the set of those linear mappings u : A → B such that there are S1, S2 ∈ CP (A,B) forwhich the preceding map V is in CP (A,M2(B)).

Remark 6.1. Let λ ∈ C. Consider the matrices a =

(0 11 0

)and b =

(1 00 λ

). Let V be as in (6.2).

Then V ′ : x 7→ a∗V (x)a and V ′′ : x 7→ b∗V (x)b are also c.p. Note

(6.3) V ′ =

(S2 u∗u S1

)V ′′ =

(S1 λuλu∗ |λ|2S2

)We will show that actually:

Lemma 6.2. D(A,B) = D′(A,B) and D(A,B) is a Banach space for the norm ‖ ‖dec.

Proof. Let u ∈ D′(A,B) with V as in (6.2). Note that we have u(x) =(1 0

)V (x)

(01

). Therefore

by the polarization formula this implies u ∈ D(A,B). Thus D′(A,B) ⊂ D(A,B).We now claim that D′(A,B) is a vector space. Clearly it is stable by addition. A look at V ′′ in(6.3) shows that u ∈ D′(A,B) ⇒ λu ∈ D′(A,B), proving the claim. Now if u ∈ CP (A,B), and ifwe denote χ =

∑e1i, the mapping

(6.4) x 7→(u(x) u(x)u(x) u(x)

)= (∑

1≤i,j≤2eij)⊗ u(x) = χ∗χ⊗ u(x)

is clearly c.p. Therefore CP (A,B) ⊂ D′(A,B). But since D′(A,B) is a vector space, this impliesD(A,B) ⊂ D′(A,B).The easy verification that ‖ ‖dec is a norm for which D(A,B) is complete is left to the reader.

Remark 6.3. It is easy to show that the infimum in the definition (6.1) of the dec-norm is a minimum(i.e. this infimum is attained) when the range B is a von Neumann algebra, or when there is acontractive c.p. projection from B∗∗ to B. Haagerup raises in [104] the (apparently still open)question whether it is always a minimum.

Lemma 6.4. Let u : A→ B be “self-adjoint” i.e. such that u∗ = u, and let S1, S2 ∈ CP (A,B).

If V =

(S1 uu S2

)∈ CP (A,M2(B)) in other words if 0 4 V (see (2.30)) then

−(S1 + S2)/2 4 u 4 (S1 + S2)/2.

Proof. For any a ∈ A+ we have V (a) ≥ 0, and hence ±u(a) ≤ (S1(a) + S2(a))/2 by (1.24).Therefore, the two mappings a 7→ (S1(a) + S2(a))/2 ∓ u(a) are positive mappings (i.e. positivity

preserving). But since Vn =

((S1)n unun (S2)n

)is assumed positive for any n, we conclude that the

same two mappings are c.p., which proves the lemma.

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Lemma 6.5. The following simple properties hold:

(i) If u ∈ CP (A,B), then ‖u‖dec = ‖u‖cb = ‖u‖.

(ii) If u(x) = u(x∗)∗ (i.e. u is “self-adjoint”) then

(6.5) ‖u‖dec = inf‖u1 + u2 ‖ | u1, u2 ∈ CP (A,B), u = u1 − u2.

(iii) To any u : A→ B we associate the self-adjoint mapping u =

(0 uu∗ 0

).

Then u ∈ D(A,B) if and only if u ∈ D(A,M2(B)) and ‖u‖dec = ‖u‖dec.

Proof. (i) If u ∈ CP (A,B), then

(u uu u

)∈ CP (A,M2(B)) and hence ‖u‖dec ≤ ‖u‖. Conversely,

for any x ≥ 0 in the unit ball of A, if V is as in (6.2), then

(S1(x) u(x)u(x) S2(x)

)≥ 0 and hence, by

Lemma 1.37, ‖u(x)‖ ≤ max‖S1(x)‖, ‖S2(x)‖. Therefore ‖u‖ ≤ max‖S1‖, ‖S2‖ by (1.21) andhence ‖u‖ ≤ ‖u‖dec. Since u is c.p. we already know (see (1.22)) that ‖u‖ = ‖u‖cb.

(ii) Assume u = u1 − u2 with u1, u2 ∈ CP (A,B). Then V1 =

(u1 u1

u1 u1

)∈ CP (A,M2(B)) and (use

(6.3) with λ = −1) V2 =

(u2 −u2

−u2 u2

)∈ CP (A,M2(B)), and hence V1 + V2 ∈ CP (A,M2(B)).

This shows ‖u‖dec = ‖u1 − u2‖dec ≤ ‖u1 + u2‖, and hence ‖u‖dec ≤ inf‖u1 + u2‖ | u = u1 − u2.

Conversely, if u = u∗ and if

(S1 uu S2

)∈ CP (A,M2(B)), let T = (S1 +S2)/2. Then by Lemma 6.4

we have −T 4 u 4 T and hence we can write u = u1−u2 with u1 = (u+T )/2 and u2 = (−u+T )/2.Then u1, u2 ∈ CP (A,B) and u1 + u2 = T . Thus ‖u1 + u2‖ = ‖(S1 + S2)/2‖ ≤ max‖S1‖, ‖S2‖.So we find inf‖u1 + u2‖ | u = u1 − u2 ≤ ‖u‖dec.(iii) Assume u = U1 − U2 with U1, U2 ∈ CP (A,M2(B)). Note that U1, U2 coincide on the diagonaland are self-adjoint. Let (S1, S2) be their diagonal coefficients, which are clearly in CP (A,B). Wehave then mappings u1 : A→ B and u2 : A→ B such that

U1 =

(S1 u1

u1∗ S2

)and U2 =

(S1 u2

u2∗ S2

).

This implies ‖u1‖dec ≤ max‖S1‖, ‖S2‖ and ‖u2‖dec ≤ max‖S1‖, ‖S2‖. Therefore ‖u‖dec ≤‖u1‖dec + ‖u2‖dec ≤ 2 max‖S1‖, ‖S2‖ ≤ ‖U1 + U2‖, where for the last inequality we used theclassical inequality

max‖a‖, ‖d‖ =

∥∥∥∥(a 00 d

)∥∥∥∥ ≤ ∥∥∥∥(a bc d

)∥∥∥∥ .This shows that ‖u‖dec ≤ inf‖U1 + U2‖ | u = U1 − U2 = ‖u‖dec, where for the last = we use (ii)for u.

Conversely, if ‖u‖dec < 1 there are c.p. maps S1, S2 with

(S1 uu∗ S2

)c.p. and max‖S1‖, ‖S2‖ < 1.

Let S =

(S1 00 S2

). Then (recall (6.3) with λ = −1) S ± u is c.p. and hence u = U1 − U2 with

U1 = (S+u)/2 and U2 = (S−u)/2, and ‖U1+U2‖ = ‖S‖ < 1. This shows that ‖u‖dec ≤ ‖u‖dec.

Proposition 6.6. The following additional properties hold:

82

(i) We have D(A,B) ⊂ CB(A,B) and

(6.6) ∀u ∈ D(A,B) ‖u‖cb ≤ ‖u‖dec.

(ii) If u ∈ D(A,B) and v ∈ D(B,C) then vu ∈ D(A,C) and

(6.7) ‖vu‖dec ≤ ‖v‖dec‖u‖dec.

Proof. (i) Assume first that u is self-adjoint, i.e. u = u∗ and ‖u‖dec < 1. Then, by part (ii) inLemma 6.5, u = u1 − u2 with u1, u2 c.p. and ‖u1 + u2‖ < 1. We claim that

(6.8) sup‖u(x)‖ | x = x∗, ‖x‖ ≤ 1 ≤ 1.

Indeed, first consider x ≥ 0 in the unit ball of A. We have then ±u(x) ≤ (u1 + u2)(x) andhence ‖u(x)‖ ≤ 1. But now if x = x∗ and ‖x‖ ≤ 1, we have ±x ≤ |x| = x+ + x−, and hence±u(x) = ±(u1 − u2)(x) ≤ (u1 + u2)(|x|), which implies ‖u(x)‖ ≤ ‖|x|‖ ≤ 1, proving the claim.But (6.8) is valid also for IdMn ⊗ u = IdMn ⊗ u1 − IdMn ⊗ u2 for any n. In particular, using n = 2

since the matrix

(0 xx∗ 0

)is self-adjoint in M2(A), we have for any x ∈ A

‖u(x)‖ ≤∥∥∥∥( 0 u(x)u(x∗) 0

)∥∥∥∥ ≤ ∥∥∥∥( 0 xx∗ 0

)∥∥∥∥ = ‖x‖,

which implies ‖u‖ ≤ 1. Since we may replace u by IdMn ⊗ u for any n, we conclude ‖u‖cb ≤ 1.By homogeneity, this proves (i) for self-adjoint u’s. But by part (iii) in Lemma 6.5, we have‖u‖dec = ‖u‖dec, and by what we just proved ‖u‖cb ≤ ‖u‖dec. Since we have obviously ‖u‖cb ≤ ‖u‖cb,we obtain (i).

(ii) Assume that

(S1 uu∗ S2

)∈ CP (A,M2(B)) and

(T1 vv∗ T2

)∈ CP (B,M2(C)). Then by Lemma

1.40 we have

(T1S1 vuv∗u∗ T2S2

)∈ CP (A,M2(C)). Therefore, observing that v∗u∗ = (vu)∗, we have

‖vu‖dec ≤ max‖T1S1‖, ‖T2S2‖ ≤ max‖T1‖, ‖T2‖max‖S1‖, ‖S2‖,

and (ii) follows.Note that for self-adjoint mappings there is a very direct argument: if u = u1 − u2 and v =

v1 − v2 we have vu = (v1u1 + v2u2) − (v1u2 + v2u1) (a difference of two c.p. maps) and hence‖vu‖dec ≤ ‖(v1u1 +v2u2)+(v1u2 +v2u1)‖ = ‖(v1 +v2)(u1 +u2)‖ ≤ ‖v1 +v2‖‖u1 +u2‖, and recalling(6.5), this yields (ii) for self-adjoint maps u, v.

The preceding results are valid with an arbitrary range. However, the special case when therange is B(H) (or is injective) is quite important:

Proposition 6.7. If B = B(H) or if B is an injective C∗-algebra, then

D(A,B) = CB(A,B)

and for any u ∈ CB(A,B) we have

(6.9) ‖u‖dec = ‖u‖cb.

83

Proof. Assume B = B(H). By the factorization of c.b. maps (see Theorem 1.50) we can write

u(x) = V ∗π(x)W (x ∈ A) with ‖V ‖ = ‖W‖ = ‖u‖1/2cb . Let S1(x) = V ∗π(x)V and S2(x) =W ∗π(x)W . Note u∗(x) = W ∗π(x)V . Then the map(

S1 uu∗ S2

)=

(V ∗ 0W ∗ 0

)(π 00 π

)(V W0 0

)is c.p. (by Remark 1.21) and hence ‖u‖dec ≤ max‖V ‖2, ‖W‖2 = ‖u‖cb. Equality holds by (6.6).If B ⊂ B(H) is injective there is a contractive c.p. projection P : B(H)→ B. Note that by (i) inLemma 6.5 ‖P‖dec = 1. Then by (ii) in Proposition 6.6

‖u : A→ B‖dec ≤ ‖u : A→ B(H)‖dec‖P : B(H)→ B‖dec = ‖u‖cb.

Again equality holds by (6.6).

In analogy with (1.15) we have:

Lemma 6.8 (Decomposable maps into a direct sum). Let A and (Bi)i∈I be C∗-algebras and letB = (⊕

∑i∈I Bi)∞. Let u : A → B. We denote ui = piu : A → Bi. Then u ∈ D(A,B) if only if

all the ui’s are decomposable with supi∈I ‖ui‖dec <∞ and we have

(6.10) ‖u‖dec = supi∈I ‖ui‖dec.

Proof. Assume supi∈I ‖ui‖dec < 1. We then have c.p. maps Vi : A → M2(Bi) such that Vi12 = uiand such that max‖Vi11‖, ‖Vi22‖ < 1. By (6.6) the mapping V : A → (⊕

∑i∈IM2(Bi))∞

associated to (Vi)i∈I is well defined and clearly c.p. Since we may identify (⊕∑

i∈IM2(Bi))∞ withM2(B) so that u = V12, we obtain ‖u‖dec ≤ 1. By homogeneity this proves ‖u‖dec ≤ supi∈I ‖ui‖dec.The converse follows from (6.7).

In the von Neumann algebra setting, the next lemma will be useful.

Lemma 6.9 (Decomposability extends to the bidual). Let u : A → M be a linear map from aC∗-algebra A to a von Neumann algebra M . Then u ∈ D(A,M) ⇒ u ∈ D(A∗∗,M) and ‖u‖dec =‖u‖dec.

Proof. By Lemma 1.43 we know u ∈ CP (A,M) ⇒ u ∈ CP (A∗∗,M). Assume ‖u‖dec < 1. Then

let S1, S2 ∈ CP (A,B) with ‖S1‖ < 1, ‖S2‖ < 1 be such that V =

(S1 uu∗ S2

)is c.p. and note

that V =

(S1 u

u∗ S2

). This implies ‖u‖dec < 1, and hence by homogeneity, ‖u‖dec ≤ ‖u‖dec. The

converse is obvious (say by (ii) in Proposition 6.6).

The next statement provides us with simple examples of decomposable maps (note that we willshow in Lemma 6.24 that (6.12) is somewhat optimal).

Proposition 6.10. Let A be a C∗-algebra.

(i) Fix a, b in A. Let u : A→ A be defined by u(x) = a∗xb, then

‖u‖dec ≤ ‖a‖ ‖b‖.

84

(ii) Let u : Mn → A be a linear mapping into a C∗-algebra. Assume that u(eij) = a∗i bj with ai, bjin A. Let ‖a‖C = ‖

∑a∗i ai‖

1/2. Then

(6.11) ‖u‖dec ≤ ‖a‖C ‖b‖C .

(iii) More generally, if u(eij) =∑

1≤k≤m a∗kibkj with aki, bkj in A then

(6.12) ‖u‖dec ≤ ‖∑

kia∗kiaki‖1/2 ‖

∑kjb∗kjbkj‖1/2.

Proof. (i) Let V : A→M2(A) be the mapping defined by

V (x) =

(a∗xa a∗xbb∗xa b∗xb

).

An elementary verification shows that V (x) = t∗(x 00 x

)t where t = 2−1/2

(a ba b

). Clearly this

shows that V is c.p. hence by definition of the dec-norm we have

‖u‖dec ≤ max‖V11‖, ‖V22‖

where V11(x) = a∗xa and V22(x) = b∗xb. Thus we obtain ‖u‖dec ≤ max‖a‖2, ‖b‖2. Applying thisto the mapping x→ u(x)‖a‖−1‖b‖−1 we find ‖u‖dec ≤ ‖a‖ ‖b‖.(ii) Let a∗ = (a∗1, a

∗2, . . . , a

∗n), b∗ = (b∗1, b

∗2, . . . , b

∗n) viewed as row matrices with entries in A (so that

a and b are column matrices). Then, for any x in Mn, u(x) can be written as a matrix product:

u(x) = a∗xb.

We again introduce the mapping V : Mn → M2(A) defined by V (x) =

(a∗xa a∗xbb∗xa b∗xb

). Again we

note V (x) = t∗(x 00 x

)t where t = 2−1/2

(a ba b

)∈ M2n×2(A) which shows that V is c.p. so

we obtain ‖u‖dec ≤ max‖V11‖, ‖V22‖ ≤ max‖b‖2, ‖a‖2 = max‖∑b∗jbj‖, ‖

∑a∗i ai‖ and by

homogeneity this yields (6.11).(iii) We have u =

∑uk where uk : Mn → A is defined by uk(eij) = a∗kibkj . Let Vk : Mn → M2(A)

be associated to uk as in (ii). Let V =∑Vk. Clearly V is c.p. and hence

‖u‖dec ≤ max‖V11‖, ‖V22‖ = max‖V11(1)‖, ‖V22(1)‖,

which yields (6.12).

Proposition 6.11. Let A,B,C be C∗-algebras. For any u ∈ D(A,B)

(6.13) ∀x ∈ C ⊗A ‖(IdC ⊗ u)(x)‖C⊗maxB ≤ ‖u‖dec‖x‖C⊗maxA.

Moreover, the mapping IdC ⊗ u : C ⊗max A→ C ⊗max B is decomposable and its norm satisfies

(6.14) ‖IdC ⊗ u‖D(C⊗maxA,C⊗maxB) ≤ ‖u‖dec.

85

Proof. Since by (6.6) ‖IdC ⊗ u‖ ≤ ‖IdC ⊗ u‖dec, it suffices to prove (6.14). We already saw inCorollary 4.18 the analogous property for c.p. maps. Let u ∈ D(A,B). Assume that the map

V =

(S1 uu∗ S2

)is in CP (A,M2(B)). Then by Corollary 4.18 the mapping

IdC ⊗ V : C ⊗max A→ C ⊗max M2(B)

is c.p. and using C ⊗max M2(B) ∼= M2(C ⊗max B), we may view it as

(IdC ⊗ S1 IdC ⊗ uIdC ⊗ u∗ IdC ⊗ S2

).

Since IdC ⊗ u∗ = (IdC ⊗ u)∗ this implies that IdC ⊗ u ∈ D(C ⊗max A,C ⊗max B) with dec-norm

≤ max‖IdC ⊗ S1 : C ⊗max A→ C ⊗max B‖, ‖IdC ⊗ S2 : C ⊗max A→ C ⊗max B‖

but, by (4.30) since S1, S2 are c.p. , this is the same as max‖S1‖, ‖S2‖. Taking the infimum overall possible S1, S2, we obtain (6.14).

Corollary 6.12. Let uj ∈ D(Aj , Bj) (j = 1, 2) be decomposable mappings between C∗-algebras.Then u1 ⊗ u2 extends to a decomposable mapping in D(A1 ⊗max A2, B1 ⊗max B2) such that

(6.15) ‖u1 ⊗ u2‖D(A1⊗maxA2,B1⊗maxB2) ≤ ‖u1‖dec‖u2‖dec.

Proof. Just write u1 ⊗ u2 = (u1 ⊗ Id)(Id⊗ u2).

When the mapping u has finite rank then a stronger result holds. We can go min→ max:

Proposition 6.13. Let u ∈ D(A,B) be a finite rank map between C∗-algebras. For any C∗-algebraC we have

(6.16) ∀x ∈ C ⊗A ‖(IdC ⊗ u)(x)‖C⊗maxB ≤ ‖u‖dec‖x‖C⊗minA.

Proof. For any finite dimensional subspace F ⊂ B, the min and max norms are clearly equivalenton C ⊗ F . Thus since its rank is finite u defines a bounded map IdC ⊗ u : C ⊗min A→ C ⊗max B.That same map has norm at most ‖u‖dec as a map from C⊗maxA to C⊗maxB. But since we havea metric surjection q : C ⊗max A→ C ⊗min A taking the open unit ball onto the open unit ball, itfollows automatically that

‖IdC ⊗ u : C ⊗min A→ C ⊗max B‖ = ‖IdC ⊗ u : C ⊗max A→ C ⊗max B‖ ≤ ‖u‖dec.

6.2 The δ-norm

In this section, we introduce a “hybrid” tensor product E ⊗δ A of an operator space E and aC∗-algebra A, which is convenient to compute the dec-norms of certain important examples, as wedo in the next section. We will use it again in §10.3 to relate nuclearity and c.p. approximationproperties. The main point is Theorem 6.15, which provides us with a useful factorization for theelements in the unit ball of (C ⊗max A) ∩ (E ⊗A) when C is the universal C∗ algebra of E.

We need the obvious generalization of the notation (4.5): given operator spaces E,F and linearmappings θ : E → B(H) and π : F → B(H), we denote by θ · π : E ⊗ F → B(H) the linearmapping defined by

(θ · π)(∑

xj ⊗ yj)

=∑

θ(xj)π(yj) (xj ∈ E, yj ∈ F ).

86

Let A be a C∗-algebra. For any y ∈ E ⊗A we define

(6.17) ∆(y) = sup‖θ · π(y)‖B(H)

where the supremum runs over all Hilbert spaces H and all pairs (θ, π) whereπ : A→ B(H) is a ∗-homomorphism and θ : E → π(A)′ is a complete contraction.When A is unital, we claim that (6.17) remains unchanged if we restrict to unital ∗-homomorphisms.Indeed, if π is not unital, let p = π(1), then p is a projection on H and it is immediate thatθ · π(y) = θ′ · π′(y) where θ′ = pθp and π′ = pπp; thus if we replace π by π′, which we view asa unital ∗-homomorphism into B(p(H)), and θ by θ′, we find ‖θ · π(y)‖ = ‖θ′ · π′(y)‖ whence theclaim.

Lemma 6.14. We view E as embedded into C∗〈E〉 (as defined in §2.7). Then

(6.18) ∆(y) = ‖y‖C∗〈E〉⊗maxA.

Proof. Since π(A)′ is a C∗-algebra, any completely contractive map θ : E → π(A)′ extends to arepresentation θ : C∗〈E〉 → π(A)′. Hence we clearly have ∀y ∈ E ⊗A

∆(y) ≤ sup ‖θ · π(y)‖ ≤ ‖y‖C∗〈E〉⊗maxA.

The reverse inequality is clear (using an embedding C∗〈E〉 ⊗max A ⊂ B(H)).

The main motivation for introducing ∆ as in (6.17) is the following.

Theorem 6.15. Let A ⊂ B(H) be a unital C∗-algebra and let E be an operator space. Consideran element y in E ⊗A. Let

(6.19) δ(y) = inf

‖x‖Mn(E)

∥∥∥∑ aia∗i

∥∥∥1/2 ∥∥∥∑ b∗jbj

∥∥∥1/2

where the infimum runs over all possible n and all possible representations of y of the form

(6.20) y =∑n

i,j=1xij ⊗ aibj .

Then

(6.21) ∆(y) = δ(y).

Proof. We adapt a proof from [208] that actually is valid even if the subalgebra A is not assumedself-adjoint (in that case the π’s are assumed to be unital completely contractive homomorphisms).We first show the easy inequality ∆(y) ≤ δ(y). Let (θ, π) be as in the definition of ∆(y). Then wehave, assuming (6.20)

(θ · π)(y) =∑

π(ai)θ(xij)π(bj)

and hence by (2.3) ∆(y) ≤ δ(y).To show the converse, since δ and ∆ are norms (for δ this can be proved by the same idea as forγE in (1.17)), it suffices to show that

∆∗ ≤ δ∗.

So let ϕ : E ⊗A→ C be a linear form such that δ∗(ϕ) ≤ 1, or equivalently such that

∀y ∈ E ⊗A |ϕ(y)| ≤ δ(y).

87

The following Lemma 6.16 is the heart of the proof. With the notation from that lemma, we have

ϕ(y) = 〈η1, (v · π)(y)η2〉

therefore|ϕ(y)| ≤ ‖v · π(y)‖ ≤ ∆(y).

This completes the proof that ∆∗ ≤ δ∗, and hence that δ ≤ ∆.

Lemma 6.16. Let ϕ : E ⊗A→ C be a linear form with δ∗(ϕ) ≤ 1. Then there are a Hilbert spaceH and a representation π : A→ B(H) together with a completely contractive map v : E → B(H)and unit vectors η1 ∈ H and η2 ∈ H such that for any a, b, c in A and any x in E we have

ϕ(x⊗ ab) = 〈η1, π(a)v(x)π(b)η2〉 and v(x)π(c) = π(c)v(x).

Proof. We first claim that there are two representations π1 : A → B(H1) and π2 : A → B(H2)together with a completely contractive map v : E → B(H2, H1) and unit vectors ξ1 ∈ H1 andξ2 ∈ H2 such that for any a, b in A and any x in E we have

(6.22) ϕ(x⊗ ab) = 〈ξ1, π1(a)v(x)π2(b)ξ2〉.

Recall the classical arithmetic/geometric mean inequality:

(6.23) ∀α, β ≥ 0 (αβ)1/2 = infs>0(α/s+ sβ)/2.

By the definition of δ, this implies that if x ∈ BMn(E)

|ϕ(∑

xij ⊗ aibj)| ≤ (1/2)(‖∑

aia∗i ‖+ ‖

∑b∗jbj‖).

Let S be the set of pairs of states (f1, f2) on A and let F : S → R be the function defined by

(6.24) F (f1, f2) = (1/2)(f1(∑

aia∗i ) + f2(

∑b∗jbj))−<(ϕ(

∑xij ⊗ aibj)).

Thensupt∈S F (t) ≥ 0.

Let F ⊂ `∞(S,R) be the set of all such functions. It is easy to check that F is a convex cone.Indeed, if F ′ is the function associated to x′, a′i, b

′j with x′ ∈ BMn′ (E) then F + F ′ is associated to(

x 00 x′

), with the sequences (ai, a

′i) and (bj , b

′j) obtained by concatenation. Moreover, S is a weak*

compact convex subset of A∗ ⊕A∗ and each F ∈ F is affine and weak* continuous. By the variantof Hahn-Banach described in Lemma 26.16 there is (f1, f2) in S such that for any x ∈ BMn(E) andai, bj ∈ A

<(ϕ(∑

xij ⊗ aibj)) ≤ (1/2)(f1(∑

aia∗i ) + f2(

∑b∗jbj)).

By (6.23) (since the left-hand side is unchanged when we replace (ai, bj) by (ai/s, sbj) for any s > 0)we find automatically

<(ϕ(∑

xij ⊗ aibj)) ≤ (f1(∑

aia∗i )f2(

∑b∗jbj))

1/2.

Changing ai to zai with z ∈ C, |z| = 1 does not change the right-hand side either, thus we also find

(6.25) |ϕ(∑

xij ⊗ aibj)| ≤ (f1(∑

aia∗i )f2(

∑b∗jbj))

1/2.

88

In particular, for any x ∈ BE and a, b ∈ A

|ϕ(x⊗ ab)| ≤ (f1(aa∗)f2(b∗b))1/2.

Let π1 : A→ B(H1) (resp. π2 : A→ B(H2)) be the GNS representation of the state f1 (resp. f2)with cyclic unit vector ξ1 ∈ H1 (resp. ξ2 ∈ H2). This gives us

|ϕ(x⊗ ab)| ≤ ‖π1(a∗)ξ1‖‖π2(b)ξ2‖.

Recall that since we use cyclic vectors, the sets π2(b)ξ2 | b ∈ A and π1(a∗)ξ1 | a ∈ A are dense inH2 and H1 respectively. Thus we can unambiguously define a linear mapping v : E → B(H2, H1)with ‖v‖ ≤ 1 such that for any x ∈ E, a, b ∈ A

ϕ(x⊗ ab) = 〈π1(a∗)ξ1, v(x)π2(b)ξ2〉.

But going back to (6.25) this shows us that actually for any x ∈ BMn(E) we have

|∑〈π1(a∗i )ξ1, v(xij)π2(bj)ξ2〉| ≤ (

∑‖π1(a∗i )ξ1‖2)1/2(

∑‖π2(bj)ξ2‖2)1/2.

In other words, we actually have ‖vn‖ ≤ 1 for any n ≥ 1 and hence ‖v‖cb ≤ 1, which proves ourclaim.Then, writing (ac)b = a(cb) into (6.22) and using the density just mentioned, we find for all c in A

(6.26) ∀x ∈ E v(x)π2(c) = π1(c)v(x).

Let π : A→ B(H1 ⊕H2) and v : E → B(H1 ⊕H2) be defined by

π(a) =

(π1(a) 0

0 π2(a)

)and v(x) =

(0 v(x)0 0

).

Then (6.26) implies v(x)π(c) = π(c)v(x), for x ∈ E, c ∈ A, and π is a unital representation on A.

Lastly, letting η2 =

(0ξ2

)and η1 =

(ξ1

0

)we find ϕ(x⊗ ab) = 〈η1, π(a)v(x)π(b)η2〉.

Remark 6.17. Let E be an operator space and A a unital C∗-algebra. Consider y ∈ E∗ ⊗ A. Letu : E → A be the associated linear map. If δ(y) ≤ 1 then for any ε > 0 u admits a factorizationof the form

Ev−→Mn

w−→A,with ‖v‖cb ≤ 1+ε and ‖w‖dec ≤ 1. Indeed, we can write y as y =

∑xij⊗a∗i bj with ‖x‖Mn(E∗) ≤ 1+ε

and ‖a‖C‖b‖C ≤ 1 (recall the notation ‖a‖C = ‖∑a∗i ai‖

1/2). Define w : Mn → A by w(eij) = a∗i bj ,and let v : E → Mn be the map associated to x, so that ‖v‖cb = ‖x‖Mn(E∗) ≤ 1 + ε. Then by(6.11) we have ‖w‖dec ≤ 1.

Remark 6.18. The space E ⊗δ A can be described as a quotient of the Haagerup tensor productA⊗h E ⊗h A via the map a⊗ e⊗ b 7→ e⊗ ab, see [208, p. 241].

6.3 Decomposable extension property

We previously described (see §1.2) an analogue of the Hahn-Banach extension theorem for c.b.maps. In this section we give an analogue for the maximal tensor product, where decomposablemaps replace the c.b. ones.

At this point, we advise the reader to review Lemma 1.38 and Corollary 5.3 on bimodule maps.We will use a generalization of Lemma 1.38 for bimodule maps on “operator modules”, as follows(this result appears in [237], see also [197]).

89

Lemma 6.19. Let C ⊂ B(K) be a unital C∗-algebra given with a representation π : C → B(H).Let E ⊂ B(K) be a C-bimodule, i.e. an operator space stable by (left and right) multiplication byany element of C. Consider a bimodule map w : E → B(H), i.e. a map satisfying w(c1xc2) =π(c1)w(x)π(c2), (c1, c2 ∈ C, x ∈ E). Let S ⊂ M2(B(K)) be the operator system consisting of all

matrices of the form

(λ ab∗ µ

)with λ, µ ∈ C, a, b ∈ E. Let W : S →M2(B(H)) be defined by

W

((λ ab∗ µ

))=

(π(λ) w(a)w(b)∗ π(µ)

).

Then ‖w‖cb ≤ 1 if and only if W is c.p.

Proof. The proof of Lemma 1.38 can be easily modified to yield this.

We now give an extension property (one more version of Hahn-Banach) for maps defined on asubspace of the maximal tensor product. We will repeat the same trick later on in §8.3.

Theorem 6.20. Let A be a unital C∗-algebra, E ⊂ A an operator space and M ⊂ B(H) a vonNeumann algebra. Let u : E →M be a bounded linear map. Let u : M ′ ⊗ E → B(H) be the linearmap defined when x′ ∈M ′, x ∈ E by u(x′ ⊗ x) = x′u(x). Let

M ′ ⊗max E = M ′ ⊗ E ⊂M ′ ⊗max A

denote the closure of M ′ ⊗ E in M ′ ⊗max A equipped with the norm induced by M ′ ⊗max A.Then u extends to a c.b. map on M ′ ⊗max E with ‖u : M ′ ⊗max E → B(H)‖cb ≤ 1 if and only ifthere is u ∈ D(A,M) with ‖u‖dec ≤ 1 extending u. In other words

‖u : M ′ ⊗max E → B(H)‖cb = inf‖u : A→M‖dec | u|E = u

and the infimum is attained.

Proof. Assume there is an extension u with ‖u : A→M‖dec ≤ 1. By (6.14) and (6.6) ‖IdM ′ ⊗ u :M ′⊗maxA→M ′⊗maxM‖cb ≤ 1 and hence a fortiori ‖IdM ′⊗u : M ′⊗maxE →M ′⊗maxM‖cb ≤ 1.Since the product map p : M ′ ⊗M → B(H) defines trivially a ∗-homomorphism on M ′ ⊗max Mwe have ‖p : M ′ ⊗max M → B(H)‖cb ≤ 1, and hence ‖u : M ′ ⊗max E → B(H)‖cb ≤ 1 sinceu = p(IdM ′ ⊗ u).Conversely, assume ‖u : M ′ ⊗max E → B(H)‖cb ≤ 1. Let K be a suitable Hilbert space so thatM ′ ⊗max A ⊂ B(K), let C = M ′ ⊗ 1 ⊂ B(K), let π : C → B(H) be the natural identificationM ′ ⊗ 1 ' M ′ and let E = M ′ ⊗max E. Note that E is a C-bimodule, and that u : E → B(H)is a C-bimodule map. Hence, with the same notation as in Lemma 6.19 that we apply here tow = u the mapping W : S →M2(B(H)) must be completely positive (and unital). Recall that byTheorem 1.39 any unital c.p. map V : A1 → B(H) on a (unital) operator system A1 ⊂ A2 admitsa (unital) c.p. extension V : A2 → B(H). Thus, let

W : M2(M ′ ⊗max A)→M2(B(H))

be a completely positive extension of W : S → M2(B(H)), and let T be the restriction of W toM2(1⊗A). Identifying A to 1⊗A ⊂ B(K), we may view T as a mapping from M2(A) to M2(B(H)).

Note T (1) = W (1) = 1. In the present case of modular maps we claim that T has the followingspecial form

(6.27) ∀x ∈M2(A) T (x) =

(T11(x11) T12(x12)T21(x21) T22(x22)

)90

with T12|E = u and moreover that T (M2(A)) ⊂M2(M).Taking this claim temporarily for granted, we will now conclude the proof. Since T is unital andc.p. we have max‖T11‖, ‖T22‖ ≤ ‖T‖ = 1, the maps T11 and T22 are c.p. and moreover the

mapping R : a 7→ T (

(a aa a

)) is clearly c.p. on A, and a fortiori self-adjoint. Let u = T12 (and as

usual u∗(a) = u(a∗)∗ for any a ∈ A). Then we have

∀a ∈ A R(a) =

(T11(a) u(a)u∗(a) T22(a)

).

Therefore, by definition of the dec-norm, we have

‖u‖dec ≤ max‖T11‖, ‖T22‖ = 1.

Thus it only remains to prove the claim. For this purpose, observe that, by its definition, W is a

∗-homomorphism on the algebra of matrices

(c1 00 c2

)with c1, c2 in C = M ′⊗ 1. Let D denote the

set of all such matrices. By Corollary 5.3 the map W must be D-bimodular, i.e. we have

(6.28) W (y1xy2) = W (y1)W (x)W (y2), ∀y1, y2 ∈ D ∀x ∈M2(B(H)).

Applying this with y1, y2 equal to either

(1 00 0

)or

(0 00 1

), we find that W is necessarily such

that W (x)ij depends only on xij . A fortiori the same is true for T = W|M2(A). Thus we can write

a priori T in the form (6.27). Moreover, since W extends W , we know W12 extends W12 = w = u,and hence restricting this to A ' 1⊗A (on which u = u) we see that u extends u.Lastly, it remains to check that all Tij ’s take their values in M . Equivalently it suffices to check thatall the terms Tij(xij) (i, j = 1, 2, xij ∈ A) commute with M ′. But this is an easy consequence of(6.28). Indeed, since 1⊗A trivially commutes with M ′⊗1, any x ∈M2(A) commutes with any y ∈ D

of the form y =

(m′ ⊗ 1 0

0 m′ ⊗ 1

)∈ D with m′ ∈M ′, and hence by (6.28) W (y)W (x) = W (yx) =

W (xy) = W (x)W (y). Equivalently W (y)T (x) = T (x)W (y) and since W (y) =

(m′ 00 m′

), this

implies that Tij(xij) all commute with any m′ ∈M ′, and hence take their values in M ′′ = M . Thiscompletes the proof of the claim, and of Theorem 6.20.

Corollary 6.21 (Infinite multiplicity). In the situation of Theorem 6.20, if the embedding M ⊂B(H) has infinite multiplicity, which means we assume H = `2 ⊗2 H and there is an isomorphismπ : M →M with M⊂ B(H) such that the embedding M ⊂ B(H) is of the form x 7→ Id`2 ⊗ π(x)(x ∈M), then

(6.29) ‖u : M ′ ⊗max E → B(H)‖ = ‖u : M ′ ⊗max E → B(H)‖cb.

Proof. We have M ′ = B(`2)⊗M′. Let v = πu : E → M. A simple verification shows that urestricted to [B(`2) ⊗M′] ⊗ E ⊂ M ′ ⊗ E can be identified with IdB(`2) ⊗ v. Moreover, we mayuse this idenfication with Mn ⊂ B(`2) in place of B(`2) and the isomorphism Mn(M ′) ⊗max E =Mn(M ′ ⊗max E) to show that ‖u‖ = ‖v‖cb. But since π is an isomorphism we also have

inf‖u : A→M‖dec | u|E = u = inf‖v : A→M‖dec | v|E = v.

By Theorem 6.20 this last equality implies ‖u‖cb = ‖v‖cb, so that we obtain ‖u‖ = ‖v‖cb = ‖u‖cb.

91

Corollary 6.22 (Case E = `n∞). In the situation of Theorem 6.20, assume either that E = A orthat E ⊂ A is a C∗-subalgebra for which there is a contractive c.p. projection P : A→ E. Then

(6.30) ‖u‖D(E,M) = ‖u‖cb.

In particular, this holds for E = `n∞, E = Mn or when E is an injective C∗-algebra.

Proof. We claim ‖u‖dec = inf‖u : A→M‖dec | u|E = u. The case E = A is trivial, and the otherone very simple. Indeed, by (6.7) for any extension u we have ‖u : E →M‖dec = ‖u|E‖dec ≤ ‖u‖decand if we choose u = uP we have ‖u‖dec ≤ ‖u‖dec‖P‖dec = ‖u‖dec.

If M ′ admits a cyclic vector, or equivalently if M admits a separating vector (see Lemma 26.62)then the mere boundedness of u on M ′ ⊗max E ensures that it is c.b. Actually this is a generalphenomenon, as the next lemma shows.

Corollary 6.23 (Cyclic case). In the situation of Theorem 6.20, assume that M ′ ⊂ B(H) admitsa cyclic vector. Let ‖ · ‖α be a C∗-norm on M ′ ⊗ A and let M ′ ⊗α A be the resulting C∗-algebra(after completion). Let

M ′ ⊗α E = M ′ ⊗ Eα ⊂M ′ ⊗α A

equipped with the norm induced by M ′ ⊗α A.Then if u defines (by density) a bounded map on M ′ ⊗α E, the latter is (automatically) c.b. andits c.b. norm is equal to its norm.

Proof. We may assume M ′⊗αA ⊂ B(H). The operator space E = M ′⊗αE ⊂M ′⊗αA ⊂ B(H) isclearly a bimodule with respect toM ′⊗1 viewed as a C∗-subalgebra ofB(H). Let π : M ′⊗1→ B(H)be the natural embedding. If u defines (by density) a bounded map on M ′ ⊗α E, the latter is (likeu itself) bimodular with respect to π : M ′ ⊗ 1 → B(H). Therefore, by Theorem 2.6 the resultingmap u : M ′ ⊗α E → B(H) is c.b. with equality of the norm and the c.b. norm.

6.4 Examples of decomposable maps

We end this section with several important examples. First we invite the reader to recall Remark1.30. We will now refine Proposition 6.10.

Lemma 6.24. Consider a linear mapping u : Mn → A into a C∗-algebra A. Let a ∈Mn(A) be thematrix defined by aij = u(eij). Then

(6.31) ‖u‖dec = inf‖∑

k,ja∗kjakj‖1/2‖

∑k,jb∗kjbkj‖1/2 | a, b ∈Mn(A), a = a∗b.

Proof. By (iii) in Proposition 6.10 we know that ‖u‖dec is ≤ the right hand side of (6.31). Con-versely, assume ‖u‖dec < 1. Let V ∈ CP (Mn,M2(A)) be such that V12 = u and ‖Vjj‖ < 1 forj = 1, 2. Let αij = V (eij) ∈ M2(A). We denote its entries by αij11, αij12, · · · ∈ A. By Remark1.30, α ∈Mn(M2(A))+, and hence α = β∗β for some β ∈Mn(M2(A)). Then

aij = αij12 =∑

kβki∗11βkj12 + βki

∗21βkj22.

Moreover, ∑k,jβkj∗11βkj11 + βkj

∗21βkj21 =

∑jαjj11 = V (1)11

and hence‖∑

k,jβkj∗11βkj11 + βkj

∗21βkj21‖ = ‖V (1)11‖ = ‖V11‖ < 1.

92

Similarly

‖∑

k,jβkj∗12βkj12 + βkj

∗22βkj22‖ ≤ ‖V22‖ < 1.

Thus we “almost” conclude as desired that the right hand side of (6.31) is < 1. The only trouble isthat we obtain a representation of the form a = a∗b with matrices a, b of size 2n×n instead of n×n.This is easy to fix using the following elementary factorization (essentially the polar decomposition):assuming A unital for simplicity, for any ε > 0 we set γ = (a∗a + ε1)−1/2a∗b(b∗b + ε1)−1/2. Letx = (a∗a+ε1)−1/2a∗ and y = b(b∗b+ε1)−1/2. Then clearly ‖x‖ = ‖xx∗‖1/2 ≤ 1, ‖y‖ = ‖y∗y‖1/2 ≤ 1and hence ‖γ‖ = ‖xy‖ ≤ 1. Then we have

a∗b = (a∗a+ ε1)1/2γ(b∗b+ ε1)1/2 = a′∗b′,

where a′ = (a∗a+ ε1)1/2 and b′ = γ(b∗b+ ε1)1/2. But now a′, b′ are both in Mn(A) and such thata′∗a′ ≤ a∗a + ε1 and b′∗b′ ≤ b∗b + ε1. It follows that ‖

∑kj a′∗kja′kj‖ ≤ ‖

∑kj a∗kjakj‖ + n2ε and

‖∑

kj b′∗kjb′kj‖ ≤ ‖

∑kj b∗kjbkj‖ + n2ε. Since ε > 0 is arbitrary this proves that the right hand side

of (6.31) is < 1. By homogeneity, this completes the proof.

For emphasis, we single out the next example, which will play an important role in the sequel.The reader should compare this to the earlier description of the unit ball of CB(`n∞, A) in (3.7).

Lemma 6.25. Consider a linear mapping T : `n∞ → A into a C∗-algebra A. Let xj = T (ej)(1 ≤ j ≤ n). Then

(6.32) ‖T‖dec = inf‖∑

ja∗jaj‖1/2‖

∑jb∗jbj‖1/2 | aj , bj ∈ A, xj = a∗jbj.

Proof. We may identify `n∞ with the C∗-subalgebra of diagonal matrices in Mn. We know there is acontractive c.p. projection P : Mn → `n∞. By (i) in Lemma 6.5 both P and the inclusion `n∞ →Mn

have dec-norm = 1. By (6.7) we have

(6.33) ‖T‖D(`n∞,A) = ‖TP‖D(Mn,A).

Using this it is easy to deduce (6.32) from (6.31). We leave the details to the reader.

Lemma 6.26. In the situation of Theorem 6.15, assume that A is a unital C∗-algebra, let F beanother operator space and B another C∗-algebra. Consider u1 ∈ CB(E,F ) and u2 ∈ D(A,B).Then for all y in E ⊗A, we have

(6.34) δ((u1 ⊗ u2)(y)) ≤ ‖u1‖cb‖u2‖decδ(y).

Proof. Assume ‖u1‖cb = 1. Note that u1 : E → F extends to a C∗-representation from C∗〈E〉 toC∗〈F 〉. Then (6.34) is an immediate consequence of (6.15) and (6.18).

In the particular cases when E = M∗n or E = `n∗∞ , Theorem 6.15 becomes:

Corollary 6.27. In Theorem 6.15, assume A is a unital C∗-algebra and let E = M∗n (resp. E =`n∗∞ ) for some n ≥ 1, viewed as dual operator spaces in the sense of §2.4. Then for all y in E ⊗A,with associated linear map y : Mn → A (resp. y : `n∞ → A), we have

(6.35) δ(y) = ‖y‖D(Mn,A) (resp. δ(y) = ‖y‖D(`n∞,A)).

93

Proof. Let y ∈ E ⊗ A when E = M∗n. Let y(eij) = aij . Let (ξij) be biorthogonal to the standardbasis (eij) in Mn, so that y =

∑ξij ⊗ aij . Assume a = a∗b as in (6.31). Then

y =∑

i,k,jξij ⊗ a∗kibkj =

∑i,k,l,j

1k=lξij ⊗ a∗kiblj =∑

i,k,l,jz((i, k), (j, l))a∗kiblj

with z((i, k), (j, l)) = 1k=lξij . By definition of δ(y) we have

δ(y) ≤ ‖∑

i,k,l,jeij ⊗ ekl ⊗ z((i, k), (j, l))‖Mn⊗minMn⊗minE∗‖(aki)‖C‖(blj)‖C

and

‖∑∑

i,k,l,jeij⊗ekl⊗z((i, k), (j, l))‖Mn⊗minMn⊗minE∗ = ‖

∑i,jeij⊗(

∑kekk)⊗ξij‖Mn⊗minMn⊗minE∗

= ‖∑

i,jeij ⊗ ξij‖Mn⊗minE∗ = ‖IdE‖cb = 1.

Thus we obtain δ(y) ≤ ‖y‖dec by (6.31). We now turn to the reverse inequality (which incidentallyis valid for any E). If we have y =

∑Ni,j=1 xijaibj for some N ≥ 1, then let v : E∗ → MN be the

map defined by v(ξ) = (ξ(xij)) (ξ ∈ E∗). Note ‖v‖cb = ‖x‖MN (E) by (2.14). Let w : MN → A bedefined by w(eij) = aibj . Then by (6.11)

(6.36) ‖w‖dec ≤∥∥∥∑ aia

∗i

∥∥∥1/2 ∥∥∥∑ b∗jbj

∥∥∥1/2.

Whence since y = wv by part (ii) in Proposition 6.6 and Proposition 6.7 :

‖y‖dec ≤ ‖v‖dec‖w‖dec = ‖v‖cb‖w‖dec ≤ ‖x‖MN (E)

∥∥∥∑ aia∗i

∥∥∥1/2 ∥∥∥∑ b∗jbj

∥∥∥1/2.

Taking the infimum over all N and all possible (ai) and (bj), we obtain ‖y‖dec ≤ δ(y).The case when E = `n∞ is proved similarly but using (6.32) instead of (6.31).

We invite the reader to compare the following fact with Lemma 3.10.

Lemma 6.28. Let F be a free group with (free) generators (gi)i∈I and let Ui = UF(gi) ∈ C∗(F)(i ∈ I). We augment I by one element by setting formally I = I ∪ 0, and we set g0 equal to theunit in F so that U0 = UF(g0) = 1. Let (xi)i∈I be a finitely supported family in a C∗-algebra A and

let T : `∞(I)→ A be the mapping defined by T ((αi)i∈I) =∑

i∈I αixi. Then we have

(6.37)∥∥∥∑

i∈IUi ⊗ xi

∥∥∥C∗(F)⊗maxA

= ‖T‖dec.

Proof. Let E = span[Ui, | i ∈ I]. Let y =∑Ui ⊗ xi ∈ E ⊗ A. Recall that, by Theorem 6.15,

δ(y) = ∆(y), and hence by Corollary 6.27, we have ‖T‖dec = ∆(y), and by definition (recallingLemma 3.9)

∆(y) = sup ‖∑

viπ(xi)‖

where the supremum runs over all representations π : A→ B(H) and all families (vi) of contractionsin π(A)′ ⊂ B(H). By Remark 3.11 (Russo-Dye), the latter supremum remains unchanged if we letit run only over all the families (vi) of unitaries in π(A)′. Equivalently, by (4.6) this means:

∆(y) =∥∥∥∑

i∈IUi ⊗ xi

∥∥∥C∗(F)⊗maxA

,

which establishes (6.37).

94

When A is a von Neumann algebra with a separating vector the preceding lemma can besignificantly reinforced and it gives us, in the case |I| = n, a rather pretty formula for the dec-normof a mapping T : `n∞ →M .

Theorem 6.29. In the situation of the preceding lemma, assume that A = M where M is avon Neumann algebra with either infinite multiplicity (in the sense of Corollary 6.21) or with aseparating vector. Then

(6.38)∥∥∥∑

i∈IUi ⊗ xi

∥∥∥C∗(F)⊗maxM

= ‖T‖dec = sup∥∥∥∑

i∈Iuixi

∥∥∥ | ui ∈ U(M ′).

Proof. Since (xi) is assumed finitely supported, we may assume |I| < ∞, say |I| = n. Let T :M ′ ⊗ `∞(I) → B(H) be the mapping defined as in Theorem 6.20 by T (

∑yi ⊗ ei) =

∑yixi. By

(6.29) and (6.30) and Corollary 6.23 we have ‖T‖dec = ‖T‖, which is the second equality in (6.38).The first one just repeats (6.37).

Remark 6.30. Note that the second equality in (6.38) does not hold in general. For instance ifM = B(`2) then M ′ = C1, and the third term in (6.38) is equal to ‖T‖, which in general is< ‖T‖cb, and a fortiori < ‖T‖dec (see Remark 1.4).

Remark 6.31. Let C = Mn ∗ C∗(Z). By Proposition 2.18, we have C ' Mn ⊗min Bn = Mn(Bn)where Bn is a unital C∗-algebra called the Brown algebra for Larry Brown who introduced it in[36] (see [54] for more on Bn). In the isomorphism C ' Mn ⊗min Bn = Mn(Bn), the embeddingMn ⊂ C becomes x 7→ x ⊗ 1. Let U =

∑eij ⊗ Uij ∈ U(Mn(Bn)) be the unitary corresponding

to the (single) generator of Z in C∗(Z) ⊂ C. Let A be a C∗-algebra. As before, let E = M∗n(operator space dual), let ξij ∈ E be biorthogonal to the usual basis (eij) of Mn, let aij ∈ A andy =

∑ni,j=1 ξij ⊗ aij ∈ E ⊗A. Then

(6.39) ‖y‖D(Mn,A) = ‖∑

ξij ⊗ aij‖C∗<E>⊗maxA = ‖∑

Uij ⊗ aij‖Bn⊗maxA.

The first equality is the same as (6.35). We will prove the second one. For any unital C∗-algebraD ⊂B(H), there is a 1-1 correspondence π 7→

∑eij⊗π(Uij) between the set of unital ∗-homomorphisms

π : Bn → D and U(Mn(D)). Indeed, π ↔ IdMn ⊗ π : Mn(Bn) → Mn(D) and since Mn(Bn) 'Mn ∗C∗(Z), each IdMn ⊗π is determined by a ∗-homomorphism ρ : C∗(Z)→Mn(D) coupled withx 7→ x⊗1 on Mn, and of course ρ is determined by its value on the single generator of Z, and henceby a single element of U(Mn(D)). Therefore for any x ∈Mn(B(H))

sup‖∑

π(Uij)xij‖ | π : Bn → D = sup‖∑

uijxij‖ | u ∈ U(Mn(D)).

By Remark 3.11 (Russo-Dye) applied to Mn(D), we have

sup‖∑

uijxij‖ | u ∈ U(Mn(D)) = sup‖∑

zijxij‖ | z ∈ BMn(D).

Let σ : A→ B(H) be a ∗-homomorphism. Applying this to D = σ(A)′ and xij = σ(aij), we find

sup‖∑

π(Uij)σ(aij)‖ | π : Bn → σ(A)′ = sup‖∑

zijσ(aij)‖ | z ∈ BMn(σ(A)′),

and since Mn(D) = E∗ ⊗min D = CB(E,D) isometrically when E = M∗n (see (2.17)) the last termis the same as ‖

∑ξij ⊗ aij‖C∗<E>⊗maxA. This proves (6.39).

In the situation of Theorem 6.29 (with A = M) the norms in (6.39) are equal to

sup‖∑

uijaij‖ | u ∈ U(Mn(M ′)).

This follows from Theorem 6.20 just like for (6.38).

95

Remark 6.32 (Computing some dec-norms). Assume |I| = n (and hence |I| = n+ 1).In the preceding theorem, consider the case when A = C∗λ(FI) and xi = λFI (gi). Then by (4.27)we have ‖T‖dec = n+ 1, and by (3.7),(4.22) and (3.21) ‖T‖cb = 2

√n, so that ‖T‖dec 6= ‖T‖cb when

n > 1. The same equalities clearly hold if C∗λ(FI) is replaced by MFI .More generally, assume that A = M is a finite von Neumann algebra, as defined in §11.2. Then if(xi) is any family of unitaries in M we have

‖T‖dec = n+ 1.

Indeed, we have clearly∥∥∑

i∈I Ui ⊗ xi∥∥C∗(F)⊗maxM

≥ ‖∑

i∈I x∗i ⊗ xi‖Mop⊗maxM and using the left

and right multiplications L and R on L2(τ) we find

‖∑

i∈Ix∗i ⊗ xi‖Mop⊗maxM ≥

∑i∈I〈1, R(x∗i )L(xi)1〉 = n+ 1.

Actually, the same reasoning shows that for any family of scalars (αi)i∈I we have∥∥∥∑i∈I

αiUi ⊗ xi∥∥∥C∗(F)⊗maxM

≥ ‖∑

i∈Iαix∗i ⊗ xi‖Mop⊗maxM ≥

∑i∈I

αi,

and by the triangle inequality this becomes an equality when αi ≥ 0 for all i ∈ I.

Remark 6.33 (The exceptional case when |I| = 2). The case n = 2 is in sharp contrast with thepreceding remark : Any linear map T : `2∞ → A into an arbitrary C∗-algebra satisfies

‖T‖dec = ‖T‖cb = ‖T‖.

We already observed ‖T‖cb = ‖T‖ in Remark 3.13. Since the C∗-algebra generated by the unit anda single unitary is commutative and hence nuclear, we can replace the max-norm by the min-normin (6.37), then the first equality in (3.7) shows that ‖T‖dec = ‖T‖cb.Remark 6.34. By Proposition 6.7 if M is injective then ‖T‖dec = ‖T‖cb for any n and any T : `n∞ →M . At the end of [104] Haagerup asks whether the converse holds, and even simply for n = 3. Wereturn to this open problem later on in Corollary 23.5 and Remark 23.4.

For the record, we now turn to decomposable multipliers on C∗(G). They turn out to be thesame as the bounded ones, as the next remark shows.

Remark 6.35. [Decomposable multipliers on C∗(G)] Recall that a bounded linear map u : C∗(G)→C∗(G) is called a multiplier if there is a (necessarily bounded) function ϕ : G → C such thatu(UG(t)) = ϕ(t)UG(t). In that case, we claim that u ∈ D(C∗(G), C∗(G)) and ‖u‖dec = ‖u‖.Indeed, assuming ‖u‖ = 1, by Proposition 3.3 we can write ϕ(t) = 〈η, π(t)ξ〉 where π is a unitaryrepresentation on G and ‖η‖ = ‖ξ‖ = 1. Let S1 (resp. S2) be the bounded multiplier mapping onC∗(G) associated to the function ϕ1(t) = 〈η, π(t)η〉 (resp. ϕ2(t) = 〈ξ, π(t)ξ〉). By Proposition 3.3‖S1‖ = ‖S2‖ = 1. Let A = C∗(G). It is easy to check using (6.4) that the linear map from A to

M2(A) that takes a = UG(t) to

(S1(a) u(a)u(a∗)∗ S2(a)

)= (η ξ)

(π(a) π(a)π(a) π(a)

)(ηξ

)⊗ a is c.p. and hence

we have ‖u‖dec ≤ maxj=1,2‖Sj‖ = 1, which proves the claim.

The next two results are the analogue of Proposition 3.25 but with C∗(G) in place of C∗λ(G).

Proposition 6.36. Let G be a discrete group. Let T ∈ D(C∗(G),MG). Define ψT : G→ C by

ψT (t) = 〈δt, T (UG(t))(δe)〉.

Then the linear mapping that takes UG(t) to ψT (t)UG(t) extends to a decomposable multiplier T ∈D(C∗(G), C∗(G)) with ‖T‖ = ‖T‖cb = ‖T‖dec ≤ ‖T‖dec.

96

Proof. Let QG : C∗(G)→MG be the natural ∗-homomorphism taking UG(t) to λG(t). Let

TG ∈ D(C∗(G)⊗max C∗(G),MG ⊗max MG)

denote the extension of T ⊗ QG given by (6.15) so that ‖TG‖dec ≤ ‖T‖dec. Let T = PGTGJGwhere PG and JG are as in Theorem 4.11. The definitions of PG and JG show that T is themultiplier corresponding to ψT and ‖T‖dec ≤ ‖T‖dec by (6.7). By (3.4) we have ‖T‖ = ‖T‖cb, and‖T‖ = ‖T‖dec by Remark 6.35.

Corollary 6.37. In the preceding situation, there is a contractive projection Q from D(C∗(G), C∗(G))onto the subspace formed of all the multipliers in D(C∗(G), C∗(G)).

Proof. Let u ∈ D(C∗(G), C∗(G)), then T = QGu ∈ D(C∗(G),MG) by (6.7). Let Q(u) = T . Then‖Q(u)‖dec ≤ ‖u‖dec by (6.7) and Q(u) = u if u is a multiplier.

6.5 Notes and Remarks

The results of §6.1 on the dec-norm come mainly from [104]. Those of §6.2 come from [208]. Theδ-norm was developed there to present the equivalence nuclear ⇔ CPAP (due to Choi-Effros andKirchberg) in a framework better suited for linear maps on operator spaces. We exploit this in thesequel in §10.2. Lemma 6.28 which appeared in [204] was directly inspired by a previous result ofHaagerup in [104, Lemma 3.5], which was essentially the second equality in (6.38). The examplesof §6.4 are variations suggested by the δ-norm with roots in Haagerup’s ideas in [104]. Haagerupin [104, Prop. 3.4] states and proves Remark 6.33 for maps on `2∞ with values in a von Neumannalgebra.

7 Tensorizing maps and functorial properties

In this chapter, we introduce a major tool to study tensor products. We will try to identify thelinear mappings u : A → B between C∗-algebras that are “tensorizing” meaning by this that, forany other C∗-algebra C, IdC⊗u gives rise to a bounded map that is bounded from C⊗A to C⊗Bwhen the domain and the range are equipped with given C∗-norms, respectively ‖ · ‖α and ‖ · ‖β.In order for this to make sense, of course we need ‖ · ‖α and ‖ · ‖β defined on C ⊗ A and C ⊗ Bfor any C. More generally, we will consider the case of maps u that are defined only on a subspaceE ⊂ A (with E ⊗ C equipped with the induced norm), but for simplicity we will restrict ourselvesto the minimal and maximal C∗-norms.

7.1 (α→ β)-tensorizing linear maps

This is meant as a quick preliminary overview of the topic. Some of the main statements are provedin detail later on in this volume in §10.2.

Definition 7.1. Let A,B be C∗-algebras. Let α, β be one of the symbols min or max. LetE ⊂ A be an operator subspace of a C∗-algebra A. We will say that a linear map u : E → B is(α→ β)-tensorizing if for any C∗-algebra C we have

∀x ∈ C ⊗ E ‖IdC ⊗ u(x)‖β ≤ ‖x‖α.

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When this holds IdC ⊗ u extends to a contraction as indicated in the following diagram.

C ⊗α A∪

C ⊗ E‖.‖αIdC⊗u

−−−−−−−−−→C ⊗β B.

Remark. It should be emphasized that this notion, when α = max depends on the particularembedding E ⊂ A under consideration.

The preceding definition equivalently means that:

(7.1) sup ‖IdC ⊗ u : C ⊗ E‖.‖α → C ⊗β B‖ ≤ 1

where the sup runs over all possible C’s. This can be automatically strengthened to

(7.2) sup ‖IdC ⊗ u : C ⊗ E‖.‖α → C ⊗β B‖cb ≤ 1

Indeed, replacing C by Mn(C) gives control of the cb-norm, and Mn(C)⊗αA = Mn(C⊗αA) wheneither α = min or α = max, and similarly for β.

By Proposition 1.11, the case (min,min) is clear:

Proposition 7.2. The map u is (min→ min)-tensorizing if and only if ‖u‖cb ≤ 1.

Note that for u : E → B to be (min → min)-tensorizing, by Proposition 1.11, it suffices toconsider the min-tensor product with the C∗-algebra C = B.

The case of (max → max) is given by the following remarkable Theorem 7.6 due to Kirchberg[161].

Let us denote by iB : B → B∗∗ the canonical inclusion map of B into B∗∗ viewed as a von Neu-mann algebra as usual. See §26.16 for background on this. Anticipating a little on the subsequentProposition 7.26 and Corollary 7.27, we will need the following preliminary fact on the bidual.

Lemma 7.3. Let B be any C∗-algebra. Then for any C∗-algebra C we have

∀x ∈ C ⊗B ‖x‖C⊗maxB = ‖x‖C⊗maxB∗∗ .

Proof. Let (π1, π2) be a pair of representations with commuting ranges taking values in some B(H)with π1 (resp. π2) defined on C (resp. B). Let π2 : B∗∗ → π1(C)′ is as in §26.16. By (4.6), wehave ‖(π1 · π2)(x)‖B(H) = ‖(π1 · π2)(x)‖B(H) ≤ ‖x‖C⊗maxB∗∗ , whence ‖x‖C⊗maxB ≤ ‖x‖C⊗maxB∗∗ ,and the converse is obvious by (4.6).

Theorem 7.4. The map u is (max → max)-tensorizing if and only if u admits a decomposableextension u : A→ B∗∗ with ‖u‖dec ≤ 1, as in the following commutative diagram.

Au

$$E?

OO

u // BiB // B∗∗

Remark 7.5. We will see in Corollary 7.16 that for u : E → B to be (max → max)-tensorizing itsuffices to consider the max-tensor product with the C∗-algebra C = C .

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Proof of Theorem 7.4. The idea is to apply Theorem 6.20 to the map iBu. To prove the if partassume we have an extension u : A→ B∗∗ with ‖u‖dec ≤ 1. By (6.13), u and a fortiori its restrictioniBu is (max → max)-tensorizing. By Lemma 7.3, the map u itself is (max → max)-tensorizing.Conversely, assume u is (max→ max)-tensorizing. We will use C = M ′ in order to apply Theorem6.20. Let M = B∗∗ viewed as a von Neumann algebra embedded in B(H) for some H, so thatB ⊂ B∗∗ ⊂ B(H). Let u : M ′ ⊗max E → B(H) be as defined in Theorem 6.20. Since (7.1) implies(7.2) we have

‖IdC ⊗ u : C ⊗ E‖.‖max → C ⊗max B‖cb ≤ 1

for all C. Then, choosing C = M ′, we observe that u is the composition of the map IdM ′ ⊗ u withthe ∗-homomorphism σ : M ′ ⊗max B → B(H) defined by σ(c ⊗ b) = cb = bc. This shows that‖u : M ′ ⊗max E → B(H)‖cb ≤ 1. Then Theorem 6.20 applied to the map iBu : E → M = B∗∗

shows that there is an extension u : A→ B∗∗ with ‖u‖dec ≤ 1.

In the particular case when E = A, we must have u = iBu, whence:

Theorem 7.6 ([161]). Let A,B be C∗-algebras and let u : A → B be a linear map. Then u is(max→ max)-tensorizing if and only if iBu : A→ B∗∗ is decomposable with ‖iBu‖dec ≤ 1.

Remark 7.7. It is easy to check that this holds if and only if ‖u∗∗‖D(A∗∗,B∗∗) ≤ 1. Note also that

u∗∗ = ¨iBu with the notation in §26.16.

We state here for future reference a consequence of (6.13):

Corollary 7.8. Any c.p. map u with ‖u‖ ≤ 1 between C∗-algebras is (max→ max)-tensorizing.

Proof. This follows from Corollary 4.18 (and ‖u‖dec = ‖u‖ by (ii) in Proposition 6.6).

Remark 7.9. Let A,B,G be C∗-algebras. Let u : A → B be an (α → β)-tensorizing linear map(where α or β can be either min or max). If u is c.p. then the mapping IdG ⊗ u extends to a c.p.map from G⊗α A to G⊗β B. This extends Corollary 1.26.Indeed, by Corollary 4.18 for any t ∈ G ⊗ A that is of the form t = a∗a with a ∈ G ⊗ A wehave (IdG ⊗ u)(t) ∈ (G ⊗max B)+ and a fortiori (IdG ⊗ u)(t) ∈ (G ⊗β B)+. But since IdG ⊗ u is(α→ β)-bounded and the set of such t’s is clearly dense in G⊗αA, it follows that IdG⊗u extendsto a positive map from G⊗α A to G⊗β B. Replacing G by Mn(G) the assertion follows.

The case (min→ max) is closely related to the notion of nuclearity: a C∗-algebra A is nuclear ifand only if the identity map on A is (min→ max)-tensorizing. Kirchberg and Choi-Effros [154, 45]independently showed that this implies a strong approximation property for A called the CPAP(see Definition 4.8). The following statement for more general linear mappings (in place of IdA)originates in their work. We postpone its proof to §10.2 (see Theorem 10.14).

Theorem 7.10. Let u : E → B be a linear mapping from an operator space to a C∗-algebra. Thefollowing assertions are equivalent.

(i) The map u is (min→ max)-tensorizing.

(ii) There is a net of finite rank maps ui : E → B admitting factorizations through matrix algebrasof the form

Mniwi

!!E

vi==

ui // B

with ‖vi‖cb‖wi‖dec ≤ 1 such that ui = wivi converges pointwise to u.

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(iii) There is a net ui : A→ B of finite rank maps with sup ‖ui‖dec ≤ 1 that tends pointwise to uwhen restricted to E.

In particular when E = A we get (see Corollary 10.16 for a detailed proof):

Theorem 7.11. Let u : A → B be a completely positive and unital linear mapping between twounital C∗-algebras. The following assertions are equivalent.

(i) u is (min→ max)-tensorizing.

(ii) There is a net of finite rank maps (ui) admitting factorizations through matrix algebras of theform

Mniwi

!!A

vi==

ui // B

where vi, wi are c.p. maps with ‖vi‖‖wi‖ ≤ 1 such that ui = wivi converges pointwise to u.

(iii) There is a net ui : A→ B of finite rank c.p. maps that tends pointwise to u.

Lastly, when E = A = B, we obtain the classical characterization of nuclear C∗-algebras:

Corollary 7.12. The following properties of a C∗-algebra A are equivalent.

(i) A is nuclear,

(ii) A has the CPAP, i.e. the identity on A is the pointwise limit of a net of finite rank c.p. maps.

As for the remaining case max→ min, we leave it as an exercise for the reader. The answer is thesame as for min→ min (hint: take C = Mn).

Let us illustrate this with group C∗-algebras:

Corollary 7.13 (Nuclearity versus amenability). The following properties of a discrete group Gare equivalent.

(i) G is amenable,

(ii) C∗λ(G) is nuclear,

(iii) C∗(G) is nuclear,

(iv) The canonical quotient map QG : C∗(G)→ C∗λ(G) is (min→ max)-tensorizing.

(v) The natural ∗-homomorphism QG : C∗(G)→MG is (min→ max)-tensorizing.

Note that the group G is amenable if and only if C∗λ(G) = C∗(G), by Theorem 3.30.

Proof. (i) ⇒ (ii) (resp. (i) ⇒ (iii)) follow from the preceding corollary since by Lemma 3.34 C∗λ(G)(resp. C∗(G)) has the CPAP when G is amenable. Assume (ii). Then we claim QG : C∗(G) →C∗λ(G) is (min → max)-tensorizing; indeed, being a ∗-homomorphism, it is clearly (min → min)-tensorizing (and also (max → max)-tensorizing), so composing QG with IdC∗λ(G) which is (min →max)-tensorizing gives the claim. Thus (ii) ⇒ (iv).

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A similar argument using IdC∗(G) (and (max→ max) for QG) shows (ii) ⇒ (iv).

(iv) ⇒ (v) is trivial. Assume (v). Then for any f ∈ C[G], since QG(UG(t)) = λG(t) we have

‖∑

f(t)λG(t)⊗ λG(t)‖C∗λ(G)⊗maxMG≤ ‖

∑f(t)λG(t)⊗ UG(t)‖C∗λ(G)⊗minC∗(G)

and hence by (4.22) and (4.25) we have

|∑

f(t)| ≤ ‖∑

f(t)λG(t)‖

and G is amenable by Theorem 3.30. This shows (v) ⇒ (i) which completes the proof.

7.2 ‖ ‖max is projective (i.e. exact) but not injective

We first observe that for the algebraic tensor product we have no problem. It is both injective andprojective. In fact if A,B and I ⊂ A are merely vector spaces we have a linear isomorphism

(7.3) (A/I)⊗B = (A⊗B)/(I ⊗B).

WhenA,B are ∗-algebras and I ⊂ A is a self-adjoint ideal, this isomorphism is also a ∗-homomorphism.We start by a basic fact, which will be generalized in Proposition 7.19.

Lemma 7.14. Let I ⊂ A be a (closed, self-adjoint, two-sided) ideal in a C∗-algebra. We have then(isometrically) for any C∗-algebra B

(7.4) I ⊗max B ⊂ A⊗max B.

Equivalently, I ⊗max B can be identified with the closure of I ⊗B in A⊗max B.

Proof. Let (xi) denote the net formed by all xi ∈ I+ such that ‖xi‖ < 1. We view these as ageneralized sequence (i ≤ j means xi ≤ xj). It is well known (see §26.15) that ‖xxi − x‖ → 0and ‖xix − x‖ → 0 for any x ∈ I when i → ∞ (along the net). Since xi ≤

√xi (and

√xi ∈ I+

with ‖√xi‖ < 1), we also have ‖x√xi − x‖ + ‖√xix − x‖ → 0, and by the triangle inequality‖√xix

√xi − x‖ ≤ ‖

√xix√xi − x

√xi‖+ ‖x√xi − x‖ ≤ ‖

√xix− x‖+ ‖x√xi − x‖. Therefore

(7.5) ∀x ∈ I ‖√xix√xi − x‖ → 0.

Let t ∈ I⊗B. Assume ‖t‖A⊗maxB < 1. Let Pi : A→ I be the c.p. map defined by Pi(x) =√xix√xi

for x ∈ A. By (4.30) we have ‖(Pi ⊗ IdB)(t)‖I⊗maxB ≤ ‖t‖A⊗maxB < 1. By (7.5)

‖(Pi ⊗ IdB)(t)− t‖I⊗maxB → 0

and hence ‖t‖I⊗maxB ≤ 1. By homogeneity this shows that ‖t‖I⊗maxB ≤ ‖t‖A⊗maxB, and the reverseinequality is trivial, proving (7.4).

Proposition 7.15 (Exactness of the max-tensor product). Let A,B be C∗-algebras and let I ⊂ Abe a (closed, self-adjoint, two-sided) ideal. We have then (isometrically)

(7.6) (A/I)⊗max B = (A⊗max B)/(I ⊗max B).

In other words the sequence

0 → I ⊗max B → A⊗max B → (A/I)⊗max B → 0

is exact. More precisely, any x in (A/I) ⊗ B with ‖x‖max < 1 admits a lifting x in A ⊗ B suchthat ‖x‖max < 1.

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Proof. To verify (7.6), we use (7.3). Let ρ : A ⊗max B → (A/I) ⊗max B denote the naturalrepresentation (obtained from A → A/I after tensoring with the identity of B). Obviously, ρvanishes on I⊗maxB. Hence denoting by Q : A⊗maxB → (A⊗maxB)/(I⊗maxB) the quotient map,we have a factorization of ρ of the form ρ = πQ where π : (A⊗max B)/(I ⊗max B)→(A/I)⊗max Bis a ∗-homomorphism such that

(7.7) ‖π : (A⊗max B)/(I ⊗max B)→(A/I)⊗max B‖ ≤ 1.

By Lemma 4.27 we have an injective ∗-homomorphism

(A⊗B)/(I ⊗B) ⊂ (A⊗max B)/(I ⊗max B).

Thus the norm of (A ⊗max B)/(I ⊗max B) induces a C∗-norm on (A ⊗ B)/(I ⊗ B), but by (7.3)we may view it as a C∗-norm on (A/I)⊗B. By (7.7) the latter C∗-norm dominates the maximalC∗-norm on (A/I)⊗B, and hence it must coincide with it.Since we know (Lemma 7.14)) that I ⊗maxB is the closure of I ⊗B in A⊗maxB, the last assertionfollows from (4.34).

Corollary 7.16. Let E ⊂ A be an operator subspace of a C∗-algebra A and let u : E → B be alinear map into another C∗-algebra B. The following are equivalent.

(i) The map u is (max→ max)-tensorizing i.e. for any C∗-algebra C we have

∀x ∈ C ⊗ E ‖(IdC ⊗ u)(x)‖C⊗maxB ≤ ‖x‖C⊗maxA.

(ii) The same as (i) holds but restricted to C = C .

Proof. If (ii) holds, we claim that it holds when C = C∗(F) for any free group F. Indeed, since wemay assume x ∈ E1⊗E for some separable subspace E1 ⊂ C∗(F), this is an immediate consequenceof Lemma 3.8. Now, since any unital C is a quotient of C∗(F) for a suitable F (see Proposition 3.39),(i) for such C’s follows from the preceding proposition and Lemma 4.26 applied with α = max.When C is not unital, we may view it as an ideal in its unitization C and by Lemma 7.14 we haveC ⊗max D ⊂ C ⊗max D for all D in particular for both D = A and D = B. Thus (i) for C implies(i) for C. This shows (ii) ⇒ (i).

Proposition 7.15 shows that the maximal tensor product behaves well with respect to quotients.However, in sharp contrast with the minimal norm, it does not behave well at all with respect tosubalgebras, i.e. when D ⊂ A is a C∗-subalgebra of a C∗-algebra and C is another C∗-algebra, the∗-homomorphism C ⊗max D → C ⊗max A is in general NOT injective.

Indeed, if this holds and if the pair (A,C) is nuclear, it follows that the pair (D,C) is nuclear.Taking C = C , this would imply that the WEP is inherited by subalgebras, and hence that anyC∗-subalgebra A ⊂ B(H) is WEP, which is of course absurd.

We will now give an explicit example showing that this fails for the inclusion C∗λ(G) ⊂ B(`2(G))with C = C∗(G) whenever G is a nonamenable discrete group such that (C∗(G), B(H)) is a nuclearpair. With the terminology of §9.4 this means that C = C∗(G) has the LLP. In particular thisholds when G = Fn for n ≥ 2.

Let G be a discrete group and let S ⊂ G be a finite subset and let f : S → R+ be any function.We will draw this from our repertoire in §4.4. Let

x =∑

t∈Sf(t)UG(t)⊗ λG(t) ∈ C∗(G)⊗ C∗λ(G) ⊂ C∗(G)⊗B(`2(G)).

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By (4.27)

‖x‖C∗(G)⊗maxC∗λ(G) =∑

t∈Sf(t),

but since the min and max norms coincide by assumption on C∗(G)⊗B(`2(G)) we have

‖x‖C∗(G)⊗maxB(`2(G)) = ‖x‖C∗(G)⊗minB(`2(G)) = ‖x‖C∗(G)⊗minC∗λ(G),

and hence by (4.22) (i.e. by Fell’s absorption principle)

= ‖∑

t∈Sf(t)λG(t)‖B(`2(G)).

By Kesten’s criterion (see Theorem 3.30), if G is not amenable, there is a finite subset S ⊂ G suchthat |S| > ‖

∑t∈S λG(t)‖B(`2(G)), thus we obtain as announced that if D = C∗λ(G), A = B(`2(G))

and C = C∗(G), the ∗-homomorphism C ⊗max D → C ⊗max A is not injective (since it is notisometric). The same proof works for the inclusion of D = MG in B(`2(G)).

More explicitly, if we consider the inclusion C∗λ(Fn) ⊂ B(`2(Fn)) then we have:

(7.8) ‖I ⊗ I +∑n

1UFn(gj)⊗ λFn(gj)‖C∗(Fn)⊗maxB(`2(Fn)) = 2

√n

but

(7.9) ‖I ⊗ I +∑n

1UFn(gj)⊗ λFn(gj)‖C∗(Fn)⊗maxC∗λ(Fn) = n+ 1

and these are different for n > 1.More generally, using a different reasoning, we have:

Proposition 7.17. Consider D = C∗λ(G) and A = B(`2(G)). If G is not amenable, then theinclusion D ⊂ A is not max-injective (in the sense of Definition 7.18). In particular, this holdswhen G = Fn for n ≥ 2.

Proof. Indeed, assume that D ⊂ A is max-injective. Then by iteration, D ⊗max D → A⊗max D →A⊗max A is isometric. Anticipating a bit, we will see in Theorem 23.7 that, when A = B(`2(G)),the norms of A⊗max A and A⊗min A coincide on the “positive definite cone” formed of the tensorsof the form

∑n1 aj ⊗ aj , and hence by Fell’s principle ‖

∑s∈S λ(s)⊗ λ(s)‖A⊗maxA = ‖

∑s∈S λ(s)⊗

λ(s)‖A⊗minA= ‖

∑s∈S λ(s)‖. But by (4.17) we have ‖

∑s∈S λ(s)⊗ λ(s)‖D⊗maxD = |S|. Therefore,

by Kesten’s criterion (see Theorem 3.30) G is amenable.

7.3 max-injective inclusions

Motivated by the last section, we are naturally led to the following notion.

Definition 7.18. Let A be a C∗-algebra and let D ⊂ A be a C∗-subalgebra. We will say that theinclusion D ⊂ A is max-injective if for any C∗-algebra C, the max-norm on C ⊗D coincides withthe norm induced by C ⊗max A. Equivalently, this means that the map C ⊗max D → C ⊗max A isisometric (or equivalently injective).

As already observed in Proposition 7.17, this does not always hold.

Here are some examples of max-injective inclusions D ⊂ A:

Proposition 7.19. Let A be a C∗-algebra and let D ⊂ A be a C∗-subalgebra. The inclusion D ⊂ Ais max-injective in each of the following cases.

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(i) If there is a c.p. projection P : A→ D with ‖P‖ = 1.

(ii) More generally, if we have “approximate projections”, meaning by this that there is a net ofc.p. mappings Pi : A→ D with ‖Pi‖ ≤ 1 such that Pi(x)→ x ∀x ∈ D.

(iii) If D is a (closed two-sided self-adjoint) ideal of A, we already saw in Lemma 7.14 that D ⊂ Ais max-injective. Actually this still holds if D is merely a hereditary subalgebra of A (i.e. wehave x ∈ A, y ∈ D 0 ≤ x ≤ y ⇒ x ∈ D).

Proof. (i) clearly suffices, since P will be (max → max)-tensorizing by Corollary 7.8. (ii) sufficesfor essentially the same reason: the Pi’s are all (max→ max)-tensorizing. We claim that case (ii)implies case (iii). Indeed, let (Pi) be as in the proof of Lemma 7.14. Note that if x ∈ D+ we have0 ≤ Pi(x) ≤ ‖x‖xi. Thus in both cases (ideal or hereditary) we have Pi(x) ∈ D and we obtain thedesired “approximate projections”, proving the claim.

Remark 7.20. The notion of max-injectivity is relevant to the study of pairs (A,B) of C∗-algebrassuch that A⊗min B = A⊗max B, that we call nuclear pairs in the sequel.For instance, if we assume that D ⊂ A is a max-injective inclusion, then clearly:

(7.10) (A,B) nuclear ⇒ (D,B) nuclear.

Remark 7.21. Let D,C be C∗-algebras. Let A = D ⊗min C and B another C∗-algebra. Then:

(7.11) (A,B) nuclear ⇒ (D,B) nuclear.

Indeed, if C is unital we have an embedding D ' D ⊗ 1 ⊂ A and a unital c.p. projectionP : A → D ⊗ 1 defined by P (d ⊗ c) = d ⊗ (ϕ(c)1), where ϕ is any state on C (recall Remark1.24). By (i) in Proposition 7.19, the inclusion D ⊂ A is max-injective. If C is not unital, thesame conclusion can be obtained using an approximate unit and (ii) in Proposition 7.19. Therefore(7.11) is a special case of (7.10).

Remark 7.22. By Proposition 3.5, for any subgroup Γ ⊂ G of a discrete group G, the inclusionC∗(Γ) ⊂ C∗(G) is max-injective. Thus for any C∗-algebra B we have isometrically C∗(Γ)⊗maxB ⊂C∗(G)⊗maxB. Let F be any free group. For any fixed t ∈ C∗(F)⊗B there is an at most countablygenerated free subgroup Γ ⊂ F (that may depend on t) such that (viewing C∗(Γ) ⊂ C∗(F)) wehave t ∈ C∗(Γ) ⊗ B. The group Γ is isomorphic to Fn for some 1 ≤ n ≤ ∞. Thus the norm of tin C∗(F) ⊗max B can be computed in C∗(Fn) ⊗max B. In fact if there is a copy of F∞ such thatΓ ⊂ F∞ ⊂ F, we may compute the latter norm simply in C∗(F∞)⊗max B.

It is natural to wonder whether in the non-separable case one can compute the norm of the max-tensor product using separable C∗-subalgebras. The next two statements address this question.

Lemma 7.23. Let A,B be C∗-algebras. Let t ∈ A ⊗ B. Then for any ε > 0 there are separableC∗-subalgebras A1 ⊂ A, B1 ⊂ B such that t ∈ A1 ⊗B1 and

(7.12) ‖t‖A1⊗maxB1 ≤ (1 + ε)‖t‖A⊗maxB.

A fortiori t ∈ A1 ⊗B and we have

(7.13) ‖t‖A1⊗maxB ≤ (1 + ε)‖t‖A⊗maxB.

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Proof. A moment of thought shows that (7.13) implies (7.12) by iteration (recall (4.8)). Thusit suffices to find A1 separable for which (7.13) holds. Assume A unital. Then by Proposition3.39 for a suitable free group F there is an onto ∗-homomorphism q : C∗(F) → A. By (7.6) and(4.34) for any ε > 0 there is t ∈ C∗(F) ⊗ B with ‖t‖C∗(F)⊗maxB ≤ (1 + ε)‖t‖A⊗maxB such that

(q ⊗ IdB)(t) = t. We may assume t ∈ E ⊗ B for some separable (and even finite dimensional)subspace E ⊂ C∗(F). By Remark 3.6 there is a separable C∗-subalgebra C ⊂ C∗(F) containing Eand admitting a contractive c.p. projection P : C∗(F) → C. By (i) in Proposition 7.19 we have‖t‖C∗(F)⊗maxB = ‖t‖C⊗maxB. Let A1 = q(C). Then A1 is separable, t = (q ⊗ IdB)(t) ∈ A1 ⊗B andby (say) (4.30)

‖t‖A1⊗maxB ≤ ‖t‖C⊗maxB = ‖t‖C∗(F)⊗maxB ≤ (1 + ε)‖t‖A⊗maxB.

This proves the unital case. The non-unital one follows by a unitization argument.

Proposition 7.24. Let A,B be C∗-algebras. Let E ⊂ A be a separable subspace. Assume thatB is separable. There is a separable C∗-subalgebra A1 such that E ⊂ A1 ⊂ A such that the ∗-homomorphism A1 ⊗max B → A⊗max B is isometric.

Proof. Assume A unital. We may clearly assume that E is a C∗-subalgebra. We first claimthat there is a separable unital C∗-subalgebra E1 such that E ⊂ E1 ⊂ A and such that for anyt ∈ E ⊗ B we have ‖t‖E1⊗maxB = ‖t‖A⊗maxB. Indeed, let tn be a dense sequence in E ⊗ Bwith respect to the norm in E ⊗max B. A fortiori tn is dense in E ⊗ B for the (smaller) norminduced by E1⊗maxB for any E1 ⊃ E. Moreover, we may and do assume that each element in tnappears infinitely many times. By Lemma 7.23 for each n there is a separable unital C∗-subalgebraDn ⊂ A such that tn ∈ Dn ⊗ B and ‖tn‖Dn⊗maxB ≤ (1 + 1/n)‖tn‖A⊗maxB. Let E1 ⊂ A be the C∗-subalgebra generated by ∪n≥1Dn. Since Dn ⊂ E1 we have obviously ‖tn‖E1⊗maxB ≤ ‖tn‖Dn⊗maxB ≤(1+1/n)‖tn‖A⊗maxB for all n ≥ 1, and hence (since each element tn appears with n arbitrary large)we obtain ‖tn‖E1⊗maxB = ‖tn‖A⊗maxB for all n ≥ 1, and by the density of tn this proves the claim.By iteration, this claim gives us a sequence of separable unital C∗-subalgebra En with E0 = E suchthat En−1 ⊂ En ⊂ A and such that for any t ∈ En−1 ⊗ B we have ‖t‖En⊗maxB = ‖t‖A⊗maxB forall n ≥ 1. Then let A1 = ∪n≥1En. Let t ∈ A1 ⊗ B. To show that ‖t‖A1⊗maxB = ‖t‖A⊗maxB wemay assume by density that t ∈ ∪n≥1En ⊗ B or equivalently that t ∈ En ⊗ B for some n ≥ 1.Then ‖t‖En+1⊗maxB = ‖t‖A⊗maxB and again since En+1 ⊂ A1, we have ‖t‖A1⊗maxB ≤ ‖t‖En+1⊗maxB.Thus we conclude ‖t‖A1⊗maxB ≤ ‖t‖A⊗maxB which completes the proof since the reverse inequalityis obvious from the start.

Remark 7.25. In the situation of Proposition 7.24 if A ⊗min B = A ⊗max B then A1 ⊗min B =A1 ⊗max B

We now turn to a very simple but quite useful example: for any C∗-algebra A the inclusionA ⊂ A∗∗ is max-injective (see §26.17 for background on A∗∗).

Proposition 7.26.

(i) For any C∗-algebras A1, A2 we have an isometric embedding

A1 ⊗max A2 → A∗∗1 ⊗max A2.

(ii) More generally, we have an isometric embedding

A1 ⊗max A2 → A∗∗1 ⊗max A∗∗2 .

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Proof. We start by a preliminary observation.If σ1 : A1 → B(H) and σ2 : A2 → B(H) are ∗-homomorphisms with commuting ranges thenσ1 : A∗∗1 → B(H) and σ2 : A∗∗2 → B(H) still have commuting ranges. Indeed, let M1 = σ1(A1)′′

and M2 = σ2(A2)′′. The latter are mutually commuting von Neumann algebras in B(H). Since σ1

is the unique (σ(A∗∗1 , A∗1), σ(M1,M1∗))-continuous extension of σ1 and M1 = σ1(A1)

weak∗(by the

bicommutant theorem 26.46) we have σ1(A1) ⊂M1 and similarly σ2(A2) ⊂M2.For (i) (resp. for (ii)), it clearly suffices to show that if σ1 : A1 → B(H) and σ2 : A2 → B(H) are∗-homomorphisms with commuting ranges, then σ1 and σ2 (resp. σ1 and σ2) still have commutingranges, which is what the preceding observation says. Actually we can also deduce (ii) from (i) byiteration.

In other words (see Definition 7.18), (i) means:

Corollary 7.27. For any C∗-algebra A the inclusion iA : A→ A∗∗ is max-injective.

Remark 7.28. For (min → max) and (max → max)-tensorizing maps in the sense of §7.1, it isworthwhile to record here the following observation: Let u : E → B be a linear map. Then uis (min → max)-tensorizing (resp. (max → max)-tensorizing) if and only if the same is true foriBu : E → B∗∗.Indeed, this is immediate, given the preceding Corollary (applied to B instead of A).

Theorem 7.29. Consider an inclusion D ⊂ A between C∗-algebras, and the bitransposed inclusionD∗∗ ⊂ A∗∗. Let iD : D → D∗∗ denote as before the canonical inclusion. The following propertiesare equivalent:

(i) The inclusion D ⊂ A is max-injective.

(i)’ The map C ⊗max D → C ⊗max A is isometric when C = C .

(ii) There is a contractive map T : A → D∗∗ such that T|D = iD, or equivalently such that thefollowing diagram commutes.

AT

!!D?

OO

iD // D∗∗

(iii) There is a contractive and normal c.p. projection P : A∗∗ → D∗∗.

(iii)’ There is a contractive projection P : A∗∗ → D∗∗.

Proof. The equivalence of (i) and (i)’ follows from Corollary 7.16.The implication (i) ⇒ (ii) can be deduced from Theorem 7.4 (applied with E = D and u = IdD):(i) holds if and only if there is T : A → D∗∗ with ‖T‖dec = 1 extending the inclusion D ⊂ D∗∗.When this holds, a fortiori (ii) holds.Assume (ii). We will use T as defined in §26.16. Then (see Proposition 26.50) P = T : A∗∗ → D∗∗

is a normal projection onto D∗∗ with ‖P‖ = 1. By Theorem 1.45, P is automatically c.p. so that(iii) holds, and (iii) ⇒ (iii)’ is trivial.Assume (iii)’. By Theorem 1.45 again, P is automatically c.p. so that by part (i) in Proposition7.19 the inclusion D∗∗ ⊂ A∗∗ is max-injective. Let C be a C∗-algebra. Let t ∈ C ⊗ D such that‖t‖C⊗maxA ≤ 1. A fortiori we have clearly ‖t‖C⊗maxA∗∗ ≤ 1 and hence ‖t‖C⊗maxD∗∗ ≤ 1 by ourassumption. By Corollary 7.27 (applied to D) we have ‖t‖C⊗maxD = ‖t‖C⊗maxD∗∗ ≤ 1, and hence

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by homogeneity ‖t‖C⊗maxD ≤ ‖t‖C⊗maxA for all t ∈ C ⊗ D. The converse being obvious, thiscompletes the proof that (iii)’ ⇒ (i).The last step can be rephrased more abstractly like this: since the inclusions D ⊂ D∗∗ and A ⊂ A∗∗are max-injective (see Corollary 7.27), it is formally immediate that the max-injectivity of D∗∗ ⊂A∗∗ implies that of D ⊂ A, as can be read on this diagram:

D∗∗ ⊗max C // A∗∗ ⊗max C

D ⊗max C?

OO

// A⊗max C?

OO

Corollary 7.30. The properties in Theorem 7.29 are also equivalent to:

(iv) Any c.p. contraction u : D →M into a von Neumann algebra M extends to a c.p. (complete)contraction u : A→M .

(iv)’ For any von Neumann algebra M , any u ∈ D(D,M) extends to a mapping u ∈ D(A,M) with‖u‖dec = ‖u‖dec, as in the following diagram.

Au

D?

OO

u //M

Proof. Assume (iii) and let u be as in (iv)’. By Lemma 6.9, u ∈ D(D∗∗,M) and ‖u‖dec = ‖u‖dec.Consider uP : A∗∗ → M and let u = (uP )|A : A → M . By (6.7) and by part (i) from Lemma6.5, we have ‖u‖dec ≤ ‖u‖dec‖P‖dec = ‖u‖dec‖P‖ = ‖u‖dec. Thus (iii) ⇒ (iv)’. Using Lemma 1.43the same argument yields (iii) ⇒ (iv). Conversely if we assume either (iv) or (iv)’ and apply it tou = iD : D → D∗∗ we obtain (ii).

Remark 7.31. Any T : A→ D∗∗ satisfying (ii) in Theorem 7.29 is automatically c.p. and completelycontractive. This follows from Tomiyama’s Theorem 1.45 since T is a projection onto D∗∗ andT = T|A.

Remark 7.32. If there is a net of maps Ti : A → D with ‖Ti‖ ≤ 1 such that ‖Ti(x) − x‖ → 0 forany x ∈ D, then the inclusion D ⊂ A is max-injective. Indeed, U being an ultrafilter refining thenet (see §26.4), let T (x) = limU Ti(x) for all x in A with respect to σ(D∗∗, D∗), then T : A→ D∗∗

satisfies (ii) in Theorem 7.29.

The next statement will be very useful when dealing with QWEP C∗-algebras in §9.7.

Theorem 7.33. Let A,B be C∗-algebras with B unital. Let ϕ : A → B be a c.p. map such thatϕ(BA) = BB. Then the restriction to the multiplicative domain

ϕ|Dϕ : Dϕ → B

is a surjective ∗-homomorphism, and moreover the inclusion

Dϕ ⊂ A

is max-injective.

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Proof. Note that for any unitary y in B there is x ∈ BA such that ϕ(x) = y. By the Choi-Cauchy-Schwarz inequality (5.2), we have

1 = ϕ(x)∗ϕ(x) ≤ ϕ(x∗x) ≤ 1

and hence ϕ(x)∗ϕ(x) = ϕ(x∗x). Similarly, ϕ(x)ϕ(x)∗ = ϕ(xx∗), so that x ∈ Dϕ and hence

Dϕ ⊃ x ∈ BA | ϕ(x) ∈ U(B),

where U(B) denotes as usual the set of unitaries in B. It follows that ϕ(Dϕ) ⊃ U(B), so that therestriction

π = ϕ|Dϕ : Dϕ → B

is a surjective ∗-homomorphism. Let C be another C∗-algebra. Let j : Dϕ → A be the inclusion.We will now show that the ∗-homomorphism

jC : C ⊗max Dϕ → C ⊗max A

that extends IdC ⊗ j is injective.Let x ∈ C ⊗max Dϕ be such that jC(x) = 0. We will show that x = 0.Let D0 = ker(π) ⊂ Dϕ. By Remark 5.2 D0 is a hereditary C∗-subalgebra of A (and an ideal inDϕ), and hence D0 ⊂ A is max-injective by Proposition 7.19 (iii). Let us denote

ϕC : C ⊗max A→ C ⊗max B

andπC : C ⊗max Dϕ → C ⊗max B

the natural extensions. Clearly, ϕCjC = πC . Therefore πC(x) = 0. But by the projectivity (orexactness) of max (see (7.6)), we have ker(πC) = C⊗maxD

0 and hence x ∈ C⊗maxD0 ⊂ C⊗maxDϕ.

But since D0 ⊂ A is max-injective, the composition

C ⊗max D0 → C ⊗max Dϕ → C ⊗max A

is injective, and we conclude that x = 0 in C ⊗max Dϕ.

7.4 ‖ ‖min is injective but not projective (i.e. not exact)

In this section, in analogy with what we saw for the maximal tensor products in §7.2, we investigateif or when the canonical identification

(7.14) (A/I)⊗B = (A⊗B)/(I ⊗B).

remains valid, after completion, for the minimal tensor products.By its very definition, the minimal tensor product is obviously injective, even in the operator

space setting (see Remark 1.10). In the C∗-case, if Ej , Gj (j = 1, 2) are C∗-algebras and Ej ⊂ Gj(j = 1, 2) are injective ∗-homomorphisms, then E1 ⊗min E2 ⊂ G1 ⊗min G2 is also an injective∗-homomorphism.

In particular, if I ⊂ A is an ideal so that A/I is a C∗-algebra, we have for any B an embeddingI⊗minB ⊂ A⊗minB, and I⊗minB is an ideal in A⊗minB. It is somewhat natural, in analogy with(7.14), to expect that the quotient (A⊗min B)/(I ⊗min B) should be identifiable with (A/I)⊗minB.However, although it is true for many B’s, it is not so for all B.

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We will now give an explicit example showing that this fails for A = C∗(Fn) whenever n ≥ 2.See §3.1 for notation and background on operator algebras such as C∗(G) and MG associated toa discrete group G. The same argument works for A = C∗(G) assuming G non-amenable, butassuming also that C∗(G) has the LLP and that G is approximately linear (so-called hyperlinear),i.e. that the von Neumann algebra of G, namely MG = λG(G)′′ is QWEP (see Remark 9.68). WhenG = Fn we will see later on in these notes that the latter properties hold (see (9.5) and Theorem12.21).

In analogy with Proposition 7.17, we will prove (the traditional terminology for this would bethat for G = Fn, the algebra C∗(G), viewed as an extension of C∗λ(G), is not “locally split”):

Proposition 7.34. Recall that QG : C∗(G) → C∗λ(G) denotes the canonical quotient map. ForG = Fn with n > 1, if A = B = C∗(G) and I = ker(QG), the natural ∗-homomorphism

(7.15) (A⊗min B)/(I ⊗min B)→ (A/I)⊗min B

is not injective. Equivalently (just exchanging A and B), the homomorphism

(7.16) (B ⊗min A)/(I ⊗min A)→ (B/I)⊗min A

is not injective.

To prove this we need to anticipate slightly: we will review a few facts that will be proved lateron in these notes.

For a group G, we will consider the following property:

Property 7.35. The map QG : C∗(G) → MG that is the same as QG but viewed as taking valuesin MG admits a factorization QG : C∗(G)

w−→B v−→MG where v, w are c.p. maps with ‖v‖cb ≤ 1 and‖w‖cb ≤ 1, and where B is a C∗-algebra such that the pair (B,C∗(G)) is nuclear.

For further reference, we introduce a notion due to Kirchberg [155].

Definition 7.36. A group G is said to have the factorization property (or simply property (F)) iffor any x ∈ C∗(G)⊗ C∗(G) we have

(7.17) ‖[λG.ρG](x)‖B(`2(G)) ≤ ‖x‖C∗(G)⊗minC∗(G),

where [λG.ρG] : C∗(G)⊗C∗(G)→ B(`2(G)) denotes the ∗-homomorphism that takes UG(s)⊗UG(t)to λG(s)ρG(t) (s, t ∈ G).

Remark 7.37. This definition should be compared with (4.18). In particular, (4.18) shows that thefactorization property holds if G is amenable, because C∗(G) is then nuclear.

Lemma 7.38. If G satisfies Property 7.35 then G has the factorization property (7.17).

Proof. Since c.p. maps with cb-norm ≤ 1 tensorize both the minimal and maximal tensor products,the following maps are all of norm 1:

C∗(G)⊗min C∗(G)→ B ⊗min C

∗(G) = B ⊗max C∗(G)→MG ⊗max C

∗(G)→MG ⊗max MG

and the composition is equal to QG ⊗ QG on the algebraic tensor product. Thus

‖QG ⊗ QG : C∗(G)⊗min C∗(G)→MG ⊗max MG‖ ≤ 1.

Then (4.18) yields the conclusion.

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Lemma 7.39. Assume that G satisfies the factorization property (7.17). If (7.15) is injective (orequivalently isometric) with A = B = C∗(G) and I = ker(QG), then G is amenable.

Proof. By (7.17) for any x ∈ C∗(G)⊗ C∗(G) we have

‖[λG.ρG](x)‖B(`2(G)) ≤ ‖x‖C∗(G)⊗minC∗(G).

Let I = ker(QG). Clearly the corresponding mapping x 7→ [λG.ρG](x) vanishes on I ⊗min C∗(G).

Therefore, we can write

‖[λG.ρG](x)‖B(`2(G)) ≤ ‖(QG ⊗ Id)x‖C∗(G)⊗minC∗(G)

I⊗minC∗(G)

.

Therefore, if (7.15) is isometric

‖[λG.ρG](x)‖B(`2(G)) ≤ ‖(QG ⊗ Id)x‖C∗λ(G)⊗minC∗(G) = ‖∑

x(s, t)λG(s)⊗ UG(t)‖C∗λ(G)⊗minC∗(G).

Let S ⊂ G be any finite set. Applying this to x =∑

s∈S UG(s) ⊗ UG(s) so that (QG ⊗ Id)x =∑s∈S λG(s)⊗ UG(s), and using (4.19) we find

|S| ≤ ‖∑

s∈SλG(s)⊗ UG(s)‖

and hence by (4.22) (Fell’s absorption principle)

|S| ≤ ‖∑

s∈SλG(s)‖.

Thus G is amenable by Theorem 3.30.

Proof of Proposition 7.34. We will show in the sequel (see Corollary 12.22) that G = Fn verifiesthe factorization appearing in 7.35 for some B with the WEP. We will also show (see Theorem 9.6or rather Corollary 9.40) that (B,C∗(G)) is a nuclear pair. Thus G = Fn satisfies Property 7.35and consequently also the factorization property (7.17) but is not amenable. By Lemma 7.39, weobtain Proposition 7.34.

More explicitly, viewing C∗λ(Fn) = C∗(Fn)/I we have by (4.22) and the preceding proof

‖1⊗ 1 +∑n

1λFn(gj)⊗ UFn(gj)‖C∗λ(Fn)⊗minC∗(Fn) = 2

√n

but‖1⊗ 1 +

∑n

1λFn(gj)⊗ UFn(gj)‖C∗(Fn)⊗minC

∗(Fn)

I⊗minC∗(Fn)

= n+ 1

and these are different for n > 1.

7.5 min-projective surjections

Motivated by the last section, we are naturally led to the following notion.

Definition 7.40. Let q : A → C be a surjective ∗-homomorphism. Let I = ker(q) so thatC ∼= A/I. We will say that the surjection q is min-projective if for any C∗-algebra B, the min-normon B ⊗C = B ⊗ (A/I) coincides with the norm induced by (B ⊗min A)/(B ⊗min I). Equivalently,this means that the canonical map qB : B ⊗min A→ B ⊗min C (that extends IdB ⊗ q) satisfies

(7.18) ker(qB) = B ⊗min ker(q).

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Thus q : A→ A/I is min-projective if for any B

B ⊗min (A/I) = (B ⊗min A)/(B ⊗min I),

or equivalently (see Remark 1.9)

(7.19) (A/I)⊗min B = (A⊗min B)/(I ⊗min B).

Remark 7.41. In analogy with Remark 7.20, let B be another C∗-algebra. If q : A → C is min-projective, then (7.6) (with the roles of B,A interchanged) shows that

(7.20) (A,B) nuclear ⇒ (C,B) nuclear.

Remark 7.42. Obviously qB(B⊗minI) = 0. Thus we have ‖(B⊗minA)/(B⊗minI)→ B⊗minC‖ ≤ 1.Therefore, q is min-projective if and only if for any B and any t ∈ B ⊗ (A/I) we have conversely

(7.21) ‖t‖(B⊗minA)/(B⊗minI) ≤ ‖t‖B⊗min(A/I).

Or equivalently, for any t ∈ (A/I)⊗B we have

(7.22) ‖t‖(A⊗minB)/(I⊗minB) ≤ ‖t‖(A/I)⊗minB.

As observed in the preceding section, the latter does not always hold. In the subsequent §10.1 wewill study the C∗ algebras B (these are called “exact”) such that (7.18) (or (7.19)) holds for anyquotient map q. But for the moment, we content ourselves with a simple characterization of thequotient maps for which (7.18) holds for any B. We will need the following useful lemma whichrequires a specific notation. Let I ⊂ A be a (closed 2-sided) ideal in a C∗-algebra A. Let E be anoperator space. As in Lemma 4.26 we denote for simplicity

Q[E] =A⊗min E

I ⊗min E.

Then if F is another operator space and if u : E → F is a c.b. map we clearly have a boundedlinear map

u[Q] : Q[E]→ Q[F ]

naturally associated to IdA ⊗ u such that

‖u[Q]‖ ≤ ‖u‖.

Lemma 7.43. If u is an isometry, then u[Q] also is one. In particular, if E ⊂ F then Q[E] ⊂ Q[F ](isometrically).

Proof. For simplicity we assume that E ⊂ F and u is the inclusion map. Then the result is aparticular case of Lemma 4.26 applied for α = min.

Next we show that, just like A→ A/I, the quotient map E ⊗min A→ (E ⊗min A)/(E ⊗min I)takes the closed unit ball to the closed unit ball.

Lemma 7.44. Let I ⊂ A be an ideal in a C∗-algebra A. Consider an operator space E and letq[E] : A ⊗min E → Q[E] denote the quotient map. Then for any y in Q[E], there is an element yin A⊗min E that lifts y (i.e. q[E](y) = y) such that ‖y‖min = ‖y‖Q[E].

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Proof. Choose y0 in A ⊗min E such that q[E](y0) = y. It is easy to check that q[E] has the prop-erties appearing in Lemmas 26.31 and 26.32 suitably modified. Therefore, repeating the argumentappearing before Lemma 26.33, we obtain a Cauchy sequence y0, y1, . . . , yn, . . . in A ⊗min E suchthat q[E](yn) = y for all n and ‖yn‖min → ‖y‖Q[E] when n → ∞. Thus y = lim yn is a lifting withthe same norm as y.

Remark 7.45 (Description of ker(qB)). Let qB : B ⊗min A → B ⊗min A/I be as in (7.18). Lett ∈ B ⊗min A. Then t ∈ ker(qB) if and only if (ξ ⊗ IdA)(t) ∈ I for any ξ ∈ B∗. Indeed, qB(t) = 0is the same as (ξ ⊗ IdA/I)(qB(t)) = 0 for any ξ ∈ B∗ (see Corollary 1.15) and it is immediate thatξ ⊗ IdA/I = (IdC ⊗ q)(ξ ⊗ IdA)(t) and of course we may identify IdC ⊗ q with q.Thus when (7.18) fails there are t’s satisfying this which fail to be in the min-closure of B ⊗ I(although they are in that of B ⊗ A). This shows that the failure of (7.18) is closely related tonontrivial approximation problems. In [247] Tomiyama introduced the related notion of Fubiniproduct. When given operator subspaces Y ⊂ B and X ⊂ A in C∗-algebras, we may consider thesubspace F (Y,X) ⊂ B ⊗min A (called the Fubini product) formed of all those t ∈ B ⊗min A suchthat (ξ ⊗ IdA) ∈ X and (IdB ⊗ η) ∈ Y for any (ξ, η) ∈ B∗ × A∗. Obviously Y ⊗min X ⊂ F (Y,X)but in general the latter is larger. For instance when (7.18) fails, what precedes shows us thatB⊗min I 6= F (B, I). See [247] for more information on this interesting notion that we will not use.

Definition 7.46. Fix a constant c ≥ 0. Let C be a C∗-algebra (or merely an operator space). Letu : C → A/I be a linear map into a quotient C∗-algebra. Let q : A → A/I denote the quotientmap. We will say that u is c-liftable if there is v : C → A with ‖v‖cb ≤ c that lifts u in the sensethat qv = u.We will say that u is locally c-liftable (or admits a local c-lifting) if, for any finite dimensionalsubspace E ⊂ C, the restriction u|E is c-liftable, or more explicitly for any finite dimensional E ⊂ Cthere is vE : E → A with ‖vE‖cb ≤ c such that qvE = u|E .

Remark 7.47. Assume that I is completely complemented in A, by which we mean that there is ac.b. projection P from A onto I. Then it is easy to see that the identity of A/I is c-liftable forc = 1 + ‖P‖cb. We just define v : A/I → A by v(x) = (I − P )(x) where x ∈ A is any lifting ofx ∈ A/I.

See Corollary 9.47 for more on unital c.p. maps that are locally 1-liftable. In that case a unitalc.p. vE can be found.

Proposition 7.48. Let C be any operator space. Fix a constant c ≥ 0. Let u : C → A/I be alinear map into a quotient C∗-algebra. Let q : A → A/I denote the quotient map. The followingare equivalent.

(i) For any C∗-algebra B, u defines a map IdB ⊗ u : B ⊗ C → (B ⊗A)/(B ⊗ I) such that

‖IdB ⊗ u : B ⊗min C → (B ⊗min A)/(B ⊗min I)‖ ≤ c.

(ii) Same as (i) with B = B.

(iii) The map u is locally c-liftable.

(iii)’ The map u is locally (c+ ε)-liftable for any ε > 0.

Proof. (i) ⇒ (ii) is trivial. Assume (ii). Let sE ∈ E∗ ⊗C be the tensor associated to the inclusionmap jE : E ⊂ C of a finite dimensional subspace. We view E∗ ⊂ B completely isometrically (see

112

Theorem 2.15).Then (IdB ⊗ u)(sE) ∈ E∗ ⊗ (A/I) is the tensor associated to u|E , and ‖sE‖min =‖jE‖cb = 1. By (ii) we have

‖(IdB ⊗ u)(sE)‖(B⊗minA)/(B⊗minI) ≤ c‖sE‖min = c.

By Lemma 7.43 we have

‖(IdB ⊗ u)(sE)‖(B⊗minA)/(B⊗minI) = ‖(IdB ⊗ u)(sE)‖(E∗⊗minA)/(E∗⊗minI),

and since E∗ ⊗min F = CB(E,F ) isometrically, we find an isometric identity

(E∗ ⊗min A)/(E∗ ⊗min I) = CB(E,A)/CB(E, I)

and taking Lemma 7.44 into account, we find that there is v ∈ CB(E,A) with ‖v‖cb ≤ c such thatqv = u|E . This shows (ii)⇒ (iii) and (iii)⇒ (iii)’ is trivial. To complete the proof it clearly sufficesto show (iii) ⇒ (i). Assume (iii). Let t ∈ B ⊗ C. Let E ⊂ C be finite dimensional and such thatt ∈ B⊗E. Let vE : E → A with ‖vE‖cb ≤ c lifting u|E : E → A/I. Let t = (IdB⊗vE)(t) ∈ B⊗A.

We have ‖t‖B⊗minA ≤ ‖vE‖cb‖t‖min ≤ c‖t‖min. Moreover, (IdB ⊗ q)(t) = (IdB ⊗ u)(t). By (4.34)applied with α = min, this shows

‖(IdB ⊗ u)(t)‖(B⊗minA)/(B⊗minI) ≤ c‖t‖B⊗minC ,

and the latter means that (i) holds.

Remark 7.49. In the situation of Proposition 7.48, assume that B⊗minA = B⊗maxA (this is whatwe take as definition of the LLP for A in the sequel). In that case the conditions in Proposition7.48 are equivalent to

(ii)’‖IdB ⊗ u : B ⊗min C → B ⊗max (A/I)‖ ≤ c.

Indeed, this follows from (7.6) (but one has to exchange the roles of the letters A and B there).

When u = IdA/I and c = 1, Proposition 7.48 becomes:

Corollary 7.50. Let q : A→ C be a surjective ∗-homomorphism. Let I = ker(q) so that C ∼= A/I.The following are equivalent

(i) q is min-projective (i.e. (7.18) holds for any B).

(ii) We have (7.18) for B = B (i.e. for B = B(`2)).

(iii) For any ε > 0, the identity map IdA/I : A/I → A/I is locally (1 + ε)-liftable.

(iii)’ Same as (iii) with ε = 0.

Remark 7.51. Here to emphasize the analogy injective/projective, we did not conform to the existingterminology: One usually says, using the exact sequence 0 → I → A → A/I → 0, that A isa “locally split extension” (of A/I by I) to express the property (iii) in Corollary 7.50, whichequivalently means that A→ A/I is min-projective.

By Remark 7.47, Corollary 7.50 implies:

Corollary 7.52. If the ideal I is completely complemented in A the quotient map A → A/I ismin-projective.

Remark 7.53. Note that in the subsequent Corollary 9.47, we show that, in the unital case (whenA,C and q are unital), the properties in Corollary 7.50 are equivalent to

(iv) For any finite dimensional operator system E ⊂ C there is a unital c.p. map uE : E → A(with ‖uE‖cb = 1) that lifts q in the sense that quE coincides with the inclusion map E ⊂ C.

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7.6 Generating new C∗-norms from old ones

Following the path Grothendieck opened up for Banach space tensor products in [98], we now deriveseveral additional C∗-tensor products from the minimal and maximal ones using the injective andprojective universality of B(H) and C∗(F) respectively. Kirchberg already followed that same roadfirst in [155], then more systematically in [158].

Let A1, A2, B1, B2, C1, C2 be C∗-algebras. The basic idea is two-fold:(i) If we are given an embedding A1 ⊂ B1 we have a linear embedding A1 ⊗ A2 ⊂ B1 ⊗ A2.

Then given a C∗-norm α on B1 ⊗ A2 its restriction to A1 ⊗ A2 defines an a priori new C∗-normon A1 ⊗ A2, denoted by α1. If α is the min-norm then so is α1 (see Remark 1.10), but when α isthe max-norm the induced norm α1 is in general not the max-norm on A1 ⊗ A2, since by §7.2 themax-norm is not injective. Thus this produces an a priori new C∗-norm on A1 ⊗A2.

(ii) If we are given a surjective ∗-homomorphism q1 : C1 → A1 so that A1 = C1/I1 (I1 =ker(q1)), we have a surjection q1⊗ IdA2 : C1⊗A2 → A1⊗A2. Then given a C∗-norm α on C1⊗A2

we can define a new C∗-norm on A1⊗A2 as the norm induced on A1⊗A2 by the natural C∗-normin (C1⊗αA2)/I1 ⊗A2

α. The resulting C∗-norm on A1⊗A2 is denoted by α1. If α is the max-norm

then by (7.6) so is α1, but when α is the min-norm, α1 is in general not the min-norm on A1⊗A2,since by §7.4 the min-norm is not projective. Thus this produces an a priori new C∗-norm onA1 ⊗A2.One can also apply these constructions to the second factor, and produce in this way another pairof a priori new C∗-norms α2, α

2 on A1 ⊗A2 .In [98] (see also [71]) Grothendieck applied these constructions in the Banach space category

starting from the minimal and maximal norms among what he called the reasonable tensor norms.In the Banach space setting, he denoted by /α and \α (resp. α\ and α/) the analogues of α1 andα1 (resp. α2 and α2). We will now describe a couple of possibilities that this idea offers us whenwe apply it to C∗-norms.

We start by the C∗-norms derived from the (injective) universality of B(H). Assume (as wemay) Aj ⊂ B(Hj) (j = 1, 2). Then the norm induced on A1 ⊗A2 by B(H1)⊗max A2 is a C∗-normon A1 ⊗ A2, that we denote by max1. Let uj : Aj → Bj (j = 1, 2) be c.b. maps. Consideru1 ⊗ u2 : A1 ⊗A2 → B1 ⊗B2. Then

(7.23) ‖u1 ⊗ u2 : A1 ⊗max1 A2 → B1 ⊗max1 B2‖ ≤ ‖u1‖cb‖u2‖dec.

Indeed, this follows from the extension property of c.b. maps (Theorem 1.18) together with (6.15)and (6.9).Using again the extension property, one shows that the max1-norm does not depend on the embed-ding A1 ⊂ B(H1). Moreover, we have

∀t ∈ A1 ⊗A2 ‖t‖max1 = inf ‖t‖B1⊗maxA2

where the inf runs over all C∗-algebras B1 containing A1 as a C∗-subalgebra.The same construction applied to the second factor leads to another C∗-norm on A1⊗A2, that

we denote by max2. By (4.8), we have A1 ⊗max2 A2 ' A2 ⊗max1 A1, so this is not really new.The most interesting case is when we do this operation on both factors: continuing, we are led

to denote by max12 the norm induced on A1⊗A2 by B(H1)⊗maxB(H2). Doing the two operationsin reverse order would lead us to define max21 but this is clearly the same as max12. Thus, tolighten the notation, we denote it in the sequel by ‖ ‖M and the completed tensor product byA1⊗M A2 (see Definition 20.11). It has the supplementary advantage that, like the minimal norm,it yields at the same time a tensor product of operator spaces. Indeed, by the same argument as

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for (7.23), this time we have

(7.24) ‖u1 ⊗ u2 : A1 ⊗M A2 → B1 ⊗M B2‖ ≤ ‖u1‖cb‖u2‖cb,

and the definition of ‖ ‖M makes sense even when A1, A2 are just operator spaces. Moreover, wehave

∀t ∈ A1 ⊗A2 ‖t‖M = inf ‖t‖B1⊗maxB2

where the inf runs over all C∗-algebras Bj containing Aj as a C∗-subalgebra (j = 1, 2), or even justcompletely isometrically.

We now turn to the C∗-norms derived from the (projective) universality of C∗(F).Assume that Aj = Cj/Ij with each Cj of the form Cj = C∗(Gj) for some free group Gj . Let

qj : Cj → Cj/Ij be the quotient map. Let t ∈ A1 ⊗A2. We define

‖t‖min1 = ‖t‖(C1⊗minA2)/(I1⊗minA2).

Clearly this is a C∗-norm on A1⊗A2. Let ut : A∗2 → A1 be the finite rank linear map associated tot. Using the property (2.15) of the dual operator space together with (4.34) we immediately obtainthe following reinterpretation of ‖t‖min1 in terms of c.b. liftings:

(7.25) ‖t‖min1 = inf‖v‖cb | v : A∗2 → C1, weak* continuous, rk(v) <∞, q1v = ut.

Obviously we could do the same on the right hand side to define ⊗min2 but by (4.8) this boilsdown to the same notion. However, if we do it twice, then we obtain a genuinely new tensorproduct. More precisely, using ⊗L for what should be denoted ⊗min12 , we set

‖t‖A1⊗LA2 = ‖t‖(C1⊗minC2)/(I1⊗minC2+C1⊗minI2).

Note A∗2 ⊂ C∗2 . Again we have

(7.26) ‖t‖A1⊗LA2 = inf‖v‖cb | v : C∗2 → C1, weak* continuous, rk(v) <∞, q1v|A∗2 = ut.

We will show later on in Remark 9.45 when discussing the LLP that (7.25) and (7.26) areindependent of the choice of C1, C2 as long as they are of the required form C∗(G) with G a freegroup (or simply as long as they have the LLP).

7.7 Notes and Remarks

This chapter is inspired mainly from Kirchberg’s ideas, but we introduce special terms (such as“max-injective” and “min-projective”) in order to emphasize as much as possible properties of linearmaps in the spirit of operator space theory. We feel some features become much clearer. Thereare analogies with the situation in Banach space theory according to Grothendieck’s viewpoint in[98]. We explain this in §7.6. For more in this direction, see Kirchberg’s presentation of his worksin [158], where he systematically adopts a category theory standpoint. Kirchberg communicatedTheorem 7.6 to the author with permission to include them in [208]. What we call min-projectivesurjection is very much the center of attention in Effros and Haagerup’s paper [77], but they donot give it a name. Proposition 7.48 is essentially there (see [77, Th. 3.2]), except they use finitedimensional operator systems and approximate c.p. liftings, but, by part (ii) in Proposition 9.42,this is equivalent to our formulation with locally liftable maps. Incidentally, Effros and Haagerupmention a 1971 paper by Douglas and Howe [73] on Toeplitz operators as an early source for thefact that the existence of liftings implies that the quotient map is min-projective as in Proposition7.48. The latter is the C∗-algebraic analogue of a well known principle in homological algebra. It isamusing to observe, as a historical curiosity, that in [73, Prop. 2] Douglas and Howe only assumethat the lifting is bounded while their proof uses its complete boundedness; this defect is observedin the later paper [20] (also quoted in [77]) and repaired by invoking [5, Th. 7].

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8 Biduals, injective von Neumann algebras and C∗-norms

In this chapter, we review the main results involving C∗-algebras and their biduals proved in theaftermath of Connes’s breakthroughs [61] on injective von Neumann algebras. In particular, weshow in Theorem 8.16 that the nuclearity of a C∗-algebra A is equivalent to the injectivity of A∗∗.

8.1 Biduals of C∗-algebras

We will use here the basic facts and notation introduced in §26.16: when A is a C∗-algebra and Ma von Neumann one, for all u : A→M we recall that

u = (u∗|M∗)∗ : A∗∗ →M.

The following statement is merely a recapitulation.

Theorem 8.1. Let u : A→M be a linear map from a C∗-algebra to a von Neumann algebra.

(i) If u is a ∗-homomorphism then u : A∗∗ →M is a normal ∗-homomorphism.

(ii) u ∈ CP (A,M)⇒ u ∈ CP (A∗∗,M) and ‖u‖ = ‖u‖.

(iii) u ∈ CB(A,M)⇒ u ∈ CB(A∗∗,M) and ‖u‖cb = ‖u‖cb.

(iv) u ∈ D(A,M)⇒ u ∈ D(A∗∗,M) and ‖u‖dec = ‖u‖dec.

Proof. We recall that, by density, u can be viewed as the unique (σ(A∗∗, A∗), σ(M,M∗))-continuousextension of u. (i) is a well known consequence of the very definition of A∗∗ (see Theorem 26.55).(ii) (resp. (iii)) was proved in Lemma 1.43 (resp. Lemma 1.61) and (iv) in Lemma 6.9.

8.2 The nor-norm and the bin-norm

Let A be a C∗-algebra and M a von Neumann algebra. In this “hybrid” situation, one defines aC∗-norm on A⊗M as follows. For any t ∈ A⊗M we set

(8.1) ‖t‖nor = sup ‖(σ.π)(t)‖

where the sup runs over all H’s and all commuting pairs of ∗-homomorphisms σ : A → B(H),π : M → B(H) with π assumed normal (i.e. continuous with respect to both weak* topologies onM and B(H)). It is easy to check that this is indeed a C∗-norm intermediate between the minimaland maximal norms. We denote by A⊗nor M the corresponding completion.

Remark 8.2. There is obviously a similar definition of the nor-norm on M ⊗ A. In some situation(say if A happens to be a von Neumann algebra too) this may lead to some confusion, so that weshould use a notation that distinguishes both cases (such as nor1 and nor2), but for simplicity weprefer not to do that. In the cases we consider in the sequel there is no risk of confusion.

Remark 8.3. Let A1,M1 be respectively a C∗-algebra and a von Neumann algebra. Let σ : A→ A1

and π : M → M1 be ∗-homomorphisms (resp. isomorphisms). Then if π is normal, the mappingσ⊗ π obviously defines (by density) a ∗-homomorphism (resp. an isomorphism) from A⊗nor M toA1 ⊗nor M

1. See Remark 26.38 for clarification.

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We now turn to a more symmetric situation. Let M,N be von Neumann algebras. On M ⊗None defines the “binormal” norm of t ∈M ⊗N by

(8.2) ‖t‖bin = sup‖(π.σ)(t)‖

where the sup runs over all pairs of normal ∗-homomorphism π : M → B(H) and σ : N → B(H)with commuting ranges.

This is clearly a C∗-norm. We denote by M ⊗bin N the completion.Using normal embeddings M ⊂ B(H1), N ⊂ B(H2), H = H1 ⊗2 H2 and the usual pair

π(x) = x⊗ 1, σ(y) = 1⊗ y, we find for any t ∈M ⊗N

(8.3) ‖t‖min ≤ ‖t‖bin.

When M = A∗∗ and N = B∗∗ are biduals, then we may clearly write (by the extension propertyof biduals, see Theorem 8.1 (i))

(8.4) ‖t‖bin = sup‖(π.σ)(t)‖

where the sup runs over all ∗-homomorphisms π : A→ B(H) and σ : B → B(H) with commutingranges.

Remark 8.4. Let M be a von Neumann algebra and A a C∗-algebra. Then the norm induced onA ⊗M by the bin-norm on A∗∗ ⊗M coincides with the nor-norm on A ⊗M as defined in (8.1).This is easy to check using again part (i) in Theorem 8.1.Moreover, for any t ∈ A∗∗ ⊗M , we have clearly ‖t‖bin ≤ ‖t‖nor (where for the nor-norm we viewA∗∗ as a C∗-algebra).

8.3 Nuclearity and Injective von Neumann algebras

We will need the following basic fact:

Proposition 8.5. A von Neumann algebra M ⊂ B(H) is injective if and only if its commutantM ′ is injective.

The (not so simple) proof is based on the following

Lemma 8.6. Let M ⊂ B(H) and N ⊂ B(H) be isomorphic von Neumann algebras. If N ′ isinjective then M ′ is injective.

Proof. By Theorem 26.61 we may write the isomorphism T : M → N as the product of threeisomorphisms of three different kinds: amplification, compression and spatial. Clearly it suffices tocheck the lemma for each of the three kinds, and this turns out to be an easy exercise that we leaveto the reader.

Proof of Proposition 8.5. By a very well known fact there is a realization of M say ψ : M ⊂ B(H)for which ψ(M) and ψ(M)′ are anti-isomorphic. This is part of what is called the standard formof M (see [102] or [242, p. 151] and the proof of the subsequent Theorem 23.30). Then clearlyψ(M) injective ⇔ ψ(M)′ injective (recall Remarks 2.9 and 2.13). Recall that by Definition 1.44injectivity is stable by completely isometric isomorphisms. Thus if M is injective, so is ψ(M) andhence ψ(M)′ is injective. By Lemma 8.6 it follows that M ′ is injective. Reversing the roles of Mand M ′ gives the converse.

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The fact that M injective ⇔ M ′ injective allows us to prove a simple, but important stabilityproperty of injective von Neumann algebras:

Proposition 8.7. If Ni | i ∈ I is a family, directed by inclusion, of injective von Neumannalgebras in B(H) then their weak* closure M = ∪i∈INi ⊂ B(H) is injective.

Proof. Note that Ni ⊂ Nj is equivalent to N ′j ⊂ N ′i . Therefore the commutants (N ′i) form adecreasing directed family. Let Pi : B(H) → N ′i be a (completely) contractive projection. Let Ube an ultrafilter refining the underlying net formed by (N ′i) (see Remark 26.6). Then the mappingP defined by P (x) = limU Pi(x) (the limit being in the weak* topology of B(H)) is clearly a(completely) contractive projection onto M ′ = ∩i∈IN ′i . Thus M ′, and hence M itself, is injective.

The von Neumann algebras M that can be written as the (weak*) closure M = ∪i∈INi ofan ascending union (directed by inclusion) of finite dimensional von Neumann algebras Ni aresometimes called “hyperfinite”, but, as already emphasized by many authors (including Conneshimself in [61, p. 113]) the term “approximately finite dimensional” (AFD in short) is moreappropriate. Thus the last statement shows that AFD ⇒ injective. The converse (say, when H isseparable) is a celebrated deep result of Connes [61], that we state without proof:

Theorem 8.8 (Finite dimensional approximation of injective von Neumann algebras). Any injec-tive von Neumann algebra M is approximately finite dimensional (AFD).

This major advance led to a number of deep characterizations of injective von Neumann algebrasand nuclear C∗-algebras. Although we do not include the proof of Theorem 8.8, we will try in thissection to include complete proofs of the latter characterizations, for which (unlike for Theorem8.8) reasonably simple proofs are now available. In particular we will now show that injectivity isequivalent to the weak* CPAP, which is a rather natural analogue of the CPAP for von Neumannalgebras. This is often called “semidiscreteness” but we prefer to use a term that emphasizes theanalogy with the CPAP.

Definition 8.9. A von Neumann algebra M has the weak* CPAP (in other words is “semidiscrete”)if the identity on M is the pointwise weak* limit of a net of finite rank normal c.p. maps, i.e. thereis a net of weak* continuous unital c.p. maps ui : M → M of finite rank such that ui(x) → x inthe weak* topology for any x ∈M .

Remark 8.10. When this holds, we may assume that ui = v∗i for some vi : M∗ → M∗, and the net(vi) converges pointwise to the identity of M∗ with respect to σ(M∗,M) (i.e. the weak topologyof M∗). By Mazur’s Theorem 26.9 after passing to convex combinations we may assume that (vi)converges pointwise to the identity of M∗ for the norm topology.

We first give a characterization of injectivity obtained by a simple but very important trickinvolving extensions of maps on the max tensor product; the latter is called “The Trick” in [39].We already used a variant of this idea previously for the extension property in Theorem 6.20, andagain for the characterization of max-injective inclusions in Theorem 7.29. Actually the main pointof the next statement (i.e. the if part) can be deduced from Theorem 6.20 (with E = A) but weprefer to repeat the argument in the present situation, thus avoiding the use of operator modulesand Lemma 6.19.

Proposition 8.11. Let A be a C∗-algebra, π : A→ B(H) a ∗-homomorphism and let M = π(A)′′ ⊂B(H) be the von Neumann algebra it generates. Let π : M ′ ⊗ A→ B(H) be the ∗-homomorphismassociated to the product, so that π(x′ ⊗ a) = x′π(a) for all a ∈ A, x′ ∈M ′. Let

M ′ ⊗max A = M ′ ⊗Amax ⊂ B(H)⊗max A

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be (as in Theorem 6.20) the closure of M ′ ⊗ A in B(H)⊗max A. Then M ′ is injective if and onlyif π extends to a continuous ∗-homomorphism from M ′⊗maxA to B(H) (equivalently π is boundedwith respect to the norm induced by B(H)⊗max A).

Proof. We first treat the case when π and hence π are unital. Assume π continuous on M ′⊗maxA,so that automatically (see Remark 1.6)

‖π : M ′ ⊗max A→ B(H)‖cb = 1.

Let ϕ : B(H) ⊗max A → B(H) be a complete contraction extending π according to Theorem1.18. Since ϕ is unital it is c.p. and its multiplicative domain D obviously includes M ′ ⊗ A. LetP : B(H) → B(H) be defined by P (b) = ϕ(b ⊗ 1) for any b ∈ B(H). Since 1 ⊗ a ∈ D for anya ∈ A, we have (see Corollary 5.3) P (b)π(a) = ϕ((b ⊗ 1)(1 ⊗ a)) = ϕ((1 ⊗ a)(b ⊗ 1)) = π(a)P (b).This shows P (b) ∈ π(A)′ = M ′. Moreover, we have P (m′) = m′ for any m′ ∈ M ′ since ϕ extendsπ. It follows that P is a completely contractive and c.p. projection from B(H) onto M ′; in otherwords M ′ is injective. This proves the “if part” in the unital case.If A is not unital, let (xi) be an approximate unit of A as in §26.15 and let Q be the weak* (orw.o.t.) limit of π(xi) in B(H). Then Qπ(a) = π(a)Q = π(a) for any a ∈ A. It follows thatQ is a (self-adjoint) projection in the center M ∩ M ′ of M . A moment of thought shows thatM ′′ = QMQ⊕B(K) with K = (I −Q)(H). Thus we are reduced to show that QMQ is injective.Then we may as well replace H by Q(H) and Q by I. In that case, if we define P (b) as thew.o.t.-limit of ϕ(b⊗ xi), the same reasoning as for the unital case leads to the desired result.Conversely, if M ′ is injective, there is a (completely) contractive c.p. projection P ′ : B(H)→M ′.By (4.30) ‖P ′ ⊗ IdA : B(H) ⊗max A → M ′ ⊗max A‖ = 1, and by definition of the maximal tensorproduct ‖π : M ′ ⊗max A→ B(H)‖ = 1. Therefore, for any t ∈M ′ ⊗A, since t = (P ′ ⊗ IdA)(t) wehave

‖π(t)‖B(H) = ‖π(P ′ ⊗ IdA)(t)‖B(H) ≤ ‖(P ′ ⊗ IdA)(t)‖M ′⊗maxA ≤ ‖t‖B(H)⊗maxA = ‖t‖M ′⊗maxA.

Theorem 8.12. All injective von Neumann algebras have the weak* CPAP.

Proof. Here we are facing an embarrassing situation. The ingredients for a complete proof arescattered in the sequel in a more general framework emphasizing the WEP instead of injectivity.Rather than move the proof to a much later chapter we choose to give it here with references to thetext coming ahead, hoping for the reader’s indulgence. We believe our (exceptional) choice gives amore focused global picture of injectivity.The first part of the proof is simply a reduction to the case when M admits a finite faithful normaltracial state τ . Indeed, if this case is settled it is very easy to deduce from it the case when M issemifinite, because semifiniteness of M implies (see Remark 11.1) the existence of a net of normalunital c.p. maps on M tending weak* to the identity and each with range in a finite von Neumannsubalgebra of M , which inherits the injectivity of M . Once the semifinite case is settled, Takesaki’sduality theorem 11.3 comes to our rescue and produces the general case, taking Remark 11.4 intoaccount.Thus it suffices to consider a tracial probability space (M, τ) (in the sense of Definition 11.6). IfM is injective, then by (22.15) for any finite set (xj) in M we have

(8.5) ‖∑

xj ⊗ xj‖M⊗maxM= ‖

∑xj ⊗ xj‖M⊗minM

.

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Since left and right multiplication are commuting representations of M on L2(τ) (see §11.3) wealways have

∑‖xj‖22 ≤ ‖

∑xj ⊗ xj‖M⊗maxM

but injectivity implies by (8.5)∑‖xj‖22 ≤ ‖

∑xj ⊗ xj‖M⊗minM

.

We assume M ⊂ B(L2(τ)). By Theorem 11.38 (with π = IdM and E = M) and (i) ⇒ (ii) inTheorem 4.21 there is a net of finite dimensional subspaces Hi ⊂ L2(τ) and Ki ⊂ L2(τ) and statesfi on B(Hi)⊗min B(Ki) such that

(8.6) ∀y, x ∈M τ(y∗x) = limi fi(PHiy|Hi ⊗ PKix|Ki).

Replacing both Hi and Ki by Hi +Ki ⊂ L2(τ) we may assume for simplicity that Ki = Hi. SinceM is dense in L2(τ) by perturbation we may assume that Hi ⊂ M . This gives us the advantage(purely for convenience in the sequel) that there is ci <∞ such that

(8.7) ∀y ∈M ‖PHiy|Hi‖ ≤ ciτ(|y|),

which is easy to deduce from (11.2). We will now describe the state ϕi ∈ (M ⊗max M)∗ defined by

∀x, y ∈M ϕi(y ⊗ x) = fi(PHiy|Hi ⊗ PHix|Hi).

By Theorem 4.16 and the discussion of M∗ ' L1(τ) in §11.2 there is a finite rank map ηi : M →L1(τ) that is c.p. in the sense of (4.28) such that

∀x, y ∈M ϕi(y ⊗ x) = τ(y∗ηi(x)).

The fact that ηi takes its values in M∗ rather that M∗ is due to the fact that ϕi is separately normalwhen considered as a bilinear form on M ×M . Its rank is finite because dim(Hi) <∞ ensures thatthe latter bilinear form is of finite rank. The condition (8.7) gives us |τ(y∗ηi(x))| ≤ ciτ(|y|)‖x‖ fromwhich ‖ηi(x)‖ ≤ ci‖x‖ follows by (11.4). Thus ηi(M) ⊂M ⊂ L1(τ). Since ηi : M → L1(τ) satisfies(4.28), one can check using (11.9) for Mn(M) that ηi ∈ CP (M,M). We now wish to modify ηi tomake it unital. Observe that by (8.6) ηi(1) → 1 for σ(M∗,M). Using Mazur’s Theorem 26.9 wemay pass to convex combinations and assume that ηi(1)→ 1 in norm in M∗. Assume for a momentthat ηi(1) is invertible in M , then we define for x ∈M

ui(x) = ηi(1)−1/2ηi(x)ηi(1)−1/2.

We have for all x, y ∈Mτ(y∗x) = limi τ(ηi(1)1/2y∗ηi(1)1/2ui(x))

but since ηi(1)1/2 → 1 in L2(τ) by the Powers-Størmer inequality (11.38) (as extended to generaltraces by Araki, Connes and Haagerup, see [242, (9) p. 143], see also [134, appendix]) we musthave as well

τ(y∗x) = limi τ(y∗ui(x)).

Thus we have obtained a net of finite rank unital c.p. maps (ui) tending to the identity in theσ(M,M)-sense but since ‖ui‖ = ‖ui(1)‖ = 1, the net being equicontinuous and M dense in M∗,we conclude that ui(x) → x for σ(M,M∗), whence the weak* CPAP. The only drawback is thatwe assumed ηi(1) invertible. This can be fixed easily by replacing ηi by ηi,ε defined by ηi,ε(x) =ηi(x) + ετ(x)1 (x ∈ M) in the preceding reasoning and letting ε > 0 tend to zero as part of ournet. We skip the details.

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Theorem 8.13. The following properties of a von Neumann algebra M ⊂ B(H) are equivalent:

(i) M is injective.

(ii) M has the weak* CPAP.

(iii) For any C∗-algebra A and any t ∈ A⊗M we have ‖t‖nor = ‖t‖min.

(iv) For any von Neumann algebra N and any t ∈ N ⊗M we have ‖t‖bin = ‖t‖min.

Proof. We already know (i) ⇒ (ii) by Theorem 8.12. Assume (ii). Let (ui) be as in Definition 8.9.Let t ∈ A⊗M . By (6.16) we have ‖(IdA ⊗ ui)(t)‖max ≤ ‖t‖min and hence for any commuting pairof ∗-homomorphisms σ : A→ B(H) and π : M → B(H) we have ‖(σ.π)((IdA ⊗ ui)(t))‖ ≤ ‖t‖min.Now if π is assumed normal it is easy to check that (σ.π)((IdA ⊗ ui)(t)) tends weak* to (σ.π)(t)and hence ‖(σ.π)(t)‖ ≤ ‖t‖min. This shows ‖t‖nor ≤ ‖t‖min. The converse inequality is obvious.(iii) ⇒ (iv) is trivial since ‖ · ‖bin ≤ ‖ · ‖nor.Assume (iv) with N = M ′. We claim that M ′ is injective and hence (i) holds by Proposition 8.5.Indeed, (iv) implies that the map π in Proposition 8.11 (with A = M and π : M → B(H) theembedding) is continuous on M ′ ⊗M , with respect to the min-norm, i.e. the norm induced byB(H) ⊗min M ; a fortiori it is continuous with respect to the norm induced by B(H) ⊗max M , soM ′ is injective by Proposition 8.11.

The equivalence between (ii) and (iii) in the next result is somewhat surprising. We state thisfor emphasis.

Theorem 8.14 (A consequence of the weak* CPAP). The following properties of a von Neumannalgebra M ⊂ B(H) are equivalent:

(i) M is injective.

(ii) The product mapping p defined on M ′ ⊗ M by p(x′ ⊗ x) = x′x defines a contractive ∗-homomorphism from M ′ ⊗min M to B(H).

(iii) The product mapping p defines a contractive ∗-homomorphism from M ′ ⊗M equipped withthe norm induced by B(H)⊗max B(H).

Proof. Assume (i). Note that p is clearly contractive on M ′ ⊗bin M , so that (ii) holds by Theorem8.13 applied with N = M ′.(ii)⇒ (iii) is clear since the norm induced on M ′⊗M by B(H)⊗maxB(H) dominates the minimalC∗-norm, i.e. the one of M ′ ⊗min M .Assume (iii). We will apply Proposition 8.11 with A = M . Note that the norm induced onM ′ ⊗M by B(H) ⊗max B(H) is clearly majorized by the norm induced by B(H) ⊗max M . ThusProposition 8.11 shows that M ′ is injective, but since we may exchange the roles of M and M ′, Mis injective.

We now derive the consequences of the preceding (major) theorems for nuclear C∗-algebras.

Corollary 8.15. Let A be a nuclear C∗-algebra. Then for any ∗-homomorphism π : A → B(H)the von Neumann algebra M = π(A)′′ generated by π is injective. In particular, A∗∗ is injective.

Proof. Since A is nuclear, we have an isometric embedding

M ′ ⊗max A ⊂ B(H)⊗max A,

and by definition of M ′ ⊗max A the map π in Proposition 8.11 clearly satisfies ‖π : M ′ ⊗max A →B(H)‖ ≤ 1. Thus M ′ and also M by Proposition 8.5 are injective.

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Actually the converse of the preceding corollary also holds, as the next statement shows.

Theorem 8.16. A C∗-algebra A is nuclear if and only if its bidual A∗∗ is injective.

Proof. Assume A∗∗ injective. By Theorem 8.12 it has the weak* CPAP. By Remark 8.10 there isa net of finite rank maps (vi) on A∗ that are preadjoints of unital c.p. maps and tend pointwiseto the identity on A∗. Let B be any C∗-algebra. Let u : B → A∗ be a c.p. map. Composingwith ui gives us a net of c.p. maps tending pointwise to u. It follows from the description of theset of states on B ⊗max A and B ⊗min A given in §4.5 and §4.6 that they must coincide. Thus weconclude that B ⊗max A = B ⊗min A, which means A is nuclear. The converse was already part ofthe preceding statement.

Corollary 8.17 ([46]). Nuclearity is preserved under quotients.

Proof. Let A/I be a quotient C∗-algebra. By (26.37) we have A∗∗ ' (A/I)∗∗⊕I∗∗. Therefore, A∗∗

injective implies (A/I)∗∗ injective.

Note that there is no known really simple and direct proof of Corollary 8.17.

Corollary 8.18 ([46]). Nuclearity is preserved under “extensions” . This means that if I ⊂ A isan ideal in a C∗-algebra, and if both I and A/I are nuclear, then A is nuclear.

Proof. This is a corollary of Proposition 7.15 (on the exactness of the max-tensor product).This assertion can also be seen as an easy consequence of Theorem 8.16 and the fact that for anyideal I ⊂ A, we have by (26.37) a C∗-isomorphism

A∗∗ ' (A/I)∗∗ ⊕ I∗∗.

Indeed, the latter isomorphism shows that A∗∗ is injective if and only if both (A/I)∗∗ and I∗∗ areinjective.

As we will see in the next chapter, injectivity is equivalent to the WEP for von Neumannalgebras (see Corollary 9.26). Thus the reader will find more conditions equivalent to injectivitythere, as well as in §11.7 on hypertraces.

8.4 Local reflexivity of the maximal tensor product

In the next section we will study local reflexivity. A C∗-algebra A is locally reflexive if for anyB and any t ∈ A∗∗ ⊗ B we have ‖t‖(A⊗minB)∗∗ ≤ ‖t‖A∗∗⊗minB (see what follows for clarification).Equivalently this means that for any finite dimensional operator space E we have CB(E,A)∗∗ =CB(E,A∗∗) isometrically. We will soon show (see Remarks 8.32 and 8.33) that this does not alwayshold, even when E = `n∞ (n > 2). In sharp contrast, we show in the present section that a propertyanalogous to local reflexivity does hold for the max-tensor product, and moreover we always haveD(E,A)∗∗ = D(E,A∗∗) isometrically when E = Mn or E = `n∞ (n ≥ 1).

Let A,B be unital C∗-algebras. We first need to clarify how we embed A∗∗ ⊗ B∗∗ into thebiduals (A⊗max B)∗∗ and (A⊗min B)∗∗.

Let us denote simply by i0 : A ⊗max B → (A ⊗max B)∗∗ (resp. i1 : A ⊗min B → (A ⊗min B)∗∗)the natural inclusion. Define π0 : A → (A ⊗max B)∗∗ (resp. π1 : A → (A ⊗min B)∗∗) and σ0 :B → (A ⊗max B)∗∗ (resp. σ1 : B → (A ⊗min B)∗∗) by π0(a) = i0(a ⊗ 1) (resp. π1(a) = i1(a ⊗ 1))and σ0(b) = i0(1 ⊗ b) (resp. σ1(b) = i1(1 ⊗ b)). Then π0, σ0 (resp. π1, σ1) are ∗-homomorphismswith commuting ranges such that i0 = π0 · σ0 (resp. i1 = π1 · σ1). Actually we can define similar

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pairs (π0, σ0) (resp. (π1, σ1)) in the nonunital case using Remark 4.2 and the observation thatthe universal representations of A ⊗max B and A ⊗min B, being direct sums of cyclic ones, arenondegenerate.

Let q : A⊗max B → A⊗min B be the quotient map. Then q∗∗ : (A⊗max B)∗∗ → (A⊗min B)∗∗

is a normal ∗-homomorphism onto (A⊗min B)∗∗.Note that we have canonical linear embeddings

A∗ ⊗B∗ ⊂ (A⊗max B)∗ (resp. A∗ ⊗B∗ ⊂ (A⊗min B)∗)

that, for any (f, g) ∈ A∗ × B∗, take f ⊗ g to the linear map f ⊗ g : A ⊗max B → C (resp.f ⊗ g : A⊗min B → C).

Proposition 8.19. There are natural inclusions Jmax : A∗∗ ⊗ B∗∗ → (A ⊗max B)∗∗ and Jmin :A∗∗ ⊗B∗∗ → (A⊗min B)∗∗ such that for all (a′′, b′′) ∈ A∗∗ ×B∗∗ and (f, g) ∈ A∗ ×B∗ we have

〈Jmax(a′′ ⊗ b′′), f ⊗ g〉 = a′′(f)b′′(g) = 〈Jmin(a′′ ⊗ b′′), f ⊗ g〉.

Moreover Jmin = q∗∗Jmax.

Proof. Let π0 : A∗∗ → (A⊗max B)∗∗ (resp. π1 : A∗∗ → (A⊗min B)∗∗) and σ0 : B∗∗ → (A⊗max B)∗∗

(resp. σ1 : B∗∗ → (A⊗minB)∗∗) be the normal ∗-homomorphisms extending π0 and σ0 (resp. π1 andσ1), still with commuting ranges. This gives us a ∗-homomorphism Jmax : A∗∗⊗B∗∗ → (A⊗maxB)∗∗

(resp. Jmin : A∗∗ ⊗B∗∗ → (A⊗min B)∗∗) defined by

Jmax = π0 · σ0 and Jmin = π1 · σ1.

Claim: For any t ∈ A∗∗ ⊗B∗∗ and any F ∈ (A⊗min B)∗ we have

(8.8) 〈Jmax(t), q∗(F )〉 = 〈Jmin(t), F 〉.

Moreover if F = f ⊗ g with (f, g) ∈ A∗ ×B∗, then

(8.9) 〈Jmin(t), F 〉 = 〈t, f ⊗ g〉

where the last pairing is the canonical one between A∗∗ ⊗B∗∗ and A∗ ⊗B∗.Proof of the Claim: It suffices to prove (8.8) for any t of the form t = a′′⊗b′′ with (a′′, b′′) ∈ A∗∗×B∗∗.Equivalently, it suffices to prove

(8.10) 〈Jmax(a′′ ⊗ b′′), q∗(F )〉 = 〈Jmin(a′′ ⊗ b′′), F 〉.

It is easy to verify going back to the definitions that, for any fixed F ∈ (A⊗min B)∗, both sides of(8.10) are separately weak* continuous bilinear forms on A∗∗ ×B∗∗, which coincide (and are equalto F ) on A×B. Therefore they coincide on A∗∗×B∗∗. This proves (8.8) and hence Jmin = q∗∗Jmax.Now if F = f ⊗ g with (f, g) ∈ A∗ × B∗, then (a′′, b′′) 7→ 〈t, F 〉 = f(a′′)g(b′′) is also a separatelyweak* continuous bilinear form on A∗∗ × B∗∗ coinciding with the preceding two on A × B. Thisimplies (8.9), completing the proof of the claim. By the second part of Remark 26.1, (8.9) showsthat Jmin is injective, and since Jmin = q∗∗Jmax so is Jmax.

It will be convenient to record here a simple observation.

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Lemma 8.20. Let A,B be C∗-algebras. Let π : A→ B(H) and σ : B → B(H) be representationswith commuting ranges such that π · σ : A⊗B → B(H) extends to a contractive ∗-homomorphismT : A⊗min B → B(H). Then T : (A⊗min B)∗∗ → B(H) satisfies

(8.11) ∀a′′ ∈ A∗∗, b′′ ∈ B∗∗ T (a′′ ⊗ b′′) = π(a′′)σ(b′′),

where the embedding A∗∗ ⊗B∗∗ ⊂ (A⊗min B)∗∗ is implicitly meant to be Jmin.

Proof. Note that by definition the mapping (a′′, b′′) 7→ Jmin(a′′⊗b′′) is separately normal. Thereforeboth sides of (8.11) are separately normal bilinear maps on A∗∗ ×B∗∗. Since they clearly coincideon A×B, and A (resp. B) is weak* dense in A∗∗ (resp. B∗∗), they must coincide on A∗∗×B∗∗.

We now come to the version of local reflexivity satisfied by the maximal tensor product. In thenext two statements, the embedding A∗∗ ⊗B∗∗ ⊂ (A⊗max B)∗∗ is implicitly meant to be Jmax.

Theorem 8.21. For any B and any t ∈ A∗∗ ⊗B

(8.12) ‖t‖(A⊗maxB)∗∗ ≤ ‖t‖A∗∗⊗maxB.

More precisely, we have

(8.13) ‖t‖(A⊗maxB)∗∗ = ‖t‖A∗∗⊗binB∗∗ ≤ ‖t‖A∗∗⊗maxB∗∗ = ‖t‖A∗∗⊗maxB.

We start by proving first a more precise version of (8.13):

Theorem 8.22. The norm induced by (A⊗max B)∗∗ on A∗∗ ⊗B∗∗ coincides with the bin-norm.

Proof. By definition Jmax : A∗∗ ⊗ B∗∗ → (A ⊗max B)∗∗ is of the form Jmax = π0.σ0. From thisfollows, by definition of the bin-norm that for any t ∈ A∗∗ ⊗B∗∗ we have

‖Jmax(t)‖ ≤ ‖t‖bin.

Actually, the reverse inequality also holds. To show this consider an isometric embedding

ϕ : A∗∗ ⊗bin B∗∗ ⊂ B(H)

such that the restriction to each factor is normal. (This obviously exists, just consider the directsum of all π.σ as in (8.4).) Note that, by definition, the bin norm of A∗∗⊗B∗∗ restricted to A⊗Bcoincides with the max-norm of A⊗B. Thus we have a ∗-homomorphism

ψ : A⊗max B → B(H)

obtained by restricting ϕ to A⊗B. But now we have a normal extension ψ : (A⊗maxB)∗∗ → B(H)(with ‖ψ‖ = 1 of course). We claim that

(8.14) ∀a ∈ A∗∗, ∀b ∈ B∗∗, ψJmax(a⊗ b) = ϕ(a⊗ b),

and hence∀t ∈ A∗∗ ⊗B∗∗, ψJmax(t) = ϕ(t).

Indeed, both sides of (8.14) are separately normal bilinear maps on A∗∗ × B∗∗, which coincide onA×B. By weak* density again, (8.14) follows. From this we deduce

∀t ∈ A∗∗ ⊗B∗∗, ‖t‖A∗∗⊗binB∗∗ = ‖ϕ(t)‖ ≤ ‖Jmax(t)‖(A⊗maxB)∗∗ .

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Remark 8.23. We cannot replace the bin-norm by the max-norm in Theorem 8.22. Indeed, considerA = B = K(H) (compact operators). Now A,B are nuclear so A⊗maxB = A⊗minB = K(H⊗2H).Thus (A ⊗max B)∗∗ = B(H ⊗2 H), and A∗∗ = B∗∗ = B(H). The norm induced by (A ⊗max B)∗∗

on B(H) ⊗ B(H) is now the min norm. But it is known (see §18.1) that the min and max normare not equivalent on B(H)⊗B(H).

Proof of Theorem 8.21. By the maximality of the max-norm, (8.12) is clear. Moreover, since theinclusion B ⊂ B∗∗ is max-injective (see Corollary 7.27) we have ‖t‖A∗∗⊗maxB = ‖t‖A∗∗⊗maxB∗∗ forany t ∈ A∗∗ ⊗B and the rest follows by Theorem 8.22.

Lemma 8.24. Let C,A be C∗-algebras. Let u : C → A∗∗ and let (ui) be a net in the unit ball ofD(C,A) such that ui(x) → u(x) with respect to σ(A∗∗, A∗) for any x ∈ C. Then u ∈ D(C,A∗∗)with ‖u‖dec ≤ 1.

Proof. By definition of ‖ui‖dec there are Vi ∈ CP (C,M2(A)) of the form

Vi : x→(Si1(x) ui(x)ui(x

∗)∗ Si2(x)

)with ‖Si1‖ ≤ 1 ‖Si2‖ ≤ 1.

Passing to a subnet we may assume that Si1(x) and Si2(x) are σ(A∗∗, A∗)-convergent for any x ∈ Cto S1(x) ∈ A∗∗ and S2(x) ∈ A∗∗, so that ‖S1‖ ≤ 1, ‖S2‖ ≤ 1. Then the limit V of (Vi) is clearly in

CP (C,M2(A∗∗)). Since V =

(S1(x) u(x)u(x∗)∗ S2(x)

), we have u ∈ D(C,A∗∗) and ‖u‖dec ≤ 1.

In sharp contrast with (8.20), we have

Theorem 8.25. For any n and any C∗-algebra A, we have natural isometric identifications

D(Mn, A∗∗) = D(Mn, A)∗∗ and D(`n∞, A

∗∗) = D(`n∞, A)∗∗.

Proof. Note that the spaces D(Mn, A∗∗) and D(Mn, A)∗∗ are setwise identical. The inclusion

D(Mn, A)∗∗ → D(Mn, A∗∗) has norm ≤ 1 by Lemma 8.24. For the reverse inclusion, we use

the description of the unit ball of D(Mn, A) given in (6.31). Let u be in the open unit ball ofD(Mn, A

∗∗). Define a ∈ Mn(A∗∗) by aij = u(eij). We can find a, b ∈ Mn(A∗∗) such that a = a∗band (by homogeneity) max‖

∑kj a∗kjakj‖1/2, ‖

∑kj b∗kjbkj‖1/2 < 1. Since Mn(A∗∗) = Mn(A)∗∗

(see Proposition 26.58), there are nets (aγ) and (bδ) in Mn(A) that are σ(A∗∗, A∗)-convergent to a

and b and such that max‖∑

kj aγkj∗aγkj‖

1/2, ‖∑

kj bδkj∗bδkj‖1/2 ≤ 1. Let aγ,δij =

∑k a

γki∗bδkj . Let us

now assume that A∗∗ ⊂ B(H) (as a von Neumann subalgebra). We then have for all h′, h ∈ H

limγ limδ〈h′, aγ,δij h〉 =∑

klimγ limδ〈aγkih

′, bδkjh〉 =∑

k〈akih′, bkjh〉 = 〈h′, aijh〉.

Thus limγ limδ aγ,δij = aij in the w.o.t. of B(H). Then since the adjoint of the embedding A∗∗ ⊂

B(H) takes B(H)∗ onto A∗, the set of functionals on A of the form x 7→ 〈h′, xh〉 (h′, h ∈ H)

is total in A∗ and hence since the aγ,δ’s are uniformly bounded we must have limγ limδ aγ,δij =

aij for σ(A∗∗, A∗). This shows ‖u‖D(Mn,A)∗∗ ≤ 1. Thus we conclude that the reverse inclusion:D(Mn, A

∗∗)→ D(Mn, A)∗∗ has norm ≤ 1.If we replace Mn by `n∞ the same proof works using (6.32). Alternatively we can use the realizationof `n∞ as diagonal matrices in Mn and (6.33) to deduce the case of `n∞ from that of Mn.

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8.5 Local reflexivity

Following [77] a C∗-algebra A is called locally reflexive if for any B and any t ∈ A∗∗ ⊗B we have

(8.15) ‖t‖(A⊗minB)∗∗ ≤ ‖t‖A∗∗⊗minB,

or equivalently (see Remark 1.9) for any t ∈ B ⊗A∗∗ we have

(8.16) ‖t‖(B⊗minA)∗∗ ≤ ‖t‖B⊗minA∗∗ .

In (8.15) and throughout this section, we (implicitly) use the map Jmin : A∗∗⊗B∗∗ → (A⊗min B)∗∗

from Proposition 8.19 to view A∗∗ ⊗B∗∗ as included in (A⊗min B)∗∗.In sharp contrast with the Banach space analogue of local reflexivity, briefly described in §26.8(from which the terminology comes) this property does not always hold (see Remark 8.33). It isimplied by “exactness” but the converse is an open problem.

Remark 8.26. [Reversing (8.16) and (8.15)] Let A be an arbitrary C∗-algebra. Then the reverseinequalities to (8.16) or (8.15) hold: for any t ∈ B ⊗A∗∗ we have

(8.17) ‖t‖B⊗minA∗∗ ≤ ‖t‖(B⊗minA)∗∗ .

This is immediate by the minimality of the min−norm among C∗-norms on B ⊗ A∗∗. However,we find it instructive to include a direct “hands on” proof, as follows. Let E be an n-dimensionaloperator space such that t ∈ E⊗A∗∗. The space E⊗minA

∗∗ is isomorphic to [A∗∗]n. The unit ballof the space E ⊗min A

∗∗ is closed for the weak* topology (i.e. the topology induced by σ(A∗∗, A∗))(we leave this as an exercise). Therefore E ⊗min A

∗∗ is isometrically a dual Banach space. LetJ : E⊗minA→ E⊗minA

∗∗ denote the isometric inclusion. Equivalently J = IdE ⊗ iA. Then, withthe notation in (26.32), we have ‖J : [E ⊗min A]∗∗ → E ⊗min A

∗∗‖ ≤ 1, and J is the identity on[E ⊗min A]∗∗ ' E ⊗min A

∗∗ ' [A∗∗]n.

Remark 8.27. Let t ∈ B⊗A∗∗ and let E ⊂ B be finite dimensional such that t ∈ E⊗A∗∗. Clearly,since E ⊗min A

∗∗ ⊂ B ⊗min A∗∗ and [E ⊗min A]∗∗ ⊂ [B ⊗min A]∗∗ are isometric inclusions, (8.16)

holds if and only if for any finite dimensional E ⊂ B and any such t ∈ E ⊗A∗∗ we have

(8.18) ‖t‖[E⊗minA]∗∗ ≤ ‖t‖E⊗minA∗∗ .

Using the identification CB(E∗, A) = E ⊗min A (see §2.4), in which we may exchange the roles ofE and E∗, and the preceding remark we see that A is locally reflexive if and only if for any finitedimensional operator space E we have CB(E,A)∗∗ = CB(E,A∗∗) isometrically.

Let us record this important fact.

Proposition 8.28. A C∗-algebra A is locally reflexive if and only for any u ∈ CB(E,A∗∗)

(8.19) ‖u‖CB(E,A)∗∗ ≤ ‖u‖CB(E,A∗∗),

or more explicitly, any u ∈ BCB(E,A∗∗) is the pointwise-weak* limit of a net in BCB(E,A).

The latter reformulation explains the analogy with (26.10). Actually the reverse of (8.19) alwaysholds by Remark 8.27, and hence we have equality in (8.19). This shows that when A is locallyreflexive, we have an isometric embedding A∗∗ ⊗min B ⊂ [A⊗min B]∗∗ for any B.

Theorem 8.29. Any nuclear C∗-algebra is locally reflexive.

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Proof. Let A,B be C∗-algebras. Assume A nuclear. Then

A⊗min B = A⊗max B ⇒ (A⊗min B)∗∗ = (A⊗max B)∗∗.

By Theorem 8.22 we get ‖A∗∗⊗binB∗∗ → (A⊗maxB)∗∗‖ ≤ 1. By Corollary 8.15 the algebra A∗∗ is

injective and hence A∗∗⊗binB∗∗ = A∗∗⊗minB

∗∗. Thus we obtain ‖A∗∗⊗minB∗∗ → (A⊗minB)∗∗‖ ≤ 1

(this is called property (C) in Remark 8.34), which implies a fortiori the local reflexivity of A.

In the opposite direction, the next statement will allow us to produce explicit examples failinglocal reflexivity.

Proposition 8.30. If a C∗-algebra A is locally reflexive then for any ideal I ⊂ A and any C∗-algebra B we have

B ⊗min (A/I) = (B ⊗min A)/(B ⊗min I).

In other words, the quotient map A→ A/I is min-projective in the sense of Definition 7.40.

Proof. Recall that the canonical map [(B ⊗min A)/(B ⊗min I)] → B ⊗min (A/I) has unit norm.Thus it suffices to prove the same for its inverse. Since B ⊗min I is an ideal in B ⊗min A we havecanonically (see (26.37))

(B ⊗min A)∗∗ ' [(B ⊗min A)/(B ⊗min I)]∗∗ ⊕ (B ⊗min I)∗∗.

Moreover since A∗∗ ' (A/I)∗∗ ⊕ I∗∗ (again by (26.37)) we have an embedding

B ⊗min (A/I)∗∗ ⊂ B ⊗min A∗∗.

By the local reflexivity of A we may write (with maps of unit norm)

B ⊗min (A/I)∗∗ ⊂ B ⊗min A∗∗ → (B ⊗min A)∗∗ → [(B ⊗min A)/(B ⊗min I)]∗∗,

and a fortiori B ⊗min (A/I) → [(B ⊗min A)/(B ⊗min I)]∗∗ has unit norm. But the range of thelatter ∗-homomorphism is included in (B ⊗min A)/(B ⊗min I), therefore we find that the map

B ⊗min (A/I)→ [(B ⊗min A)/(B ⊗min I)]

also has unit norm.

Remark 8.31 (Local reflexivity and injectivity). Let E ⊂ F be an inclusion of operator spaces withE finite dimensional. Assume that a C∗-algebra A has the following extension property: for anyu : E → A there is u : F → A extending u with ‖u‖cb = ‖u‖cb.If A is locally reflexive then A∗∗ has the same property.

Fu

%%E?

OO

u // A ⊂ A∗∗

Indeed, given u ∈ CB(E,A∗∗) we have a net ui ∈ CB(E,A) with ‖ui‖cb ≤ ‖u‖cb tending weak* tou. Then the map u ∈ CB(F,A∗∗) equal to a pointwise-weak* cluster point of the net (ui) is thedesired extension of u.When A is a von Neumann algebra the preceding property holds for all inclusions E ⊂ F withdim(E) <∞ if and only if A is injective. Indeed, this follows by a simple weak* limit argument.

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Remark 8.32 (Local reflexivity is inherited by subalgebras and quotients). Local reflexivity passesto C∗-subalgebras. Indeed, this follows directly from the definition, once one recalls that forany closed subspace Y ⊂ X of a Banach space X we have an isometric canonical embeddingY ∗∗ ⊂ X∗∗ (see Remark 26.54). When A1 ⊂ A is a C∗-subalgebra, we have A∗∗1 ⊂ A∗∗ (completely)isometrically, and the latter embedding realizes A∗∗1 as a von Neumann subalgebra of A∗∗. We alsohave similarly (A1 ⊗min B)∗∗ ⊂ (A⊗min B)∗∗ isometrically. Therefore, if A is locally reflexive, forany t ∈ A∗∗1 ⊗B ⊂ A∗∗ ⊗B we have

‖t‖(A1⊗minB)∗∗ = ‖t‖(A⊗minB)∗∗ ≤ ‖t‖A∗∗⊗minB = ‖t‖A∗∗1 ⊗minB,

and we conclude that A1 is locally reflexive.Thus B(H) must fail local reflexivity, otherwise any C∗-algebra would be locally reflexive. Onequick way to see that B(H) fails local reflexivity is to observe that if it were locally reflexive thenB(H)∗∗ would be injective, and hence B(H) would be nuclear (see Remark 8.31).Local reflexivity passes to quotient C∗-algebras. We briefly sketch the easy argument for this. Letq : A → A/I be the quotient map. By (26.37) we have a ∗-homomorphism r : (A/I)∗∗ → A∗∗

such that q∗∗r = Id(A/I)∗∗ . Let B be another C∗-algebra, we have ‖r ⊗ IdB : (A/I)∗∗ ⊗min

B → A∗∗ ⊗min B‖ = 1 and if A is locally reflexive ‖A∗∗ ⊗min B → (A ⊗min B)∗∗‖ = 1, whileclearly ‖(q ⊗ IdB)∗∗ : (A ⊗min B)∗∗ → ((A/I) ⊗min B)∗∗‖ = 1. By composition it follows that‖(A/I)∗∗ ⊗min B → ((A/I) ⊗min B)∗∗‖ = 1, which means that A/I is locally reflexive. Thus, forsome free group F (resp. for F = F∞) C∗(F) (resp. C ) must fail local reflexivity, otherwise any(resp. any separable) C∗-algebra would be locally reflexive.

Remark 8.33 (C and B fail local reflexivity (quantitative estimate)). By Proposition 7.34 thequotient map C∗(Fn) → C∗λ(Fn) is not min-projective when 1 < n ≤ ∞. By Proposition 8.30 thismeans that C∗(Fn) is not locally reflexive, and since the latter embeds in B, a fortiori B is notlocally reflexive.

More explicitly, let G = Fn, A = C∗(G), H = `2(G) and M = MG ⊂ B(H). Then theextension of λG defines a ∗-homomorphism π : A → M , such that π : A∗∗ → M is a surjectivenormal ∗-homomorphism (see Remark 26.49). Let I = ker(π). Then A∗∗ ' M ⊕ I by (26.24), sothat we have a natural embedding M ⊂ A∗∗, which we denote by Φ : M → A∗∗, that lifts π so thatπΦ = IdM . Let Uj = UG(gj) as usual and U0 = 1. Consider the tensor

t =∑n

0Uj ⊗ Φ(πUj) ∈ A⊗A∗∗.

Then‖t‖A⊗minA∗∗ = 2

√n and ‖t‖(A⊗minA)∗∗ = n+ 1.

Indeed, ‖t‖A⊗minA∗∗ = ‖∑n

0 Uj ⊗ πUj‖A⊗minM = 2√n by (4.27). By Lemma 8.20, the inequality

‖t‖(A⊗minA)∗∗ ≥ n+ 1 follows from the factorization property of the free groups that will be provedlater on in Corollary 12.23. Indeed, let σ : A→ B(H) be the representation associated to ρG. Bythe factorization property (see Definition 7.36) we know that T = σ · π : A ⊗min A → B(H) iscontractive. By Lemma 8.20 we have

T (t) =∑

σ(Uj)π(Φ(π(Uj))) =∑

σ(Uj)π(Uj), and ‖T : (A⊗min A)∗∗ → B(H)‖ ≤ 1.

This gives us n+ 1 = ‖∑σ(Uj)π(Uj)‖ ≤ ‖T (t)‖ ≤ ‖t‖(A⊗minA)∗∗ .

Using (2.14) this can be reformulated using the linear operator u : `n+1∞ → A∗∗ (corresponding to

t) defined by u(ej) = Φ(Uj), as follows

(8.20) ‖u‖CB(`n+1∞ ,A∗∗) = 2

√n and ‖u‖CB(`n+1

∞ ,A)∗∗ = n+ 1.

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This argument shows that if a group G has the factorization property then G is amenable if andonly if C∗(G) is locally reflexive.

Remark 8.34 (Properties C and C ′). The origin of local reflexivity for C∗-algebras lies in [10].There Archbold and Batty introduced two properties that they named C and C ′.A has property C if for any C∗-algebra B we have an isometric embedding

A∗∗ ⊗min B∗∗ ⊂ (A⊗min B)∗∗.

A has property C ′ if for any C∗-algebra B we have an isometric embedding

A⊗min B∗∗ ⊂ (A⊗min B)∗∗.

They showed in [10] that property C implies exactness, as defined in the sequel in §10.1, and it isobvious that C ⇒ C ′. Kirchberg showed later on that C and C ′ are actually equivalent properties,and each is equivalent to exactness. Clearly property C implies local reflexivity but (as we alreadymentioned) the converse remains open. We refer the reader to [208, Ch. 18] for more commentsand to [39, Ch. 9] for a complete proof of the equivalence of C and C ′, which is much more delicatethan the one of C ′ with exactness that we prove later on in Proposition 10.12.

Remark 8.35. The definition of local reflexivity makes sense equally well for operator spaces. Sur-prisingly, it turns out that any predual of a von Neumann algebra is locally reflexive when viewedas an operator space. See [78].

8.6 Notes and Remarks

The results of §8.1 on biduals are all well known facts, while those on decomposable maps are basedon [104]. (6.14) and (6.15) appear in [141].Concerning injectivity in §8.3 again the main references are Connes’s [61], Lance’s [165] and theChoi-Effros papers [45, 46, 47, 48]. The original proof that injective factors on a separable Hilbertspace are approximately finite dimensional (i.e. “hyperfinite”) is an outstanding achievement ofConnes [61].The case of general von Neumann algebras was deduced from Connes’s results by Elliott. See[82, 83, 84, 262] for clarifications on that question.Later on, simpler proofs of Connes’s result that injective implies AFD were given by Uffe Haagerup[105] and Sorin Popa [218]. See also chapter XVI in [242], [39, p. 333] and [4, chap. 11] for morerecent detailed expositions.The proof that injective ⇒ semidiscrete is more accessible. A simpler proof of that implicationappears in S. Wassermann’s [257]. Before Connes’s work and the Choi-Effros papers a numberof implications between injectivity and semidiscreteness (or in other words the weak* CPAP) ap-peared in the Effros-Lance paper [79] which was already circulating as a preprint around 1974. Inparticular, they proved the equivalence of injectivity and semidiscreteness for the von Neumannalgebra of a discrete group. They also proved the equivalence of semidiscreteness with either (iii)or (iv) in Theorem 8.13 and also with (ii) in Theorem 8.14.

Local reflexivity originates in Archbold-Batty’s [10] (see Remark 8.34) where they prove Theo-rem 8.29, but the subject owes a lot to a subsequent paper by Effros and Haagerup [77], and toSimon Wassermann’s work in connection with exactness (see [259] for more references).See [78] for more recent important work on operator space local reflexivity.

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9 Nuclear pairs, WEP, LLP and QWEP

We start by a few general remarks around nuclearity for pairs.

Definition 9.1. A pair of C∗ algebras (A,B) will be called a nuclear pair if

A⊗min B = A⊗max B,

or equivalently if the min- and max-norm are equal on the algebraic tensor product A⊗B.

Remark 9.2. If the min- and max-norm are equivalent on A⊗B, then they automatically are equalby Corollary 26.26.

Remark 9.3. Let A1 ⊂ A and B1 ⊂ B be C∗-subalgebras. In general, the nuclearity of thepair (A,B) does not imply that of (A1, B1). As the sequel will demonstrate, this “defect” is amajor feature of the notion of nuclearity. However, if (A1, B1) admit contractive c.p. projections(conditional expectations) P : A → A1 and Q : B → B1 then (A1, B1) inherits the nuclearity of(A,B). This is an immediate application of Corollary 7.8 (see also Remark 7.20) and Proposition7.19.

More generally, the following holds:

Lemma 9.4. Let A, D be C∗-algebras. We assume that IdD factors through A in a certain “local”sense as follows: For any finite dimensional subspace E ⊂ D and any ε > 0 there is a factorizationE

v→A w→D of the inclusion map with ‖v‖cb‖w‖dec ≤ 1 + ε.Let B be another C∗-algebra. Then, if (A,B) is nuclear, the same is true for (D,B).

Proof. Let x ∈ D ⊗ B. We may assume x ∈ E ⊗ B with dim(E) < ∞. Then, since x =(w ⊗ IdB)(v ⊗ IdB)(x) we have by (6.13)

‖x‖D⊗maxB ≤ ‖w‖dec‖(v⊗IdB)(x)‖A⊗maxB = ‖w‖dec‖(v⊗IdB)(x)‖A⊗minB ≤ ‖v‖cb‖w‖dec‖x‖E⊗minB.

Thus, since ‖x‖E⊗minB = ‖x‖D⊗minB, we obtain ‖x‖D⊗maxB = ‖x‖D⊗minB.

Remark 9.5. Since iD : D → D∗∗ is max-injective, the preceding argument works equally well ifwe only assume that iD (instead of IdD) factors through A in the same local sense as described inLemma 9.4.

Recall that A is called nuclear if (A,B) is nuclear for all B.The basic examples of nuclear C∗-algebras (see §4.2) include all commutative ones, the algebraK(H) of all compact operators on an arbitrary Hilbert space H, C∗(G) for all amenable discretegroups G and the Cuntz algebras.

While the meaning of nuclearity for a C∗-algebra seems by now fairly well understood, it is notso for pairs, as reflected by Kirchberg’s fundamental conjecture from [155] that we discuss in detailin §13.

9.1 The fundamental nuclear pair (C∗(F∞), B(`2))

A large part of the sequel revolves around the two fundamental examples

B = B(`2) and C = C∗(F∞)

Note that these are both universal but in two different ways, injectively for B = B(`2) (by thiswe mean that every separable C∗-algebra embeds in B), projectively for C = C∗(F∞) (by this wemean that every separable unital C∗-algebra is a quotient of C , see Proposition 3.39 for details).

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Kirchberg’s conjecture is simply that the pair (C ,C ) is nuclear. The main goal of these notesis to introduce the reader to the state of the art related to this conjecture, and in particular itsequivalence with a problem posed back in 1976 by Alain Connes [61].

In this context, Kirchberg [155] proved the following striking result. We will give the simplerproof from [204].

Theorem 9.6 (The fundamental pair). The pair (C∗(F∞), B(`2)) = (C ,B) is a nuclear pair, aswell as (C∗(F∞), B(H)) for any H.

The following simple fact is essential for our argument.

Proposition 9.7. Let A,B be two unital C∗-algebras. Let (ui)i∈I be a family of unitary elementsof A generating A as a unital C∗-algebra (i.e. the smallest unital C∗-subalgebra of A containingthem is A itself). Let E ⊂ A be the linear span of (ui)i∈I and 1A. Let T : E → B be a linearoperator such that T (1A) = 1B and taking each ui to a unitary in B. Then ‖T‖cb ≤ 1 suffices toensure that T extends to a (completely) contractive ∗-homomorphism T : A→ B.Moreover, T : A → B is the unique completely contractive (or equivalently the unique unital c.p.)map extending T .Lastly, if T is completely isometric and T (E) generates B (as a C∗-algebra) then T is a ∗-isomorphism from A to B.

Proof. The first variant uses multiplicative domains (see §5.1). Consider B as embedded in B(H).By Arveson’s extension Theorem 1.18, T extends to a complete contraction T : A→ B(H). SinceT is assumed unital, T is unital, and hence completely positive by Corollary 1.52. Now for anyunitary U in the family (ui)i∈I , since T (U) = T (U) is unitary by assumption, we have

ui | i ∈ I ⊂ DT,

and since DT

is a C∗-algebra (see Theorem 5.1) this implies automatically A = DT, so that T is

actually a (contractive) ∗-homomorphism into B(H). Since T (ui) = T (ui) and the ui’s generate A,we must have T (A) ⊂ B. Moreover, any other complete contraction T ′ : A→ B extending T mustbe a ∗-homomorphism equal to T on E, and hence must be equal to T .Lastly, let F = T (E) ⊂ B. Note that T (A) is the C∗-algebra generated by F . Assume T completelyisometric and T (A) = B. Then we can apply the first part of the proof to T−1 : F → A. This givesus a ∗-homomorphism σ : B → A with ‖σ‖ ≤ 1 that is inverse to T and proves the last assertion.This completes the proof.Alternate argument: The reader who so wishes can avoid the use of multiplicative domains byarguing like this: By Theorem 1.22 we can find an embeddingH ⊂ K and a unital ∗-homomorphismπ : A→ B(K) such that T (a) = PHπ(a)|H. Then an elementary argument shows that if a unitaryU on K is such that PHU|H is still unitary, then U must commute with PH. Thus, by our assumptionthis commutation is true for π(ui) and hence for π(A) since the π(ui)’s generate it. This shows thata 7→ PHπ(a)|H (which is the same as T ) is a ∗-homomorphism, necessarily with range in B.

The main idea of our proof of Kirchberg’s Theorem 9.6 is that if E is the linear span of 1 and thefree unitary generators of C∗(F∞), then it suffices to check that the min- and max-norms coincideon E ⊗B(H). More generally, we will prove

Theorem 9.8. Let A1, A2 be unital C∗-algebras. Let (ui)i∈I (resp. (vj)j∈J) be a family of unitaryoperators that generate A1 (resp. A2). Let E1 (resp. E2) be the closed span of (ui)i∈I (resp. (vj)j∈J).Assume 1 ∈ E1 and 1 ∈ E2. Then the following assertions are equivalent:

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(i) The inclusion map E1 ⊗min E2 → A1 ⊗max A2 is completely isometric.

(ii) A1 ⊗min A2 = A1 ⊗max A2.

Proof. The implication (ii) ⇒ (i) is trivial (since ∗-homomorphisms are completely contractive),so we prove only the converse. Assume (i). Let E = E1 ⊗min E2. We view E as a subspace ofA = A1⊗minA2. By (i), we have an inclusion map T : E1⊗minE2 → A1⊗maxA2 with ‖T‖cb ≤ 1. ByProposition 9.7, T extends to a (contractive) ∗-homomorphism T from A1 ⊗min A2 to A1 ⊗max A2.Clearly T must preserve the algebraic tensor products A1⊗1 and 1⊗A2, hence also A1⊗A2. Thuswe obtain (ii).

Remark 9.9. Let us denote by E1⊗1+1⊗E2 the linear subspace e1⊗1+1⊗e2 | e1 ∈ E1, e2 ∈ E2.Then, in the situation of Theorem 9.8, E1⊗1+1⊗E2 generates A1⊗minA2, so that it suffices for theconclusion of Theorem 9.8 to assume that the operator space structures induced on E1⊗1 + 1⊗E2

by the min and max norms coincide.

Proof of Kirchberg’s Theorem 9.6. LetA1 = C∗(F∞), A2 = B(H). We may clearly assume dim(H) =∞. Let U0 = 1 and let (Ui)i≥1 denote the free unitary generators of C = C∗(F∞). We takeE2 = B(H) and let E1 be the linear span of (Ui)i≥0.Consider x ∈ E1 ⊗ E2, with ‖x‖min < 1. By Lemma 3.10 we can write x =

∑i≥0 Ui ⊗ xi with

xi ∈ B(H), (xi)i≥0 finitely supported, admitting a decomposition as xi = aibi with∥∥∑ aia

∗i

∥∥ < 1,∥∥∑ b∗i bi∥∥ < 1, ai, bi ∈ B(H). Now, let π : A1 ⊗max A2 → B(H) be any faithful ∗-homomorphism.

Let π1 = π|A1⊗1 and π2 = π|1⊗A2. We have

π(x) =∑

i≥0π1(Ui)π2(xi) =

∑i≥0

π1(Ui)π2(ai)π2(bi).

Since π1 and π2 have commuting ranges we have π(x) = y with

y =∑

i≥0π2(ai)π1(Ui)π2(bi).

Now by (2.1) from Lemma 2.3 we have

‖y‖ ≤∥∥∥∑

i≥0π2(ai)π2(ai)

∗∥∥∥1/2 ∥∥∥∑

i≥0π2(bi)

∗π2(bi)∥∥∥1/2

< 1.

Thus we conclude that‖x‖max = ‖π(x)‖ < 1.

This shows that the min and max norms coincide on E1 ⊗B(H). But since dim(H) =∞ we haveMn(B(H)) ' B(H) for any n, and hence

Mn(E1 ⊗min B(H)) = E1 ⊗min Mn(B(H)) ' E1 ⊗min B(H),

and alsoMn(A1 ⊗max B(H)) = A1 ⊗max Mn(B(H)) ' A1 ⊗max B(H),

therefore the latter coincidence of norms “automatically” implies that the inclusion

E1 ⊗min B(H)→ A1 ⊗max B(H)

is completely isometric. In other words, the operator space structures associated to the min andmax norms coincide. We may clearly replace E1 by its closure. Thus, the proof is concluded byTheorem 9.8 (here E2 = A2 = B(H)).

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As a complement to his fundamental Theorem 9.6 Kirchberg observed the following generalphenomenon.

Theorem 9.10. Let F be any free group and M any von Neumann algebra. Then ‖ · ‖max = ‖ · ‖nor

on C∗(F)⊗M .

Proof. Assume F = FI . Let E = span[Ui | i ∈ I] with Ui | i ∈ I as in Lemma 6.28. Let us denoteby E ⊗max M (resp. E ⊗nor M)) the operator space generated by E ⊗M in C∗(F)⊗max M (resp.C∗(F)⊗nor M). By Proposition 9.7 it suffices to prove that the natural mapping E ⊗max M →E ⊗nor M is completely isometric. Since we may replace M by Mn(M) to pass from isometric tocompletely isometric (we skip the easy details for this point), it suffices to show that ‖t‖max = ‖t‖nor

for any t ∈ E⊗M (the max- and nor-norms being the ones induced on E⊗M by those of C∗(F)⊗M).Let π : M → B(H) be a normal embedding with infinite multiplicity (i.e. H = `2 ⊗2 K andπ(·) = Id`2 ⊗ σ(·) where σ : M → B(K) embeds M as a von Neumann subalgebra). Then (6.38)means that for any t ∈ E ⊗M

(9.1) ‖t‖max = supσ ‖(σ.π)(t)‖

where the sup runs over all ∗-homomorphisms σ : C∗(F)→M ′, and hence ‖t‖max ≤ ‖t‖nor. Bymaximality ‖t‖max = ‖t‖nor. The proof actually shows that (9.1) holds for all t ∈ C∗(F)⊗M .

In the same direction, the following variant will be useful.

Theorem 9.11. Let (Ai)i∈I be a family of C∗-algebras and let A = (⊕∑

i∈I Ai)∞. Let t ∈ C ⊗A.For each i ∈ I let ti = (IdC ⊗pi)(t) ∈ C ⊗Ai, where pi : A→ Ai is the coordinate projection. Then

(9.2) ‖t‖C⊗maxA = supi∈I ‖ti‖C⊗maxAi .

Proof. By Proposition 9.7 it suffices to check that this holds for any t ∈ E ⊗ A where E is thelinear span of the unitary generators (Uj) and the unit. (Indeed, one can replace A by Mn(A) andobserve that Mn(A) ' (⊕

∑i∈IMn(Ai))∞.) Then we may as well assume that t ∈ E ⊗ A with

E = span[I, U1, · · · , UN−1]. Let u : `N∞ → A (resp. ui : `N∞ → Ai) be the linear map associated to t(resp. ti). Then (6.37) shows us that (9.2) is equivalent to

‖u‖dec = supi∈I ‖ui‖dec,

which we observed in (6.10).

For our exposition, it will be convenient to adopt the following definitions (equivalent to themore standard ones by [155]).

Definition 9.12. Let A be a C∗-algebra.We say that A has the WEP (or is WEP) if (A,C ) is a nuclear pair.We say that A has the LLP (or is LLP) if (A,B) is a nuclear pair.We say that A is QWEP if it is a quotient (by a closed, self-adjoint, 2-sided ideal) of a WEPC∗-algebra.

Here WEP stands for weak expectation property (and LLP for local lifting property).

Remark 9.13. If A has the WEP (resp. LLP) then any C∗-subalgebra D ⊂ A such that the inclusionD ⊂ A is max-injective has the WEP (resp. LLP) by (7.10).

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Remark 9.14. By Proposition 3.5, for any subgroup Γ ⊂ G of a discrete group G, the inclusionC∗(Γ) ⊂ C∗(G) is max-injective. Therefore, for any given B, if the pair (C∗(G), B) is nuclear, thenthe same is true for the pair (C∗(Γ), B).For instance, it is well known that F∞ embeds as a subgroup in Fn for any 1 < n <∞. Therefore:

(9.3) (C∗(F∞), B) is nuclear⇔ (C∗(Fn), B) is nuclear.

In particular, C∗(Fn) is LLP for any 1 ≤ n ≤ ∞.Let F be any free group. By Lemmas 3.8 and 9.4 we can replace Fn by F in (9.3):

(9.4) (C∗(F∞), B) is nuclear⇔ (C∗(Fn), B) is nuclear⇔ (C∗(F), B) is nuclear for any F.

In particular:

(9.5) C∗(F) is LLP for any free group F.

Remark 9.15. Let A,B,C be C∗-algebras. Assume that A is nuclear. If the pair (B,C) is nuclearthen the pair (A⊗min B,C) is also nuclear. We leave the proof as an (easy) exercise (see (4.9)). Inparticular, if B has the WEP (resp. LLP) then the same is true of A⊗min B.

Remark 9.16. If A is WEP, LLP (or QWEP) then the same is true of Aop or A. This can bededuced easily from the fact that Aop ' A (or A ' A) when A = C and A = B (see Remarks 3.7and 2.10). Moreover, by (4.10) the properties WEP, LLP (or QWEP) are stable under direct sum.

9.2 C∗(F) is residually finite dimensional

A C∗-algebra is called residually finite dimensional (RFD) if for any x ∈ A with x 6= 0 there is a∗-homomorphism π : A→ B(H) with dim(H) <∞ such that π(x) 6= 0.It is easy to see that this holds if and only if there is a family of finite dimensional ∗-homomorphisms

πi : A→Mn(i) i ∈ I, n(i) <∞

such that their direct sum

(9.6) ⊕i∈Iπi : A→ (⊕∑

i∈IMn(i))∞

is an embedding. Equivalently A is RFD if and only if the ∗-homomorphism that is the direct sum,over all n ≥ 1, of all ∗-homomorphisms π : A→Mn is injective.

Remark 9.17. If A is separable (and RFD), since there is a countable subset dense in the unitsphere, we can always find a countable family (Mn(i))i∈I for which (9.6) is an embedding.

In this section we prove the following result (due to Choi).

Theorem 9.18. Let G be any free group with free generators gi | i ∈ I. Then C∗(G) is residuallyfinite dimensional.

Proof. Let π | π ∈ G0 denote the collection of all the finite dimensional unitary representationsof G (without repetitions). We define

∀ t ∈ G σ(t) = ⊕π(t) | π ∈ G0.

Clearly, σ extends to a (contractive) ∗-homomorphism

u : C∗(G)→ C∗σ(G) ⊂ (⊕∑

π∈G0

B(Hπ))∞,

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taking UG(t) to σ(t). Let E ⊂ C∗(G) be the linear span of 1 and UG(gi) | i ∈ I. To prove thetheorem, we will show that u is isometric. By Proposition 9.7 it suffices to show that u|E : E →C∗σ(G) is completely isometric. Let I = 0 ∪ I (disjoint union) and let xi | i ∈ I be any finitelysupported family in B(H) (with H arbitrary). Then a typical element of B(H)⊗E is of the form

x = x0 ⊗ I +∑

i∈Ixi ⊗ UG(gi),

and we have‖x‖B(H)⊗minE = sup ‖x0 ⊗ I +

∑i∈I

xi ⊗ ui‖,

where the sup runs over all possible families ui | i ∈ I of unitaries (including infinite dimensionalones). But we saw in (3.11) that this supremum remains the same if we let it run over all possiblefamilies of finite dimensional unitaries. Thus (3.11) tells us that

‖x‖B(H)⊗minE = ‖[IdB(H) ⊗ u](x)‖B(H)⊗minC∗σ(G),

which shows that the restriction of u to E is completely isometric. By Proposition 9.7, u gives usan embedding of C∗(G) into (⊕

∑π∈G0

B(Hπ))∞.

Remark 9.19. If A,B are residually finite dimensional C∗-algebras, then A⊗minB is residually finitedimensional. Indeed, if A ⊂ (⊕

∑i∈IMn(i))∞ and B ⊂ (⊕

∑j∈JMm(j))∞ then

A⊗min B ⊂ (⊕∑

(i,j)∈I×JMn(i) ⊗min Mm(j))∞ = (⊕

∑(i,j)∈I×J

Mn(i)m(j))∞.

Let us say for short that a state f on a C∗-algebra A is “a finite dimensional state” (resp. “afinite dimensional vector state”) if there is a ∗-homomorphism π : A→ B(H) on a finite dimensionalHilbert space H and a state (resp. a vector state) F on B(H) such that f(a) = F (π(a)) for alla ∈ A.Using multiplicity, we observe that any finite dimensional state is actually a finite dimensional vectorstate: indeed, any state f on B(H) with dim(H) <∞ is of the form f(x) = tr(a∗xa) = 〈a, xa〉S2(H)

for some unit vector a ∈ S2(H), and hence it becomes a vector state when we embed B(H) inB(S2(H)) by left multiplication. Thus, for later reference, we state the next result only for vectorstates.

Proposition 9.20. A C∗-algebra A is residually finite dimensional if and only if any state on Ais the pointwise limit of a net of finite dimensional vector states.In particular this holds for the states on C ⊗min C .

Proof. Assume A RFD, so that (9.6) is an embedding. Let Hi = `n(i)2 and H = ⊕i∈IHi. Any state

f on A extends to a state on B(H), which is a limit of normal states on B(H). The latter statesare limits of states fγ in the convex hull of those of the form a 7→ 〈ξ, π(a)ξ〉 where ξ is a unit vectorin H, which, after truncation and renormalization, we may assume to be all in ⊕i∈I(γ)Hi for somefinite subset I(γ) ⊂ I. Then fγ is in the convex hull of states of the form

a 7→ 〈ξ, (⊕i∈I(γ)πi(a))ξ〉.

Since dim(⊕i∈I(γ)Hi) <∞, each fγ is a finite dimensional state. By the preceding observation, thisproves the only if part.Conversely, for any a ∈ A we have ‖a‖2 = ‖a∗a‖ = sup f(a∗a) where the sup is over all states. Ifany such f is the pointwise limit of finite dimensional states we can restrict the sup to the latter,and then we find ‖a‖2 ≤ sup ‖π(a∗a)‖ where the sup runs over all ∗-homomorphisms π : A→ B(H)with H finite dimensional. Since the reverse inequality is obvious, we conclude that A is RFD.The last assertion follows from Theorem 9.18 and Remark 9.19.

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9.3 WEP (Weak Expectation Property)

We defined the WEP for A by the equality A ⊗min C = A ⊗max C . We will now see that it isequivalent to a weak form of extension property (a sort of weakening of injectivity), which is theoriginal and more traditional definition of the WEP. Let A ⊂ B(H) be a C∗-subalgebra. If A isinjective, there is a completely contractive projection P : B(H) → A, satisfying the properties of

a conditional expectation by Theorem 1.45. Recall that the weak* closure Aweak∗

of A in B(H),

is equal to A′′ by Theorem 26.46. A unital c.p. mapping T : B(H) → Aweak∗

is called a weakexpectation if T (a) = a for any a ∈ A. This concept goes back to Lance [165]. We will show that

the WEP is equivalent to the existence of a weak expectation T : B(H) → π(A)weak∗

= π(A)′′ forany H and any embedding π : A → B(H) (see Remark 9.23). But for our broader framework, itwill be convenient to enlarge Lance’s concept, as follows.

Definition 9.21. Let A ⊂ B be a C∗-subalgebra of another one. A linear mapping V : B → A∗∗

will be called a generalized weak expectation if ‖V ‖ ≤ 1 and V (a) = a for any a ∈ A.

If V (B) ⊂ A then V is a conditional expectation in the usual sense as in Theorem 1.45.In Theorem 7.29 we already gave an important characterization of the inclusions that admit ageneralized weak expectation, as those such that A ⊂ B is max-injective.

We will show that the WEP of A is equivalent to the existence of a generalized weak expectationV : B → A∗∗ whenever A embeds in B. Indeed, this reduces to the case B = B(H) treated inTheorem 9.31. Note that if the embedding A ⊂ B(H) is the universal representation of A then

π(A)weak∗

= A∗∗ (see §26.16) and hence the generalized notion of weak expectation coincides inthis case with Lance’s original one. See Remark 9.32 for more on generalized weak expectations.

Theorem 9.22. Let A ⊂ B(H) be a C∗-algebra. The following are equivalent.

(i) A has the WEP (i.e. (A,C ) is a nuclear pair).

(ii) The inclusion A ⊂ B(H) is max-injective.

(iii) Any ∗-homomorphism u : A → M into a von Neumann algebra M extends to a completelypositive and (completely) contractive mapping from B(H) to M .

(iii)’ Any ∗-homomorphism u : A→M into a von Neumann algebra M factors completely positivelyand (completely) contractively through B(H) for some H.

(iv) The inclusion iA : A → A∗∗ factors completely positively and (completely) contractivelythrough B(H) for some H.

Proof. (i) ⇒ (ii). Assume (i). Then

A⊗max C = A⊗min C ⊂ B(H)⊗min C = B(H)⊗max C

where the last equality is from Theorem 9.6. By (i) ⇔ (i)’ in Theorem 7.29, (ii) holds.Assume (ii). Then (iii) holds by Corollary 7.30.(iii) ⇒ (iii)’ is obvious, and using u = iA we see that (iii)’⇒(iv).Assume (iv). Consider a completely positive and (completely) contractive factorization

iA : A→ B(H)→ A∗∗.

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By Theorem 9.6 (recalling that by Corollary 7.8 contractive c.p. maps are (max→ max)-tensorizing)we find a contractive factorization

iA ⊗ IdC : A⊗min C → B(H)⊗min C = B(H)⊗max C → A∗∗ ⊗max C

and since iA is max-injective we conclude that A ⊗min C = A ⊗max C . In other words we obtain(i).

Remark 9.23 (On Lance’s WEP). Lance’s definition of the WEP for a C∗-algebra is different buteasily seen to be equivalent to ours. Lance [165] says that a ∗-homomorphism π : A → B(H) has

the WEP if the von Neumann algebra it generates, i.e. the weak* closure π(A)weak∗

, admits a weakexpectation. He then says that A has the WEP if every faithful π has the WEP. We claim thatLance’s WEP is the same as our WEP in Theorem 9.22 (i). Indeed, by (i)⇒ (iii) in Theorem 9.22,A has Lance’s WEP if it has our WEP (because if A has our WEP so does π(A) for any faithful π).Conversely, Lance’s WEP applied to the embedding π : A ⊂ A∗∗ ⊂ B(H) (assuming A∗∗ embeddedas a von Neumann algebra in B(H)) implies the existence of a weak expectation from B(H) to A∗∗

that is equal to π on A and hence by (iv) ⇒ (i) in Theorem 9.22 that A has our WEP.

Using the extension properties of B(H) described in Theorems 1.18 and 1.39, the following isan immediate consequence of (iv):

Corollary 9.24. If A is WEP then it has the following weak forms of injectivity into the bidual:For any operator space X and any subspace E ⊂ X, any u ∈ CB(E,A) admits an “extension”u ∈ CB(X,A∗∗) with ‖u‖cb = ‖u‖cb such that u|E = iAu.If E,X are operator systems, then any u ∈ CP (E,A) admits an “extension” u ∈ CP (X,A∗∗) with‖u‖ = ‖u‖ such that u|E = iAu.

Remark 9.25. It is obvious that the second property in Corollary 9.24 characterizes WEP: justsubstitute to E ⊂ X the inclusion jA : A→ B(H). That the first one also does is less obvious, butit will be shown by Theorem 23.7.

Corollary 9.26. A von Neumann algebra M is injective if and only if it has the WEP.

Proof. If M has the WEP, then the identity of M factors through some B(H) as in (iii) in Theorem9.22 (take A = M and u = IdM ). Therefore M is injective. Conversely, if M is injective the identityof M factors completely positively and (completely) contractively through B(H), so (iv) in Theorem9.22 (with A = M) follows immediately.

Recall that it is obvious by our definition that nuclear implies WEP, thus

nuclear ∪ injective ⊂ WEP.

By Theorem 8.16 we can deduce from Corollary 9.26:

Corollary 9.27. Let C = C∗(F∞). A C∗-algebra A is nuclear if and only if

C ⊗min A∗∗ = C ⊗max A

∗∗,

i.e. if and only if the pair (C , A∗∗) is nuclear.

Corollary 9.28. A C∗-algebra A is both WEP and locally reflexive if and only if it is nuclear.

137

Proof. Assume that A has the WEP. By our definition of the WEP, C ⊗minA = C ⊗maxA and hence(C ⊗minA)∗∗ = (C ⊗maxA)∗∗. If A is locally reflexive, then ‖C ⊗minA

∗∗ → (C ⊗minA)∗∗‖ = 1, andhence ‖C ⊗minA

∗∗ → (C ⊗maxA)∗∗‖ = 1. By Theorem 8.22 we have ‖C ⊗minA∗∗ → C ∗∗⊗binA

∗∗‖ =1. But the norm induced on C ⊗ A∗∗ by the bin-norm on C ∗∗ ⊗ A∗∗ coincides with the nor-norm(see Remark 8.4). Therefore we have ‖C ⊗min A

∗∗ → C ⊗nor A∗∗‖ = 1. Since by Theorem 9.10

C ⊗norA∗∗ = C ⊗maxA

∗∗, we conclude that ‖C ⊗minA∗∗ → C ⊗maxA

∗∗‖ = 1, and hence A is nuclearby Corollary 9.27. The converse is immediate since nuclear implies locally reflexive by Theorem8.29.

Corollary 9.29. If the reduced C∗-algebra C∗λ(G) of a discrete group G has the WEP then G isamenable.

Proof. Assume that C∗λ(G) has the WEP. Let M ⊂ B(`2(G)) be as usual the von Neumann algebragenerated by C∗λ(G). Let T : B(`2(G))→M be the completely positive contraction extending theinclusion C∗λ(G) → M , given by (iii) in Theorem 9.22. By Corollary 5.3, T is C∗λ(G)-bimodular.This implies in particular that T (λ(g)xλ(g)∗) = λ(g)T (x)λ(g)∗ for any x ∈ B(`2(G)) and anyg ∈ G. For any x ∈ `∞(G), let Dx ∈ B(`2(G)) denote the (diagonal) operator of multiplication byx. Note that

λ(g)Dxλ(g)∗ = Dδg∗x.

Let τG : M → C be defined by τG(a) = 〈δe, aδe〉. It is easy to check that τG(λ(g)aλ(g)∗) = τG(a)for any a ∈M and g ∈ G (in other words τG is a trace on M in the sense of §11.1). Let ϕ ∈ `∞(G)∗+be the functional defined by

ϕ(x) = τG(T (Dx)) = 〈δe, T (Dx)δe〉.

Then we have

ϕ(δg ∗ x) = τG(T (λ(g)Dxλ(g)∗)) = τG(λ(g)T (Dx)λ(g)∗) = τG(T (Dx)) = ϕ(x).

Thus ϕ is an invariant mean on G.

Proposition 9.30. For any separable C∗-algebra A ⊂ B(H) there is a separable unital C∗-algebraA1 ⊂ B(H) with the WEP such that A ⊂ A1.

Proof. This follows directly from Proposition 7.24 and Remark 7.25 applied with B = C andA = B(H).

Now that we know by Theorem 9.22 that A has the WEP if and only if the inclusion A ⊂ B(H)is max-injective, let us review what Theorem 7.29 tells us about it.

Theorem 9.31 (On weak expectations). Let A ⊂ B(H) be a C∗-algebra and let A∗∗ ⊂ B(H)∗∗ bethe embedding obtained by bitransposition. The following are equivalent.

(i) A has the WEP.

(ii) There is a contractive linear map V : B(H) → A∗∗ such that V (x) = x for any x ∈ A (inother words, V is a generalized weak expectation for A ⊂ B(H)).

(iii) There is a projection P : B(H)∗∗ → A∗∗ with ‖P‖ = 1.

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Remark 9.32 (Comparing weak expectations and projections). We repeatedly use the observationrecorded in Proposition 26.50 that, assuming A ⊂ B, A,B being here merely Banach spaces, alinear map V : B → A∗∗ is a generalized weak expectation if and only if V : B∗∗ → A∗∗ is acontractive projection. It can be shown (by a fairly easy application of the Hahn-Banach theorem)

that such a V exists if and only the natural map A∧⊗C → B

∧⊗C between the projective tensor

products is isometric for any Banach space C, and actually it suffices to have this for C = A∗.This kind of duality argument can be generalized to treat the case when V is not necessarily a

contraction. It can also be checked easily that the natural map A∧⊗C → A∗∗

∧⊗C is isometric for any

Banach space C. In analogy with the latter facts, in the C∗-algebra case, we showed in Theorem7.29 that a generalized weak expectation exists if and only if the inclusion A ⊂ B is max-injective,and in Corollary 7.27 that iA : A→ A∗∗ is always max-injective.

Remark 9.33 (Warning on a trap). To avoid possible errors, we emphasize that (in sharp contrastwith the analogue for injectivity) the existence of an embedding u : A∗∗ ⊂ B(H)∗∗ admitting ac.p. contractive projection P : B(H)∗∗ → A∗∗ does not in general imply the WEP for A. Indeed,we will show in Theorem 9.72 that this holds if and only if A is QWEP. For the WEP to hold itis essential to assume in addition that u = v∗∗ for some v : A → B(H). Then we effectively canconclude that v is max-injective so A has the WEP by Theorem 9.22.

We now turn to the stability of the WEP under infinite direct sums in the sense of `∞.

Proposition 9.34. For any family Ai | i ∈ I of WEP C∗-algebras the direct sum(⊕∑

i∈I Ai)∞

also has the WEP.

Proof. With the notation in (9.2), for any t ∈ C ⊗A we have by (9.2) and (1.14)

‖t‖C⊗maxA = supi∈I ‖ti‖C⊗maxAi = supi∈I ‖ti‖C⊗minAi = ‖t‖C⊗minA,

which means that A has the WEP.

Remark 9.35. Thus if C denotes as usual the full C∗-algebra of F∞, this means that if (Ai,C ) isnuclear for any i then (A,C ) is nuclear where A =

(⊕∑

i∈I Ai)∞. However, this is not true if we

replace C by an arbitrary C∗ algebra. Indeed, nuclearity is not preserved by infinite direct sums ofthe type A =

(⊕∑

i∈I Ai)∞. For example B = (⊕

∑n≥1Mn)∞ is not nuclear (see Corollary 18.11).

We refer to [189, Lemma 3.2] for a different proof of Proposition 9.34 based on the followingpurely Banach space result, which as its corollary, is of independent interest.

Theorem 9.36. Let B be any Banach space and let A ⊂ B be a closed subspace. Then the followingare equivalent:

(i) There is a projection P : B∗∗ → A∗∗ with ‖P‖ = 1.

(ii) For any ε > 0 and any finite dimensional subspace E ⊂ B there is a linear map ϕ : E → Awith ‖ϕ‖ ≤ 1 + ε such that

∀a ∈ E ∩A ϕ(a) = a.

Corollary 9.37. Let Bi | i ∈ I be a family of Banach spaces. Let Ai | i ∈ I be a familyof subspaces, with Ai ⊂ Bi for each i ∈ I, such that there is a projection Pi : B∗∗i → A∗∗i with‖Pi‖ = 1. Then there is a projection

P :(⊕∑

i∈IBi

)∗∗∞→(⊕∑

i∈IAi

)∗∗∞

with ‖P‖ ≤ 1.

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9.4 LLP (Local Lifting Property)

In Banach space theory, the “lifting property” of `1 is classical: for any bounded linear map ufrom `1 into a quotient Banach space X/Y and for any ε > 0, there is a lifting u : `1 → X with‖u‖ ≤ (1 + ε)‖u‖. Moreover, the so-called L1-spaces satisfy a local variant of this. There areanalogues of this lifting property for C∗-algebras and operator spaces (see [208, Ch. 16]). Wereturn to this in §9.5 and §21.2. But in this chapter we concentrate on the local variant called LLP,which is better understood. We show next that indeed the LLP as defined previously in Definition9.12 is equivalent to a certain “local” lifting property. Recall that linear maps with values in aquotient A/I (of a C∗-algebra A by an ideal I) that locally c-lift were defined in Definition 7.46.

Theorem 9.38 (Local lifting property). The following properties of a C∗-algebra C are equivalent:

(i) C has the LLP i.e. the pair (C,B) is nuclear.

(ii) Any u in the unit ball of D(C,A/I) is locally 1-liftable for any quotient A/I.

(iii) Any u ∈ CP (C,A/I) with ‖u‖ ≤ 1 is locally 1-liftable for any quotient A/I.

(iv) Any ∗-homomorphism u : C → A/I is locally 1-liftable for any quotient A/I.

Proof. Assume (i). Let u ∈ D(C,A/I) with ‖u‖dec ≤ 1. By (6.13) for any C∗-algebra B we have

(9.7) ‖IdB ⊗ u : B ⊗max C → B ⊗max (A/I)‖ ≤ 1

and by (7.6) (with the roles of A,B interchanged)

B ⊗max (A/I) = (B ⊗max A)/(B ⊗max I),

and hence a fortiori we have a “canonical” ∗-homomorphism of norm ≤ 1

B ⊗max (A/I)→ (B ⊗min A)/(B ⊗min I).

By definition of the LLP, we have B ⊗max C = B ⊗min C. Thus taking B = B in (9.7), we find

‖IdB ⊗ u : B ⊗min C → (B ⊗min A)/(B ⊗min I)‖ ≤ 1

which means, by Proposition 7.48, that u is locally 1-liftable. Thus (i) ⇒ (ii).(ii) ⇒ (iii) ⇒ (iv) are trivial. It remains to show that (iv) ⇒ (i).Assume (iv). Consider t ∈ C ⊗B. We can assume t ∈ E ⊗B with E ⊂ C finite dimensional. LetF be a free group such that C ' C∗(F)/I for some ideal I ⊂ C∗(F) (see Proposition 3.39). Letus denote A = C∗(F), let q : A → A/I ' C be the quotient map, and let u : C → A/I be theidentity mapping (here we identify C with A/I) so that u|E : E → A/I is the natural inclusion.By assumption (iv) u|E admits a completely contractive lifting v : E → A, so that qv = u|E . Wehave then t = (q ⊗ IdB)(v ⊗ IdB)(t) and hence

‖t‖C⊗maxB ≤ ‖(v ⊗ IdB)(t)‖A⊗maxB

= ‖(v ⊗ IdB)(t)‖A⊗minB (by (9.5))

≤ ‖t‖E⊗minB = ‖t‖C⊗minB (by (1.8)).

Hence we have proved (iv) ⇒ (i).Alternative proof: Combine Remark 7.41 and (iii)’ ⇔ (i) in Corollary 7.50.

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See Corollary 9.47 for more on the local liftings in points (iii) and (iv) of Theorem 9.38.

Remark 9.39 (On the OLLP). The reader is probably wondering why we do not include c.b. mapsin the list of mappings from C to A/I appearing in Theorem 9.38. The reason is the correspondingproperty of C is a much stronger one called the OLLP. An operator space X is said to have theOLLP if its universal C∗-algebra C∗u〈X〉 (in the sense of §2.7) has the LLP. The OLLP is studiedin detail by Ozawa in [186]. See also [208, p. 278]. The spaces `3∞ or Mn for n ≥ 3 are the simplestexamples of C∗-algebras with the LLP but failing the OLLP. Thus, when C is one of these, it isnot true that any u in the unit ball of CB(C,A/I) is locally liftable.

We now come to the general form of Kirchberg’s Theorem 9.6:

Corollary 9.40 (Generalized Kirchberg Theorem). For any LLP C∗-algebra C and any WEPC∗-algebra B, the pair (C,B) is nuclear.

Proof. Assume C LLP and B WEP, to prove that (C,B) is nuclear we simply invoke (9.4) andrepeat the reasoning for (iv) ⇒ (i) in Theorem 9.38 but with B in place of B.

Corollary 9.41. Let A,B,C be C∗-algebras with C = A/I. Assume that C has the LLP. If (A,B)is nuclear, then (C,B) is nuclear.In particular, if a QWEP C∗-algebra C has the LLP then it has the WEP.

Proof. Since the identity on C = A/I is locally 1-liftable (with respect to A→ A/I), Proposition7.48 implies that B ⊗min C → (B ⊗min A)/(B ⊗min I) is well defined and of norm 1 (and hence isan isomorphism). If (A,B) is nuclear

(B ⊗min A)/(B ⊗min I) = (B ⊗max A)/(B ⊗max I) = B ⊗max C

where at the last step we used (7.6). Thus (C,B) is nuclear.The second assertion corresponds to the case when A has the WEP and B = C .

It is important to emphasize that the local liftings considered in Theorem 9.38 are all for mapsdefined on a whole C∗-algebra C. In the next statement the maps that we want to lift are onlydefined on a finite dimensional operator subspace or system E ⊂ C and in general they do notcontractively extend to C.

Proposition 9.42 (On approximate liftings). Let C be a unital C∗-algebra C, E ⊂ C a finitedimensional linear subspace and u : E → A/I a linear operator. Let c ≥ 0 be a constant.

(i) Assume that u : E → A/I admits a lifting v1 : E → A with ‖v1‖ ≤ c and another onev2 : E → A that is c.p. Then, for any ε > 0, there is v ∈ CP (E,A) with ‖v‖ ≤ c such that‖qv − u‖ ≤ ε.

(ii) Assume that E is a (finite dimensional) operator system, that A is unital, that u ∈ CP (E,A/I)is unital and admits a lifting v1 : E → A with ‖v1‖cb ≤ 1. Then, for any ε > 0, there is aunital v ∈ CP (E,A) such that ‖qv − u‖ ≤ ε.

Proof. (i) Let σi be a quasi-central approximate unit as in §26.15. For any w : E → A we denotewi(x) = (1 − σi)1/2w(x)(1 − σi)1/2. By (26.19) we have (1 − σi)w − wi → 0 pointwise on E, andhence q(w − wi)→ 0 pointwise on E. This gives us that

u− qvi2 = q(v2 − vi2)→ 0 pointwise on E.

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Since dim(E) <∞ pointwise convergence implies norm convergence, and hence

(9.8) ‖u− qvi2‖ → 0.

Moreover, if w takes all its values in the ideal I, then wi → 0 pointwise on E, and hence ‖wi‖ → 0.In particular this holds for the mapping w = v2− v1 : E → I. Then vi2 = vi1 +wi, ‖vi2‖ ≤ c+ ‖wi‖,and vi2 is c.p. Let v = vi2c(c+ ‖wi‖)−1 so that ‖v‖ ≤ c and ‖v− vi2‖ ≤ ‖wi‖. Since u = qv2 we have

‖qv − u‖ ≤ ‖qv − qvi2‖+ ‖qvi2 − u‖ ≤ ‖wi‖+ ‖qvi2 − u‖.

Thus we can choose i large enough so that ‖qv − u‖ ≤ ε.(ii) We first observe that by replacing v1 by (v1 + v1∗)/2 we may assume that v1 is self-adjoint.Choose any f ∈ C∗+ so that f(1) = 1 (i.e. f is a state on C). Let wi : E → A be defined bywi(x) = vi1(x) + f(x)σi, so that qwi = qvi1 and wi(1)− 1 = (1− σi)1/2(v1(1)− 1)(1− σi)1/2. Sinceqv1(1) = u(1) = 1, we know 1− v1(1) ∈ I, and hence 1− wi(1)→ 0. Note that

wi(x) = ((1− σi)1/2 σ1/2i )

(v1(x) 0

0 f(x)1

)((1− σi)1/2

σ1/2i

)

and hence ‖wi‖cb ≤ max‖v1‖cb, ‖f‖ ≤ 1.Fix δ > 0 (to be specified). By (9.8) we can choose i far enough so that ‖qvi1 − u‖ ≤ δ, and also‖1− wi(1)‖ ≤ δ. We now set

ϕ(x) = wi(x) + f(x)(1− wi(1))

so that ϕ is self-adjoint, ϕ(1) = 1 and ‖ϕ − wi‖cb ≤ δ, and hence ‖ϕ‖cb ≤ ‖wi‖cb + δ ≤ 1 + δ. Letn = dim(E). By Theorem 2.28, there is a unital v ∈ CP (E,A) such that ‖ϕ − v‖cb ≤ 8nδ. Notethat q(wi − vi1) = 0 and hence ‖q(ϕ− vi1)‖ = ‖q(ϕ− wi)‖ ≤ ‖ϕ− wi‖ ≤ δ. Therefore

‖qv − u‖ ≤ ‖q(v − ϕ)‖+ ‖q(ϕ− vi1)‖+ ‖qvi1 − u‖ ≤ 8nδ + 2δ.

Choosing δ so that 8nδ + 2δ < ε, we obtain v with the desired property.

Remark 9.43. Actually the preceding proof in part (ii) works as well if we merely assume that uadmits a family of liftings vi ∈ CB(E,A) such that infi∈I ‖vi‖cb = 1.

To end this section we sketch a proof of the stability of the LLP under free products, generalizingthe fact that C∗(F2) = C∗(Z) ∗ C∗(Z) has the LLP, that we saw in Theorem 9.6.

Theorem 9.44. Let C1, C2 be unital C∗-algebras with the LLP. Then C1 ∗ C2 has the LLP.

Proof. Let E ⊂ C1 ∗ C2 be the linear span of a1a2 | (a1, a2) ∈ C1 × C2. By Theorem 9.8 appliedwith E1 = A1 = B and E2 = E with A2 = C1 ∗ C2, it suffices to prove that for any n ≥ 1 andt ∈Mn(B ⊗ E) with ‖t‖Mn(B⊗minE) ≤ 1 we have ‖t‖Mn(B⊗max(C1∗C2)) ≤ 1.Since Mn(B) ' B, it suffices to prove this for n = 1, i.e. for t ∈ B ⊗ E with ‖t‖min ≤ 1. ByLemma 2.22 we can factorize t as t = t1 t2 with tj ∈ B ⊗ Cj in the unit ball of B ⊗min Cj(j = 1, 2). Let π : B → B(H) and σ : C1 ∗ C2 → B(H) be ∗-homomorphisms with commutingranges. Equivalently, the restrictions σj = σ|Cj : Cj → B(H) (j = 1, 2) have range in π(B)′. Letsj = (π.σj)(tj) ∈ B(H). Then ‖sj‖ ≤ ‖tj‖max (j = 1, 2), and by the assumed commutations wehave (π.σ)(t) = (π.σ)(t1 t2) = s1s2, and hence

‖(π.σ)(t)‖ ≤ ‖s1‖‖s2‖ ≤ ‖t1‖max‖t2‖max.

If C1 and C2 have the LLP this implies ‖(π.σ)(t)‖ ≤ 1 and hence ‖t‖max ≤ 1.

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Remark 9.45. We can now show that (7.25) and (7.26) are independent of the choice of C1, C2 aslong as they have the LLP.Let D1 be another C∗-algebra with LLP admitting a surjective morphism r1 : D1 → A1 =D1/ ker(r1). Consider v : A∗2 → C1 as in (7.25). Let E ⊂ C1 be the (finite dimensional) rangeof v. By Theorem 9.38 the LLP of C1 implies that the map from C1 to D1/ ker(r1) is locally1-liftable. So there is a lifting w : E → D1 such that r1w(e) = e for all e ∈ E and ‖w‖cb = 1. Thens = wv : A∗2 → D1 is a (weak* continuous with finite rank) lifting of ut, with ‖s‖cb ≤ ‖v‖cb. Sincewe can reverse the roles of D1 and C1, this shows that the norm in (7.25) is the same if we computeit using C1, q1 or using D1, r1.Reasoning as we just did for (7.25) one can show that (7.26) does not depend on the choice of eitherC1 or C2, as long as both have the LLP.

9.5 To lift or not to lift (global lifting)

In this section we will prove several global lifting theorems, notably the well known Choi-Effroslifting Theorem [49], for c.p. maps defined on nuclear C∗-algebras taking values in a quotient C∗-algebra. We return to this theme in the later §21.2 where we will formally introduce and discussbriefly the lifting property (LP). We will also explain there why if Kirchberg’s conjecture holds thenLLP⇒ LP in the separable case. This is based on Theorem 9.46 (which heads this section) and thefact that WEP implies a certain restricted form of extension property (to be proved in §21.2): forany finite dimensional subspace E ⊂ C of an LLP C∗-algebra C, and any ε > 0 any u ∈ CB(E,W )from E to a space W with WEP admits an extension u ∈ CB(C,W ) with ‖u‖cb ≤ (1 + ε)‖u‖cb.

Our first statement, due to Arveson, that says that nicely liftable maps on separable spaces arestable by pointwise limits is a priori surprising: one would expect a stronger limit to be requiredfor this to hold. Roughly, the idea of the proof is similar to that of Lemma 26.33.

Theorem 9.46 (Pointwise limits of liftables are liftable). Let E be a separable operator space. LetI ⊂ A be an ideal in a C∗-algebra and let q : A → A/I be the quotient map. Consider a boundedlinear map u : E → A/I. Assume that there is a net of complete contractions vγ : E → A suchthat qvγ → u pointwise on E. Then:

(i) The map u admits a completely contractive lifting v : E → A, i.e. we have ‖v‖cb ≤ 1 andqv = u.

(ii) If E is a (separable) operator system and if u and all the vγ’s are c.p. (resp. unital and c.p.)then we can find a c.p. (resp. unital and c.p.) lifting v : E → A with ‖v‖cb ≤ 1.

Proof. Let xk be a dense sequence in the unit ball of E. Assume given a complete contraction(c.c. in short) wn such that

(9.9) ‖qwnxk − uxk‖ < 2−n ∀k = 1, . . . , n.

Moreover, if u is c.p. we assume that wn is c.p.We claim there is a map wn+1 : E → A (c.p. if u and wn are c.p.) with ‖wn+1‖cb ≤ 1 such that

(9.10) ‖qwn+1xk − uxk‖ < 2−n−1 ∀k = 1, . . . , n+ 1

and

(9.11) ‖(wn+1 − wn)xk‖ < 2−n+1 ∀k = 1, . . . , n.

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Taking this for granted, we may construct by induction a sequence (wn) satisfying (9.10) and(9.11) for any n. Then v(x) = limwn(x) is the desired lifting. Indeed, wn(x) is Cauchy for any x inx1, x2, . . .. Therefore v(x) = limnwn(x) exists and satisfies qv(x) = u(x). Since the norms ‖wn‖are uniformly bounded (by 1), this still holds for any x ∈ E. We have ‖v‖cb ≤ limn ‖wn‖cb ≤ 1,and in the c.p. case v is c.p. as a limit of c.p. maps.

Thus it suffices to prove the claim. Let wn be as in the claim. Going far enough in the net (vγ),we can find v : E → A (that is also c.p. in the c.p. case) with ‖v‖cb ≤ 1 such that

(9.12) ‖qvxk − uxk‖ < 2−n−2 ∀k = 1, . . . , n+ 1.

Let (σi) be an approximate unit in I as in §26.15.For a suitable choice of i (to be specified later on) we will let

wn+1(x) = σ1/2i wn(x)σ

1/2i + (1− σi)1/2v(x)(1− σi)1/2.

Note that since

wn+1(x) = [σ1/2i (1− σi)1/2]

[wn(x) 0

0 v(x)

][σ

1/2i

(1− σi)1/2

]we have

‖wn+1‖cb ≤ max‖wn‖cb, ‖v‖cb ≤ 1.

Moreover, in the c.p. case, wn+1 is also c.p. By (26.23), for any given x and ε > 0, i can be chosenlarge enough so that

(9.13) ‖wn+1(x)− [σiwn(x) + (1− σi)v(x)]‖ < ε

and hence (since σiwn(x)− σiv(x) ∈ I) we have ‖q(wn+1(x)− v(x))‖ < ε, which implies

‖qwn+1(x)− u(x)‖ < ε+ ‖qv(x)− u(x)‖.

So we can choose i large enough so that

‖qwn+1(xk)− u(xk)‖ < ε+ 2−n−2 ∀k = 1, . . . , n+ 1.

Moreover, using wn+1(x)− wn(x) = wn+1(x)− [σiwn(x) + (1− σi)wn(x)] we find by (9.13)

‖wn+1(x)− wn(x)‖ < ε+ ‖(1− σi)[v(x)− wn(x)]‖

hence for i large enough, by (26.20) we can ensure that

‖wn+1(x)− wn(x)‖ < 2ε+ ‖q[v(x)− wn(x)]‖ < 2ε+ ‖qv(x)− u(x)‖+ ‖qwn(x)− u(x)‖.

Thus if we now make this last choice of i valid for any x in x1, . . . , xn and take ε = 2−n−2 weobtain the announced estimates (9.10) and (9.11) for wn+1 (recalling (9.12) and (9.9)).Lastly, the same proof yields the unital case.

We can now add a complement to Definition 7.46 concerning locally liftable unital c.p. maps.

Corollary 9.47. Let C be a unital operator system and let u : C → A/I be locally 1-liftable. Ifu is unital and c.p. then for any finite dimensional operator system E ⊂ C there is a unital c.p.map v : E → A such that qv = u|E.

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Proof. By (ii) in Proposition 9.42 applied to u|E , for any ε > 0 there is a unital vε ∈ CP (E,A)such that ‖qvε − u|E‖ ≤ ε. By (ii) in Theorem 9.46, there is a unital v ∈ CP (E,A) such thatqv = u|E .

Definition 9.48. Let λ ≥ 1. We say that an operator space E ⊂ B(H) has the λ-completelybounded approximation property (in short λ-CBAP) if there is a net of finite rank maps ui : E → Ewith sup ‖ui‖cb ≤ λ tending pointwise to the identity of E. We say that E has the CBAP if itsatisfies this for some 1 ≤ λ <∞.

Corollary 9.49. Let E be a separable operator space. Assume that E has the λ-CBAP for someλ ≥ 1. Then any locally c-liftable u ∈ CB(E,A/I) admits a (global) lifting v ∈ CB(E,A) with‖v‖cb ≤ cλ.

Proof. Let (ui) be as in Definition 9.48. Let Ei be any finite dimensional operator space such thatEi ⊃ ui(E). Since u locally c-lifts (see Definition 7.46), there is wi ∈ CB(Ei, A) with ‖wi‖cb ≤ csuch that qwi = u. Then wiui lifts uui and ‖wiui‖cb ≤ cλ. Clearly qwiui = uui tends to u pointwise.Then the net vi = (cλ)−1wiui is formed of complete contractions such that qvi → (cλ)−1u pointwise.Thus Theorem 9.46 gives us the conclusion.

Remark 9.50. There are purely Banach space analogues of Corollary 9.49. See [261] for a survey.For instance, for any fixed n ≥ 1, there is an analogue for which the cb-norm of a map u is replacedeverywhere by the norm of un = IdMn ⊗ u. In the case n = 1 this reduces to the Banach spacecase. More explicitly, using ‖un‖ in place of ‖u‖cb in the definition of locally c-liftable maps, weget the definition of a locally (c,Mn)-liftable map u : E → A/I. Similarly, we define the Mn-BAPwith constant λ > 0. Then if E is separable with the Mn-BAP with constant λ > 0, any locally(c,Mn)-liftable map u : E → A/I admits a (global) lifting v ∈ B(E,A) with ‖vn‖ ≤ cλ. Thisfollows by a cosmetic adaptation of the proofs of Theorem 9.46 and Corollary 9.49.

Corollary 9.51. Let E be a separable operator system. If E has the CPAP (in the sense ofDefinition 4.8), in particular if dim(E) <∞, then any unital and locally 1-liftable u ∈ CP (E,A/I)admits a (global) lifting v ∈ CP (E,A) with ‖v‖ = ‖u‖.

Proof. Let (ui) be a net of finite rank c.p. maps tending pointwise to IdE . Recalling (1.27), we mayassume ‖u‖ = ‖u‖cb = 1. Let Ei be any finite dimensional operator system such that Ei ⊃ ui(E).Since u is locally 1-liftable (see Definition 7.46), by (ii) in Proposition 9.42 (or by Corollary 9.47)for any ε > 0 there is wi ∈ CP (Ei, A) with ‖wi‖ ≤ 1 such that ‖qwi − u|Ei‖ < ε. Then thecomposition vi = wiui satisfies ‖qvi − uui‖ = ‖(qwi − u|Ei)ui‖ < ε. Since uui → u pointwise (andε is arbitrary), we can arrange so that the net (vi) is such that qvi → u pointwise on E. ThenTheorem 9.46 allows us to conclude.

Remark 9.52. Actually it suffices that u itself be approximable in the following sense: it is enoughto assume that u is the pointwise limit of a net of finite rank maps ui ∈ CP (E,E) completelypositively and completely contractively factorized through some Mn(i). To check this, one uses thelifting property of the Mn(i)’s (see Remark 9.54).

We can now deduce the celebrated Choi-Effros lifting theorem.

Theorem 9.53 (Choi-Effros lifting theorem). Let C be a unital separable nuclear C∗-algebra. Thenany unital c.p. map u : C → A/I (into an arbitrary unital quotient C∗-algebra) admits a c.p. liftingv : C → A with ‖v‖ = ‖u‖.

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Proof. Since C is nuclear, it has the LLP and (by Corollary 7.12) the CPAP, so by (i) ⇔(iii) inTheorem 9.38 this can be viewed as a particular case of the preceding Corollary.

Remark 9.54 (Lifting property of Mn). In particular, the preceding Theorem shows that any unitalc.p. map u : Mn → A/I admits a unital c.p. lifting v : Mn → A.We wish to emphasize again (see Remark 9.39) that the c.b. variant of the preceding lifting theoremis generally not valid. In general one cannot lift a complete contraction u : Mn → A/I to a completecontration v : Mn → A.

9.6 Linear maps with the WEP and the LLP

It might clarify certain features of the theory to consider the WEP and the LLP for linear maps,whence the following definitions, where we restrict for simplicity to the unit ball.

Definition 9.55. Let E ⊂ B(H) be an operator space and B a C∗-algebra. Let us say that alinear mapping u : E → B is WEP if

‖IdC ⊗ u : C ⊗min E → C ⊗max B‖ ≤ 1.

The following is a corollary of Theorem 7.4.

Corollary 9.56. In the situation of Definition 9.55, the following are equivalent:

(i) u is WEP.

(ii) For some H, there are v ∈ CB(E,B(H)) and w ∈ D(B(H), B∗∗) such that wv = iBu and‖v‖cb‖w‖dec ≤ 1.

Proof. We have isometrically C ⊗min E ⊂ C ⊗min B(H) and C ⊗min B(H) = C ⊗max B(H) byTheorem 9.6. Thus we have isometrically C ⊗minE ⊂ C ⊗maxB(H), so this is indeed an immediateconsequence of Theorem 7.4 and Remark 7.5.

This allows us to formulate the following interesting variant of Theorem 9.22.

Theorem 9.57. Let u : A → B be a unital c.p. map between unital C∗-algebras. The followingare equivalent.

(i) u is WEP.

(ii) The mapping iBu : A→ B∗∗ factorizes completely positively and contractively through someB(H), i.e. there are v ∈ CP (A,B(H)) and w ∈ CP (B(H), B∗∗) such that wv = iBu and‖v‖‖w‖ ≤ 1.

(iii) For some H, there are v ∈ CB(A,B(H)) and w ∈ CP (B(H), B∗∗) such that wv = iBu and‖v‖cb‖w‖ ≤ 1.

Proof. Assume (i). Assume A ⊂ B(H). By Corollary 9.56 u admits an extension w : B(H)→ B∗∗

with ‖w‖dec ≤ 1. Since w(1) = u(1) = 1 and ‖w‖cb ≤ ‖w‖dec ≤ 1 (see (6.6)) w must be c.p. byTheorem 1.35. Thus we obtain (ii). (ii) ⇒ (iii) is obvious. Assume (iii). We have by (4.30)

‖IdC ⊗ iBu : C ⊗min A→ C ⊗max B∗∗‖ ≤ ‖w‖‖IdC ⊗ v : C ⊗min A→ C ⊗max B(H)‖.

By Theorem 9.6 this is

= ‖w‖‖IdC ⊗ v : C ⊗min A→ C ⊗min B(H)‖

146

and by (1.8) the latter is ≤ ‖w‖‖v‖cb ≤ 1. Lastly, since B ⊂ B∗∗ is max-injective (see Proposition7.26), we have

‖IdC ⊗ iBu : C ⊗min A→ C ⊗max B∗∗‖ = ‖IdC ⊗ u : C ⊗min A→ C ⊗max B‖,

and we obtain (i).

To emphasize the parallelism WEP/LLP, we also introduce the LLP for linear maps.

Definition 9.58. Let E ⊂ B(H) be an operator space and B a C∗-algebra. Let us say that alinear mapping u : E → B is LLP if

‖IdB ⊗ u : B ⊗min E → B ⊗max B‖ ≤ 1.

From this definition it is clear that u : E → B is LLP if and only if for any finite dimensionalsubspace E1 ⊂ E the restriction u|E1

: E1 → B is LLP. Thus the assumption of finite dimensionalityin the next result is not too restrictive.

Proposition 9.59. Let E ⊂ B(H) be finite dimensional and let u : E → B be a linear map into aC∗-algebra B. The following are equivalent:

(i) u is LLP.

(ii) There are v ∈ CB(E,C ) and w ∈ D(C , B) such that wv = u and ‖v‖cb‖w‖dec ≤ 1.

Proof. Assume (i). Remark 7.49 and Theorem 7.48 show that (ii) holds with C replaced by C∗(F)where F is a large enough free group so that B = C∗(F)/I for some ideal I ⊂ C∗(F). Then we canreplace C∗(F) by C using Remark 3.6.Conversely, assume (ii). Since the fundamental pair (B,C ) is nuclear, (i) is easily derived using(0.3) for v and (6.13) for w.

9.7 QWEP C∗-algebras

We start by two consequences of Theorem 7.33 for QWEP C∗-algebras.

Corollary 9.60. Let A,B be C∗-algebras, with B unital, for which there exists ϕ ∈ CP (A,B) suchthat ϕ(BA) = BB. If a C∗-algebra C is such that (A,C) is a nuclear pair, the same is true for thepair (Dϕ, C). In particular, if A is WEP, or merely QWEP, then B is QWEP.

Proof. The first assertion is obvious (by Remark 7.20). Thus, by our definition of WEP, if A isWEP, Dϕ is WEP and hence B is QWEP. If A is a quotient of a WEP C∗-algebra, we may composewith the quotient mapping, then we are reduced to the case when A is WEP.

For further reference, we spell out a particular case:

Corollary 9.61. Let D ⊂ A be a C∗-subalgebra for which there is a c.p. projection P : A → Dwith ‖P‖ = 1. If A is QWEP, then D is QWEP.

We now turn to the stability properties of the class of QWEP C∗-algebras. We first state animmediate consequence of Proposition 9.34.

Corollary 9.62. For any family Ai | i ∈ I of QWEP C∗-algebras the direct sum(⊕∑

i∈I Ai)∞

also is QWEP.

147

Remark 9.63. Let Ai | i ∈ I be a family, directed by inclusion, of C∗-subalgebras of B(H) andlet A ⊂ B(H) be the norm closure of their union. If all the Ai’s are WEP then A is WEP. Indeed,it suffices to show that ‖t‖max = ‖t‖min for any t ∈ (∪Ai)⊗C . But then t ∈ Ai⊗C for some i andhence

‖t‖A⊗maxC ≤ ‖t‖Ai⊗maxC = ‖t‖Ai⊗minC = ‖t‖A⊗minC .

Proposition 9.64. Let D ⊂ D1 be a C∗-subalgebra of a quotient one D1 = A1/I. Let q : A1 → D1

be the quotient map. Let A = q−1(D) so that I = q−1(0) ⊂ A ⊂ A1 and D = A/I.If the inclusion D ⊂ D1 is max-injective, then the inclusion A ⊂ A1 is also max-injective.

Proof. Note I = ker(q) = ker(q|A). Let C be any C∗-algebra. We must show that the natural mapJ : C ⊗max A → C ⊗max A1 is injective. Let x ∈ C ⊗max A be in its kernel. Then, if D ⊂ D1 ismax-injective, (IdC⊗q|A)(x) must vanish in C⊗maxD, as the following commuting diagram shows.

C ⊗max A

IdC⊗q|A

J // C ⊗max A1

IdC⊗q

C ⊗max D // C ⊗max D1

Therefore by (7.6), x ∈ C ⊗max I, but since I is an ideal in A1, I ⊂ A1 is max-injective, and henceJ restricted to C ⊗max I (which obviously coincides with the inclusion C ⊗max I → C ⊗max A1) isinjective. Therefore x = 0. This proves the proposition.

Corollary 9.65. Let D ⊂ D1 be a C∗-subalgebra of a QWEP C∗-algebra D1. If the inclusionD ⊂ D1 is max-injective, then D is QWEP.

Proof. Assume D1 = A1/I with A1 WEP. Let A = q−1(D) where q : A1 → D1 is again thequotient map. By Proposition 9.64 the inclusion A ⊂ A1 is max-injective. By Remark 9.13 A hasthe WEP and hence D = A/I is QWEP.

Theorem 9.66. A C∗-algebra B is QWEP if and only if its bidual B∗∗ is also QWEP.

Proof. Assume B∗∗ QWEP. By Corollary 7.27, the inclusion B → B∗∗ is max-injective, and henceB is QWEP by Corollary 9.65.To prove the converse, assume B QWEP and B∗∗ ⊂ B(H) (as a von Neumann subalgebra). Let Ibe the net of neighbourhoods of 0 for the weak* topology of B(H). Let U be an ultrafilter refiningthis net (see Remark 26.6). By the weak* density of BB, for any x ∈ BB∗∗

(9.14) ∀i ∈ I ∃x(i) ∈ (x+ i) ∩BB.

This implies that x(i)→ x (weak*). For any i ∈ I we set Ai = B. We define a linear map

ϕ :(⊕∑

i∈IAi

)∞→ B∗∗

by setting, for any x = (xi) in the unit ball of(⊕∑

i∈I Ai)∞

ϕ(x) = limU xi,

where the limit is meant in the weak* sense. Clearly, ϕ is positive (meaning positivity preserving),and similarly (using Mn

((⊕∑

i∈I Ai)∞) ∼= (

⊕∑

i∈IMn(Ai))∞) we see that ϕn = IdMn ⊗ ϕ is

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positive for any n. Thus ϕ is c.p. Let A =(⊕∑

i∈I Ai)∞. By Corollary 9.62, A is QWEP. By

(9.14), ϕ(BA) = BB∗∗ . This shows that ϕ ∈ CP (A,BB∗∗) satisfies the assumption of Corollary9.60. The latter ensures that B∗∗ is QWEP.Actually, using the density of BB in BB∗∗ for the so-called strong* operator topology, one obtainsdirectly a ∗-homomorphism in place of ϕ, and then B∗∗ appears as a quotient of A.

Theorem 9.67 (Characterization of QWEP). The following properties of a unital C∗-algebra Dare equivalent:

(i) D is QWEP.

(ii) For any LLP C∗-algebras C and C1, any decomposable map u : C → D with ‖u‖dec ≤ 1,satisfies

‖IdC1 ⊗ u : C1 ⊗min C → C1 ⊗max D‖ ≤ 1.

(ii)’ Same as (ii) for C = C1 = C .

(iii) There is a free group F and a surjective unital ∗-homomorphism π : C∗(F)→ D such that

‖IdC ⊗ π : C ⊗min C∗(F)→ C ⊗max D‖ ≤ 1,

in other words such that the quotient mapping π : C∗(F)→ D is WEP.

Proof. Assume (i), so that D = A/I with A WEP. Let q : A → A/I be the quotient map. Let ube as in (ii). Let t ∈ C1⊗C. Let E ⊂ C be a finite dimensional subspace such that t ∈ C1⊗E. ByTheorem 9.38, if C is LLP u is locally 1-liftable, so there is uE ∈ CB(E,A) with ‖uE‖cb ≤ 1 suchthat quE = u|E , and hence

‖(IdC1 ⊗ uE)(t)‖C1⊗minA ≤ ‖t‖C1⊗minC .

By Corollary 9.40, the pair (C1, A) is nuclear. Therefore we have

‖(IdC1 ⊗ uE)(t)‖C1⊗maxA = ‖(IdC1 ⊗ uE)(t)‖C1⊗minA

and hence, since (obviously) (IdC1 ⊗ u)(t) = (IdC1 ⊗ u|E)(t), we find

‖(IdC1 ⊗ u)(t)‖C1⊗maxA = ‖(IdC1 ⊗ quE)(t)‖C1⊗maxA ≤ ‖(IdC1 ⊗ uE)(t)‖C1⊗minA ≤ ‖t‖C1⊗minC .

This shows (i)⇒ (ii). (ii)⇒ (ii)’ is trivial. (ii)⇒ (iii) is obvious since any D is a quotient of C∗(F)for some F and the latter has the LLP by (9.5). In addition, (ii)’ ⇒ (iii) is easy to check usingLemma 3.8 since any t ∈ C ⊗ C∗(F) lies in C ⊗ E for some finite dimensional E ⊂ C∗(F).Assume (iii). Let C = C∗(F). By Theorem 9.57, we can write π = wv with v ∈ CP (C,B(H)),w ∈ CP (B(H), D∗∗) of unit norm. By Theorem 8.1 w admits a weak* continuous extension w ∈CP (B(H)∗∗, D∗∗) with ‖w‖ = ‖w‖. Since π is onto, we know by Lemma 26.33 that π(BC) = BD,and hence w(BB(H)) ⊃ BD. This implies by weak* density that w(BB(H)∗∗) = BD∗∗ . Since B(H)∗∗

is QWEP (see Theorem 9.66), Corollary 9.60 shows that D∗∗ is also QWEP. Since D ⊂ D∗∗ ismax-injective (see Corollary 7.27), Corollary 9.65 shows that D is QWEP, and we obtain (i).

Remark 9.68. Let G be a discrete group such that C∗(G) has the LLP. If MG is QWEP then thenatural map C∗(G)⊗ C∗(G)→MG ⊗MG extends to ∗-homomorphism

C∗(G)⊗min C∗(G)→MG ⊗max MG.

A fortiori, G has the factorization property in the sense of Definition 7.36 (see also §11.8).Indeed, this follows from (ii) in Theorem 9.67 with C = C1 = C∗(G) and u = QG : C∗(G) → MG.Note that this remark typically applies when G is a free group (see Corollary 12.23).

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Taking C = D and u = IdD in (ii)’ from Theorem 9.67, we immediately derive:

Corollary 9.69. If a QWEP C∗-algebra D has the LLP then it has the WEP.

Corollary 9.70. Let A and D be C∗-algebras. Assume that there is a factorization of the identityof A of the form

IdA : Av−→D w−→A

where v, w are decomposable maps. If D is QWEP then A also is QWEP.

Proof. Let λ = ‖v‖dec‖w‖dec. Assume D is QWEP. Thus D satisfies (ii) in Theorem 9.67. Letπ : C∗(F) → A be a quotient map. By (6.13), ‖IdC ⊗ π : C ⊗min C

∗(F) → C ⊗max A‖ ≤ λ. Butsince IdC ⊗π is a ∗-homomorphism, the latter norm must be = 1. Thus A satisfies (iii) in Theorem9.67 and hence is QWEP.

Remark 9.71. In Corollary 9.70 it suffices to assume that there are nets of decomposable mapswi : A → D and vi : D → A with supi ‖vi‖dec‖wi‖dec < ∞ such that viwi tends pointwise toiA : A→ A∗∗. Indeed, the preceding proof leads to ‖IdC ⊗ iAπ : C ⊗minC

∗(F)→ C ⊗maxA∗∗‖ ≤ λ,

and Corollary 7.27 allows us to conclude.

Theorem 9.72 (Another characterization of QWEP). The following properties of a unital C∗-algebra D are equivalent:

(i) D is QWEP.

(ii) For some H there is an embedding (as a von Neumann subalgebra) D∗∗ ⊂ B(H)∗∗ admittinga contractive projection P : B(H)∗∗ → D∗∗.

(iii) There is a factorization of the identity of D∗∗ of the form

IdD∗∗ : D∗∗v−→B(H)∗∗

w−→D∗∗

where v, w are decomposable maps.

Proof. Assume D = A/I with A WEP. By Theorem 9.31 the algebra A∗∗ certainly satisfies (ii).But by (26.37) we have A∗∗ ' I∗∗ ⊕D∗∗. Therefore D∗∗ also satisfies (ii). This shows (i) ⇒ (ii).Then (ii) ⇒ (iii) is obvious (recalling that P is automatically c.p. by Theorem 1.45) and (iii) ⇒(i) follows from Corollary 9.70 and Theorem 9.66.

Remark 9.73 (Warning on a trap). It is important to understand the difference between the laststatement and Theorem 9.31. For that purpose, we urge the reader to look into Remark 9.33.

Theorem 9.74. Let Ai | i ∈ I a family, directed by inclusion, of C∗-subalgebras of B(H), letA ⊂ B(H) (resp. N ⊂ B(H)) be the norm (resp. weak* ) closure of their union. If all the Ai’s areQWEP then A and N are QWEP.

Proof. Assume A unital for simplicity. Using the unitary groups U(Ai) it is easy to find a freegroup F and a surjective ∗-homomorphism π : C∗(F) → A so that there is a family, directed byinclusion, of free subgroups Gi with union = F, such that π restricted to C∗(Gi) realizes Ai as aquotient of C∗(Gi). Assume all the Ai’s are QWEP. Then arguing as in Remark 9.63 one showsthat π satifies (iii) in Theorem 9.67 and hence A is QWEP. We may view N as generated by A.Then by Theorem 9.66 A∗∗ is QWEP, and since N is a quotient of A∗∗ (see Theorem 8.1 (i)), itis also QWEP. One can also prove directly that N is QWEP by a modification of the proof ofTheorem 9.66.

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Remark 9.75 (QWEP is separably determined). Let A1 ⊂ A and A2 ⊂ A be separable C∗-subalgebras of a C∗-algebra A. Then the C∗-algebra generated by A1 ∪ A2 is still separable.Thus the family of separable C∗-subalgebras of A forms a directed net for inclusion and the unionof all of them is equal to A. Thus, by Theorem 9.74 if all separable C∗-subalgebras are QWEPthen A is QWEP. A fortiori, if all weak* separable subalgebras of a von Neumann algebra M areQWEP then M is QWEP.

9.8 Notes and Remarks

As we already mentioned, Takesaki introduced the notion of nuclearity for C∗-algebras (initially un-der a different name), and of course implicitly also for pairs. The term was inspired by Grothendieck’swork on nuclear locally convex spaces. Later on the subject was deepened by the works of Lance[165], Choi-Effros and Kirchberg [154, 45] (who independently proved that nuclearity is equivalentto the CPAP, see §10.2) and Effros-Lance [79]. A major step was taken thanks to Connes’s workon injective factors [61] that allowed Choi and Effros to complete the proof that a C∗-algebra A isnuclear if and only if its bidual A∗∗ is an injective von Neumann algebra (see §8.3). For most of thisinitial period, not much interest was devoted to what we call “nuclear pairs”, except for Lance’squestion whether the nuclearity of (A,Aop) implies that A is nuclear, answered by Kirchberg in[155]. The latter constructed a counterexample, and at the same time started a deep study ofnuclear pairs that led him to identify the WEP and the LLP for a C∗-algebra A as the nuclearityof the pairs respectively (A,C ) and (A,B) (which in this volume we choose as defining WEP andLLP). Kirchberg’s Theorem 9.6 (that tells us that (C ,B) is a nuclear pair) was proved in [156],but we followed the simpler proof from [204]. We encourage the reader to compare the latter proofwith the original one ! Theorem 9.18 is due to Choi. The origin of the WEP is Lance’s paper[165]. Our terminology is slightly different, as explained in Remark 9.23. Major advances weremade in Kirchberg’s [155]. Our presentation of the stability properties of the WEP (due to Lanceand Kirchberg) in §9.3 is much influenced by Ozawa’s [189]. Corollary 9.28 already appears in [77].Lance raised the question whether there are examples of embeddings A ⊂ B(H) admitting a weakexpectation from B(H) to M = A

σbut nevertheless not admitting a contractive projection from

B(H) onto M (in other words non-injective). Such examples were proposed by Blackadar in [23, 24].More examples (where M is a free group factor and A a specially chosen weak*-dense Popa C∗-subalgebra) appear in later papers by Brown and Dykema [37, 38]. The definition of the LLP andthe results in §9.4 are due to Kirchberg in [155]. The stability of the LLP under free products inTheorem 9.44 is due to the author [204]. A subsequent paper by Boca [31] contains variations onthe same theme. The results of §9.5 are due to Kirchberg but are based on Arveson’s ideas in [13].Theorem 9.53 is due to Choi and Effros [49].In §9.6 and 9.7 again the main ideas come from Kirchberg’s work, but as usual we try to put forwardproperties of linear maps. Theorem 9.72 is easy to deduce from Ozawa’s viewpoint in [189], butapparently was not formulated yet.

10 Exactness and nuclearity

We already gave a brief introduction to nuclear C∗-algebras in §4.2 and we already announced someof their main properties, like their characterization by the CPAP, which will be proved in the presentchapter. Since, in the separable case, exact C∗-algebras are just C∗-subalgebras of nuclear ones,it is not surprising that many features of nuclear C∗-algebras and their approximation propertieshave analogues for exactness. We try to emphasize this “parallelism”.

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10.1 The importance of being Exact

A C∗-algebra A is called exact if the phenomenon described in (7.16) cannot happen when we takethe minimal tensor product with A. More precisely:

Definition 10.1. A C∗-algebra A is called exact if for any C∗-algebra B and any ideal I ⊂ B sothat B/I is a C∗-algebra, the sequence

0 → I ⊗min A→ B ⊗min A→ (B/I)⊗min A→ 0

is exact. Equivalently, this holds if and only if the kernel of the mapping B⊗minA→ (B/I)⊗minAcoincides with I ⊗min A. In other words, this reduces to:

(10.1) the ∗−homomorphismB ⊗min A

I ⊗min A→ (B/I)⊗min A is injective (or equivalently isometric).

Even more explicitly this boils down (equivalently) to:

(10.2) ∀x ∈ (B/I)⊗A ‖x‖(B⊗minA)/(I⊗minA) ≤ ‖x‖(B/I)⊗minA.

Remark 10.2. Let B/I be a quotient C∗-algebra by an ideal I. For brevity, let us say that A is(B, I)-exact if (10.1) or (10.2) holds. Let B0 ⊂ B be a C∗-subalgebra, let I0 = I∩B0 so that I0 ⊂ Iand B0/I0 ⊂ B/I. Assume that there is a completely contractive projection P : B → I such thatP (B0) = I0. It is easy to check that if A is (B, I)-exact then it is also (B0, I0)-exact: indeed,we have isometric embeddings (B0/I0) ⊗min A ⊂ (B/I) ⊗min A and (B0 ⊗min A)/(I0 ⊗min A) ⊂(B ⊗min A)/(I ⊗min A). For the latter we use the projection P ⊗ IdA : B ⊗min A→ I ⊗min A.The classical Calkin algebra Q = B/K is of special interest (here K is the ideal of compactoperators in B). Let (Nn) be any increasing sequence of integers, and let B0 = (⊕

∑MNn)∞ and

let I0 = b = (bn) ∈ B0 | lim ‖bn‖ = 0. We have a classical block diagonal embedding B0 ⊂ B suchthat I0 = B0 ∩K . Let q0 : B0 → B0/I0 denote the quotient map. It is easy to check that for anyb = (bn) ∈ B0 we have ‖q0(b)‖ = lim sup ‖bn‖. More generally, the same easy argument shows thatfor any finite dimensional operator space E and any b = (bn) ∈ (⊕

∑MNn ⊗min E)∞ ' B0 ⊗min E

(for the last ' see (1.14)), we have

(10.3) ‖(q0 ⊗ IdE)(b)‖(B0⊗minE)/(I0⊗minE) = lim supn→∞ ‖bn‖MNn⊗minE .

The main point of the next result is that exactness restricted to the Calkin algebra implies thegeneral exactness.

Theorem 10.3. The following properties of a C∗-algebra A are equivalent:

(i) The C∗-algebra A is (B,K )-exact.

(ii) For any finite dimensional subspace E ⊂ A and any bounded sequence of linear maps un :En → E, where En are arbitrary operator spaces, we have

(10.4) lim supn→∞ ‖un‖cb ≤ supk lim supn→∞ ‖IdMk⊗ un : Mk(En)→Mk(E)‖.

(iii) For any finite dimensional subspace E ⊂ A and any ε > 0 there are an integer N , a subspaceE ⊂MN and u : E → E such that ‖u‖cb < 1 + ε and ‖u−1‖cb ≤ 1.

(iv) The C∗-algebra A is exact.

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Proof. Assume (i). Choose εn > 0 such that εn → 0. For any n there is Nn such that ‖un‖cb ≥‖IdMNn

⊗ un‖ > ‖un‖cb − εn. We may clearly adjust so that (Nn) is increasing. For any n there isxn ∈ BMNn (En) such that

‖un‖cb ≥ ‖(IdMNn⊗ un)(xn)‖MNn⊗minE > ‖un‖cb − εn.

Let tn = (IdMNn⊗ un)(xn) ∈ MNn ⊗min E. We use the notation in Remark 10.2 for B0 and I0.

Since the ranks of the un’s are bounded we must have sup ‖un‖cb < ∞ (see Remark 1.56), andhence (tn) ∈ (⊕

∑MNn ⊗min E)∞ ' B0 ⊗min E. Let t = (q0 ⊗ IdE)((tn)) ∈ (B0/I0) ⊗min E ⊂

(B0/I0)⊗min A. By Remark 10.2, we may assume that A is (B0, I0)-exact. Recalling Lemma 4.26,this gives us

(10.5) ‖t‖(B0⊗minE)/(I0⊗minE) ≤ ‖t‖(B0/I0)⊗minE .

By (10.3) we have on one hand

‖t‖(B0⊗minE)/(I0⊗minE) = lim sup ‖tn‖MNn⊗minE = lim sup ‖un‖cb,

and on the other hand we claim that

‖t‖(B0/I0)⊗minE ≤ supk lim supn→∞ ‖IdMk⊗ un : Mk(En)→Mk(E)‖.

The last two inequalities together with (10.5) imply (10.4), which proves (ii).We now prove the claim. By (1.11) (recalling (1.6)) we have

(10.6) ‖t‖(B0/I0)⊗minE = supk sup‖(IdB0/I0 ⊗ v)(t)‖(B0/I0)⊗minMk| v ∈ BCB(E,Mk),

and since (B0/I0)⊗min Mk = (B0 ⊗min Mk)/(I0 ⊗min Mk) (indeed Mk is trivially exact), we findfor any k and v ∈ CB(E,Mk) by (10.3)(10.7)‖(IdB0/I0 ⊗ v)(t)‖(B0/I0)⊗minMk

= lim supn ‖(IdMNn⊗ vun)(xn)‖MNn⊗minMk

≤ lim supn ‖vun‖cb

and since ‖vun‖cb = ‖IdMk⊗ vun‖ by (1.18), we have

lim supn ‖vun‖cb = lim supn ‖IdMk⊗ vun‖ ≤ lim supn ‖IdMk

⊗ un‖,

and the claim now follows from (10.6) and (10.7).Assume (ii). Let E ⊂ A with dim(E) < ∞. Since E is separable, we may assume E ⊂ B(H)(completely isometrically) with H separable. Let Hn ⊂ H (n ≥ 1) be an increasing sequence ofsubspaces with dim(Hn) < ∞ whose union is dense in H. Then (as earlier for (1.9)) for any kand any x ∈ Mk(E) we have ‖x‖Mk(E) = limn ↑ ‖IdMk

⊗ vn(x)‖ where vn(e) = PHne|Hn (e ∈ E).Since the unit sphere SE of E is compact (and |‖vn(x)‖ − ‖vn(y)‖| ≤ ‖x − y‖ ∀x, y ∈ E), wehave by Ascoli’s theorem infx∈SE ‖vn(x)‖ ↑ 1 and hence vn : E → vn(E) is invertible for all nlarge enough. For such n, let un = v−1

n : vn(E) → E be its inverse. We have ‖un‖ ↓ 1. Similarly,limn ↓ ‖IdMk

⊗un‖ = 1 for any k. By (ii) we have limn ‖un‖cb ≤ 1, and hence (iii) follows: choosingn large enough so that ‖un‖cb < 1 + ε we may take N = dim(Hn) and u = un.Assume (iii). Let B/I be a quotient C∗-algebra. To prove (10.2) it suffices to show that for anyE ⊂ A with dim(E) <∞ and any ε > 0 we have ‖(B/I)⊗min E → B⊗min E/(I ⊗min E)‖ ≤ 1 + ε.Let E ⊂ MN and u be as in (iii). Since MN is exact we have isometrically (B/I) ⊗min MN =(B⊗minMN )/(I ⊗minMN ) and hence also isometrically (B/I)⊗min E = (B⊗min E)/(I ⊗min E) byLemma 4.26. Therefore using IdE = u u−1 we find ‖(B/I)⊗min E → (B ⊗min E)/(I ⊗min E)‖ ≤‖u−1‖cb‖u‖cb ≤ 1 + ε. Thus we obtain (iv) and (iv) ⇒ (i) is trivial.

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Remark 10.4. By the results of the preceding section 7.2 (see Proposition 7.15) it is clear that anynuclear C∗-algebra is exact.

Remark 10.5 (On the stability properties of exactness). By Lemma 4.26 (applied here with theroles of A and B exchanged) if A is exact then any C∗-subalgebra D ⊂ A is also exact. This is alsoimmediate from (iii) ⇔ (iv) in Theorem 10.3. Incidentally, this shows that B is not exact.Kirchberg proved that exactness also passes to quotient C∗-algebras but this lies much deeper (see[39, p. 297] for a detailed proof). However, as we will prove in §18.3, exactness is not stable underextensions.By a simple iteration argument, one shows that the minimal tensor product of two exact C∗-algebras is exact. In sharp contrast, exactness is not preserved by the maximal tensor product.Indeed, by Theorem 4.11 there is an embedding C∗(Fn) ⊂ C∗λ(Fn)⊗maxC

∗λ(Fn) and C∗λ(Fn) is exact

(see Remark 10.21) while C∗(Fn) is not exact when n > 1 (see Proposition 7.34).

Remark 10.6 (Embeddings in the Cuntz algebra). Kirchberg ([158, 162]) obtained a series of strikingresults on exact C∗-algebras culminating with his outstanding proof that any separable exact C∗-algebra A can be embedded (as a C∗-subalgebra) in a nuclear C∗-algebra, namely the Cuntz algebraO2. Moreover, he showed that if A is nuclear the embedding A ⊂ O2 can be made so that thereis a contractive c.p. projection from O2 onto A. See [3] for a nice presentation of this subject.Most of the tools for this topic, being related to C∗-algebraic K-theory, are quite far from thesubject of the present volume, which explains why we do not discuss this fundamental embeddingany further. Nevertheless, Theorem 10.3 as well as the next statement are important steps (whichwe fully prove) on the way to this achievement. In any case, with the applications related to theCBAP in Theorem 10.18, all this already shows that exactness is a certain form of “subnuclearity”.

For (min→ max)-tensorizing c.p. maps, see Theorem 7.11 (which is proved in the next section).

Theorem 10.7. A C∗-algebra A ⊂ B(H) is exact if and only if the inclusion mapping jA : A →B(H) is (min→ max)-tensorizing.

Proof. Assume A exact. Let B be any unital C∗-algebra. Then, by Proposition 3.39, for some freegroup F, B is a quotient of C = C∗(F), so that B = C/I. Then the following ∗-homomorphism iscontractive

A⊗min B =A⊗min C

A⊗min I→ B(H)⊗min C

B(H)⊗min Ibut since the pair (C,B(H)) is nuclear (see (9.4)) we have

B(H)⊗min C

B(H)⊗min I=B(H)⊗max C

B(H)⊗max I

and by (7.4) this last space coincides with B(H)⊗max (C/I) = B(H)⊗maxB. Therefore A→ B(H)is (min→ max)-tensorizing.Conversely, if the latter inclusion is (min → max)-tensorizing, then for any C and any quotientC/I we have

A⊗min (C/I)→ B(H)⊗max (C/I) =B(H)⊗max C

B(H)⊗max Iand a fortiori

A⊗min (C/I)→ B(H)⊗min C

B(H)⊗min I.

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Thus if we denote as before Q[E] = E⊗minCE⊗minI we have a contractive map A⊗min (C/I)→ Q[B(H)].

But by Lemma 7.43 we know that the norm induced by Q[B(H)] on A ⊗ (C/I) is the norm ofQ[A]. Therefore we obtain ‖A⊗min (C/I)→ Q[A]‖ ≤ 1 which means that A is exact.Alternative route: (iii) in Theorem 10.3 implies that jA factors as in (iii) in Theorem 7.10.

Corollary 10.8. If A ⊂ B(H) is exact, any complete contraction u : A → W with values in aWEP C∗-algebra W is (min→ max)-tensorizing.

Proof. If W is WEP, the inclusion iW : W → W ∗∗ factors through some B(K) via c.p. completecontractions. By the extension property of B(K) (see Theorem 1.18) we have a factorization ofiWu of the form A→ B(H)→W ∗∗, showing that iWu is (min→ max)-tensorizing. Therefore (seeRemark 7.28) u itself is (min→ max)-tensorizing.

Applying this to u = IdA (and Recalling Remark 10.4) we immediately obtain

Corollary 10.9. A C∗-algebra A is nuclear if and only if A is exact and has the WEP.

Recall that we denote by Q = B/K (where B = B(`2),K = K(`2)) the classical “Calkinalgebra”. The analogue of the last corollary for the LLP involves Q.

Corollary 10.10. For a C∗-algebra A, the pair (A,Q) is nuclear if and only if A is exact and hasthe LLP. In that case, (A,B) is nuclear for any QWEP C∗-algebra B.

Proof. If A is exact we have

(10.8) A⊗min Q = (A⊗min B)/(A⊗min K ).

If A has the LLP, (A,B) is nuclear by our definition of the LLP. Therefore

(A⊗min B)/(A⊗min K ) = (A⊗max B)/(A⊗max K ).

By (7.6) (exactness of the max-tensor product):

(10.9) (A⊗max B)/(A⊗max K ) = A⊗max Q,

and hence (A,Q) is nuclear. The same argument (with Corollary 9.40) shows that (A,B) is nuclearfor any QWEP C∗-algebraB. Conversely, assume (A,Q) nuclear. Then (10.8) holds by (10.9) (sincethe max-norm dominates the min-norm on A⊗B). By Theorem 10.3, (10.8) implies exactness. By(10.8) and (10.9) we have

(A⊗min B)/(A⊗min K ) = (A⊗max B)/(A⊗max K ),

and also A⊗min K = A⊗max K . One easily deduces from this that A⊗min B = A⊗max B, whichmeans, by our definition of the LLP, that A has it.

Remark 10.11. Kirchberg conjectured (see Proposition 13.1) that LLP ⇒ WEP, and also that allC∗-algebras are QWEP (see Proposition 13.1). By the preceding corollaries, this would show that(A,Q) nuclear ⇒ A nuclear.

We now return to the connection between exactness and local reflexivity (see Remark 8.34 forimportant additional information).

Proposition 10.12. Property C ′ is equivalent to exactness.

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Proof. Assume that A has property C ′. Let I ⊂ B be an ideal. A preliminary observation is that by(26.37) the kernel of the natural mapping B∗∗⊗minA→ (B∗∗/I∗∗)⊗minA is equal to I∗∗⊗minA (seealso Corollary 7.52). Property C ′ tells us that we have an isometry B∗∗ ⊗min A → (B ⊗min A)∗∗.Let Z = ker[q ⊗ IdA : B ⊗min A → (B/I) ⊗min A]. Viewing B as embedded in B∗∗, we viewB ⊗min A ⊂ B∗∗ ⊗min A. We also may view I∗∗ ⊂ B∗∗. Then our preliminary observation showsus that Z ⊂ I∗∗ ⊗min A. By property C ′ a fortiori Z ⊂ I∗∗ ⊗min A ⊂ (I ⊗min A)∗∗. Therefore, anyz ∈ Z is in the σ((B ⊗min A)∗∗, (B ⊗min A)∗)-closure of a bounded net in I ⊗min A. But since bothZ and I ⊗min A are included in B ⊗min A, this means that any element z ∈ Z is the weak limit inB ⊗min A of a bounded net in I ⊗min A, and hence by (Mazur’s) Theorem 26.9 the norm limit ofanother such net, which implies that z ∈ I ⊗min A. Thus Z = I ⊗min A and A is exact.Conversely, assume A ⊂ B(H) exact. Let B be a C∗-algebra. By Theorem 10.7 we have

‖A⊗min B∗∗ → B(H)⊗max B

∗∗‖ = 1.

By (8.12) (and (4.8)) we have ‖B(H)⊗maxB∗∗ → (B(H)⊗maxB)∗∗‖ = 1 and a fortiori ‖B(H)⊗max

B∗∗ → (B(H) ⊗min B)∗∗‖ = 1. Therefore ‖A ⊗min B∗∗ → (B(H) ⊗min B)∗∗‖ = 1. But since the

inclusion A⊗minB → B(H)⊗minB is isometric, so is (A⊗minB)∗∗ → (B(H)⊗minB)∗∗. Thereforewe conclude ‖A⊗min B

∗∗ → (A⊗min B)∗∗‖ = 1, which means A has property C ′.

Remark 10.13. The argument described in Remark 8.32 to show that local reflexivity (i.e. propertyC ′′) passes to quotients also shows the same for property C. However, it does not seem to work forproperty C ′. Thus it does not lead to a proof of the same for exactness.

10.2 Nuclearity, exactness and approximation properties

The aim of this section is to give a reasonably direct proof that a C∗-algebra A is nuclear if andonly if it has the approximation properties mentioned in Theorem 7.11 (but not proved yet). Weexpand on the same theme including several more general statements in the next section. Theproofs will require the dual description of the maximal C∗-norm given by Theorem 6.15.

The next statement is the analogue for general linear maps of the characterization of (min →max)-tensorizing maps stated (but not proved yet) in Theorem 7.10.

Theorem 10.14. Let λ be a positive constant. Consider two C∗-algebras A and B and an operatorsubspace E ⊂ A. Let u : E → B be a linear mapping. The following assertions are equivalent.

(i) For any C∗-algebra C, IdC ⊗ u defines a bounded linear map from C ⊗min E to C ⊗max Bwith norm ≤ λ. In other words, u is (min→ max)-tensorizing with constant λ.

(ii) Same as (i) with C = C∗〈F ∗〉 for all finite dimensional operator subspaces F ⊂ E.

(iii) For any finite dimensional subspace F ⊂ E, the restriction u|F admits, for any ε > 0, a

factorization of the form FV−→Mn

w−→B with ‖V ‖cb‖w‖dec ≤ λ+ ε.

(iv) There is a net of finite rank maps ui : E → B admitting factorizations through matrix algebrasof the form

Mniwi

!!A ⊃ E

vi::

ui // B

with ‖vi‖cb‖wi‖dec ≤ λ such that ui = wivi converges pointwise to u.

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(v) There is a net ui : A→ B of finite rank maps with sup ‖ui‖dec ≤ λ such that the restrictionsui|E tend pointwise to u.

Proof. (i) ⇒ (ii) is trivial. Assume (ii). Let F ⊂ E be an arbitrary finite dimensional subspaceand let tF ∈ F ∗ ⊗ E be the tensor associated to the inclusion jF : F → E. Let C = C∗〈F ∗〉. By(ii) we have ‖(IdC ⊗ u)(tF )‖C⊗maxB ≤ λ‖tF ‖C⊗minE . But by the injectivity of the min-norm (seeRemark 1.10) and by (2.15), we have

‖tF ‖C⊗minE = ‖tF ‖F ∗⊗minE = ‖jF ‖CB(F,E) = 1.

Hence we have ‖(IdC ⊗ u)(tF )‖C⊗maxB ≤ λ. By (6.18) and Remark 6.17, this implies that, for anyε > 0, there is a factorization of u|F of the following form

Mn

w

!!F

V==

u|F // B

with ‖V ‖cb ≤ 1 and ‖w‖dec ≤ λ+ε. This shows (ii) ⇒ (iii). Assume (iii). By the extension propertyof Mn (see Theorem 1.18), we can extend V to a mapping v : E →Mn with ‖v‖cb ≤ ‖V ‖cb. Thusif we take for index set I the set of all finite dimensional subspaces F ⊂ E (directed by inclusion)we obtain nets vi : E → Mni and wi : Mni → B such that ‖wivi(x) − u(x)‖ → 0 for all x in Eand such that (after a suitable renormalization) sup‖vi‖cb‖wi‖dec ≤ 1. This completes the proofthat (iii) ⇒ (iv). We may clearly assume (by the extension Theorem 1.18) that vi is extended toA with the same cb norm, thus recalling (6.7) and (6.9), (iv) ⇒ (v) is immediate.Finally, the proof that (v) ⇒ (i) is an immediate consequence of Proposition 6.13.

Recall that a C∗-algebra A is said to have the CPAP (for completely positive approximationproperty) if the identity on A is in the pointwise closure of the set of finite rank c.p. maps on A.To derive the known results on this approximation property, the following will be useful (here wefollow closely [208, Ch. 12]).

Lemma 10.15. Let E ⊂ B(H) be a finite dimensional operator system and let A be a C∗-algebra.Consider a unital self-adjoint mapping u : E → A associated to a tensor t ∈ E∗ ⊗ A. Fix ε > 0.Then if δ(t) < 1 + ε, we can decompose u as u = ϕ − ψ with ϕ,ψ c.p. such that ‖ψ‖ ≤ ε and ϕadmits for some n a factorization of the form

Mn

W

!!E

V==

ϕ // A

where V,W are c.p. maps with

‖V ‖ ≤ 1 + ε and ‖W‖ ≤ 1.

Proof. By the definition of the norm δ (see §6.2) and by Theorem 1.50, we can assume that u = wvwhere v : E →Mn satisfies

∀x ∈ E v(x) = v∗1π(x)v2,

where π : E → B(H) is the restriction of a ∗-homomorphism, v1, v2 are operators in B(`n2 , H)with ‖v1‖ = ‖v2‖ < (1 + ε)1/2, and w : Mn → A is defined by w(eij) = aibj with ‖

∑aia∗i ‖ =

157

‖∑b∗jbj‖ < 1. Let b be a column matrix with entries (b1, . . . , bn) and a∗ a row matrix with entries

(a1, . . . , an), so that, in matrix notation, we have u(x) = a∗ · v∗1π(x)v2 · b.Since u is self-adjoint we have u(x) = u(x∗)∗ = b∗v∗2π(x)v1a and hence (by “polarization”)

u(x) = ϕ(x)− ψ(x)

with

ϕ(x) = (1/4)[(v1a+ v2b)∗π(x)(v1a+ v2b)] and ψ(x) = (1/4)[(v1a− v2b)

∗π(x)(v1a− v2b)].

Clearly ϕ and ψ are c.p. and 1 = u(1) = ϕ(1)−ψ(1) so that ϕ(1) ≥ 1 and ‖ϕ(1)‖ ≤ ‖v1a+ v2b‖2/4 ≤(1 + ε), which implies that ‖ϕ(1) − 1‖ ≤ ε. Thus we obtain (see (1.27)) ‖ψ‖ = ‖ψ(1)‖ ≤ ε. Itremains to show that ϕ admits the announced factorization. Let (again in matrix notation)

V (x) =1

2

v∗1π(x)v1 v∗1π(x)v2

v∗2π(x)v1 v∗2π(x)v2

=1

2

(v∗1v∗2

)π(x)(v1, v2).

Clearly V is c.p. from E to M2n and ‖V ‖ ≤ (1/2)(‖v1‖2 + ‖v2‖2) ≤ 1 + ε. Moreover, if wedefine W : M2n → A by W (t) = 1

2(a∗ b∗)t(ab

)then W is c.p. with ‖W‖ ≤ 1 and we have

ϕ(x) = W (V (x)).

In the case of c.p. maps, Theorem 10.14 becomes:

Corollary 10.16. Let u : A → B be a unital c.p. map between two unital C∗-algebras. Thefollowing assertions are equivalent.

(i) The mapping u is (min→ max)-tensorizing.

(ii) For any C∗-algebra C, IdC ⊗ u defines a completely positive linear map from C ⊗min A toC ⊗max B with norm = 1.

(iii) Same as (ii) with C = C∗〈E∗〉 for all finite dimensional operator systems E ⊂ A.

(iv) There is a net of finite rank maps (ui) admitting factorizations through matrix algebras of theform

Mniwi

!!A

vi==

ui // B

with c.p. maps vi and wi satisfying ‖vi‖‖wi‖ ≤ 1 such that ui = wivi converges pointwise tou.

(v) There is a net ui : A→ B of finite rank c.p. maps that tends pointwise to u.

Proof. (i) ⇒ (ii) was already observed in Remark 7.9 and (ii) ⇒ (iii) is trivial. Assume (iii). LetE be a finite dimensional operator system inside A. Let B = C∗〈E∗〉. Let tE ∈ E∗ ⊗ A be thetensor associated to the inclusion map jE : E → A, so that ‖tE‖min = ‖jE‖cb = 1. By (ii) wehave ‖(IdC ⊗ u)(tE)‖max ≤ 1. Note that the linear map from E to B associated to the tensor(IdC ⊗ u)(tE) is nothing but u|E . Hence, by (6.18) (6.20) and by Lemma 10.15, for any ε > 0 wecan write u|E = ϕ− ψ with ϕ = WV as in Lemma 10.15. By the extension property of c.p. mapswe may as well assume that V is a c.p. mapping of norm ≤ 1 + ε from A to Mn. Then, using the

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net (directed by inclusion) formed by the finite dimensional operator systems E ⊂ A, and lettingε→ 0 we obtain (iv) (after a suitable renormalization of V ). Then (iv) ⇒ (v) is trivial.Lastly, assume (v). Note that since ‖ui‖ = ‖ui(1)‖ and ‖ui(1)‖ → ‖u(1)‖ = 1, we have “automati-cally” ‖ui‖ → 1. By (6.16) and (i) in Lemma 6.5, we have for any C∗-algebra C

‖IdC ⊗ ui‖C⊗minA→C⊗maxB ≤ ‖ui‖,

and hence since ui → u pointwise

‖IdC ⊗ u‖C⊗minA→C⊗maxB ≤ 1.

This shows that (v) ⇒ (i).

We now recover the most classical characterization of nuclear C∗-algebras, due independentlyto Choi-Effros and Kirchberg [45, 154]:

Theorem 10.17 (Nuclear ⇔ CPAP). The following properties of a unital C∗-algebra A are equiv-alent:

(i) A is nuclear.

(ii) There is a net of finite rank maps of the form Avi−→Mni

wi−→A where vi, wi are c.p. maps with‖vi‖ ≤ 1, ‖wi‖ ≤ 1, that tends pointwise to the identity.

(iii) A has the CPAP.

Proof. This follows from Corollary 10.16 with A = B and u = IdA.

Just like Theorem 10.7 the next statement is a characterization of exactness for C∗-algebras,that is parallel to that of nuclearity. We just replace the identity on A namely IdA by the inclusionjA : A → B(H). This already suggests to think of exactness as some sort of “subnuclearity”, atleast in a somewhat local sense, but actually Kirchberg proved that separable exact C∗-algebrasglobally embed in nuclear ones (see Remark 10.6).

Note however, in sharp contrast, that the preceding paralellism does not extend to (iv) inTheorem 10.18, since the CBAP for IdA does not imply nuclearity (see Remark 10.21).

Theorem 10.18. The following properties of a unital C∗-algebra A ⊂ B(H) are equivalent:

(i) A is exact.

(ii) There is a net of finite rank maps of the form Avi−→Mni

wi−→B(H) where vi, wi are c.p. mapswith ‖vi‖ ≤ 1, ‖wi‖ ≤ 1, that tends pointwise to the inclusion mapping jA : A ⊂ B(H).

(iii) There is a net of finite rank c.p. maps ui : A→ B(H) tending pointwise to jA : A ⊂ B(H)(in other words, we might say that “jA has the CPAP”).

(iv) There is a net of finite rank c.b. maps ui : A → B(H) with supi ‖ui‖cb < ∞ that tendspointwise to jA : A ⊂ B(H) (we might say that “jA has the CBAP”). .

Proof. (i) ⇔ (ii) ⇔ (iii) follows from Corollary 10.16 with B = B(H) and u = jA.(iii) ⇒ (iv) is trivial. Assume (iv). By (6.9) ‖ui‖dec = ‖ui‖cb. By (i) ⇔ (v) in Theorem 10.14,jA is (min → max)-tensorizing with constant λ = supi ‖ui‖cb. But actually since jA is a ∗-homomorphism, this automatically is true also with a constant = 1 (see Proposition 26.24). ThusA is exact (meaning (i) holds) by Theorem 10.7.

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Remark 10.19. [On unitizations] A C∗-algebra A is nuclear (resp. exact) if and only if its unitizationis nuclear (resp. exact). Indeed, A is an ideal in its unitization A so that A/A = C. By Proposition7.19, an ideal in a nuclear C∗-algebra is nuclear. Thus A nuclear implies A nuclear. The analoguefor exactness is obvious (see Remark 10.5). The converses are easy and left as an exercise to thereader. Moreover, using an approximate unit of A, one easily shows that the CPAP of A impliesthat of A. This remark allows us to extend Theorem 10.17 to the non-unital case. A similarreasoning applies for exactness and Theorem 10.18.

Remark 10.20. [On the CBAP] Let Λ(E) be the smallest λ for which E has the λ-CBAP in thesense of Definition 9.48. By Theorem 10.18 any C∗-algebra with the CBAP must be exact. Werefer the reader to Cowling and Haagerup’s [64] for an important example of a sequence of groups(Gn) (with property (T)) for which Λ(C∗λ(Gn)) = n+ 1 (n = 1, 2, ...).

Remark 10.21. [Exactness for free groups] By Haagerup’s classical paper [103], for any free groupF, the reduced C∗-algebra C∗λ(F) has the 1-CBAP, and a fortiori is exact. Thus unlike the CPAPthe CBAP does not imply nuclearity.In sharp contrast, the full C∗-algebra C∗(Fn) is not exact whenever n ≥ 2. The latter fact followsfrom (7.8) and (7.9).

Remark 10.22 (On the weak* CBAP). Let λ > 0 be a constant. We say that a von Neumannalgebra M has the weak* λ-CBAP if there is a net of normal finite rank maps Ti : M → M with‖Ti‖cb ≤ λ that tend weak* to the identity. Haagerup proved in 1986 (see [108]) that for a discretegroup G, the λ-CBAP for C∗λ(G) is equivalent to the weak* λ-CBAP for MG, and moreover whenthis holds we may always find the net (Ti) formed of finite rank multipliers (see Proposition 3.25and Remark 3.29). In particular the latter net is then formed of maps such that Ti(MG) ⊂ C∗λ(G).By Remark 10.21, MG has the weak* 1-CBAP for any free group G. See [41, 110] for furtherdevelopments on approximation properties for groups. Because of the equivalent reformulationwith multipliers, the groups G for which C∗λ(G) has the CBAP are now called weakly amenable.Haagerup (1986, unpublished) proved that SL3(Z) is not weakly amenable (see also [190]), eventhough C∗λ(G) is exact when G = SL3(Z) as well as when G is any closed discrete subgroup of aconnected Lie group. This last fact is attributed to A. Connes in [155, p.453]. By [99] the sameholds when G is any discrete linear group. See [164, 113, 114, 109] for more recent breakthroughsin this direction.

Remark 10.23. Haagerup also proved in [103] a different kind of multiplier approximation propertyfor free groups, now called the Haagerup property, for which we refer to [42].

10.3 More on nuclearity and approximation properties

In this section we present a result (due to Junge and Le Merdy) that refines Theorem 6.15 usingan original application of Kaplansky’s density theorem (see Theorem 26.47) discovered by Junge[137], as follows. We merely sketch the proof.

Lemma 10.24. Let A,B be arbitrary C∗-algebras. Then any c.b. map θ : B∗ → A can be approx-imated in the point-norm topology by a net of weak∗-continuous finite rank maps θi : B∗ → A with‖θi‖cb ≤ ‖θ‖cb.

Sketch. Let A∗∗ ⊂ B(H), and B∗∗ ⊂ B(K) be embeddings (as von Neumann subalgebras). LetM = A∗∗⊗B∗∗ denote the von Neumann algebra generated by A∗∗ ⊗ B∗∗ in B(H ⊗2 K). Thespace CB(B∗, A∗∗) can be identified isometrically with M in a natural way (see e.g. [208, p.49]for details). Let t ∈ A∗∗⊗B∗∗ be the tensor associated to iAθ : B∗ → A∗∗ (recall iA : A → A∗∗

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is the canonical inclusion). Then, by Kaplansky’s density Theorem 26.47, there is a net (ti) inA ⊗ B with ‖ti‖min ≤ ‖t‖ such that ti σ(M,M∗)-tends to t. Let θi : B∗ → A be the finite rankmap associated to ti. We have ‖θi‖cb = ‖ti‖min ≤ ‖t‖ = ‖θ‖cb. Moreover, for any ξ in B∗, θi(ξ)must σ(A∗∗, A∗)-tend to θ(ξ). But since θi(ξ) and θ(ξ) both lie in A, this means that θi(ξ) tendsto θ(ξ) weakly in A. Passing to suitable convex hulls, we obtain (by Mazur’s Theorem 26.9) a net(θi) such that, for any ξ, θi(ξ) tends to θ(ξ) in norm.

Remark 10.25. The same argument shows that, for any von Neumann algebra R, any c.b. mapθ : R∗ → A can be approximated pointwise by a net of finite rank maps θi : R∗ → A with‖θi‖cb ≤ ‖θ‖cb.

Remark 10.26. We will use the approximation property described in Lemma 10.24 via the following:Let A be a C∗-algebra. Assume that E is the dual B∗ (resp. the predual R∗) of a C∗-algebra B(resp. von Neumann algebra R). Then for any y ∈ E ⊗ A the supremum defining the norm ∆(y)in (6.17) can be restricted to pairs (θ, π) where θ : E → π(A)′ is a complete contraction of finiterank. Moreover in case E = B∗, we can restrict to weak∗-continuous finite rank maps θi : B∗ → A.Indeed, this is an immediate application of Lemma 10.24 (resp. Remark 10.25).

The next statement shows that if E is a C∗-algebra and u a finite rank map, the “approximatefactorizations” of u appearing in part (iv) of Theorem 10.14 become bona fide factorizations of u.

Theorem 10.27 ([137]). Let u : B → A be a finite rank map between two C∗-algebras. Then, forany ε > 0, there is an integer n and a factorization u = wv of the form

Bv−→Mn

w−→A

with (‖v‖cb‖w‖cb ≤) ‖v‖cb‖w‖dec ≤ ‖u‖dec(1 + ε).Therefore, if y ∈ B∗ ⊗A is the tensor associated to u : B → A, we have

‖u‖dec = δ(y).

Proof. If u = wv as in Theorem 10.27, then, by (6.7) and (6.9), we have ‖u‖dec ≤ ‖v‖dec‖w‖dec =‖v‖cb‖w‖dec, hence by Remark 6.17,

‖u‖dec ≤ δ(y).

We now turn to the converse. We may assume u(x) =∑k

1 ξj(x)aj with ξj ∈ B∗ and aj ∈ A, orequivalently y =

∑ξj ⊗ aj . By the equality δ = ∆ reinforced by the preceding Remark 10.26 it

suffices to show the following claim: for any representation π : A → B(H) and weak∗ continuousfinite rank map θ : B∗ → π(A)′ with ‖θ‖cb ≤ 1, we have∥∥∥∑ θ(ξj)π(aj)

∥∥∥ ≤ 1.

Let θ be such a map and let t ∈ π(A)′⊗B be the tensor associated to it, so that ‖t‖min = ‖θ‖cb ≤ 1.Let C = π(A)′. By (6.16), since u : B → A has finite rank, we have

‖(IdC ⊗ u)(t)‖C⊗maxA ≤ ‖u‖dec‖t‖C⊗minB = ‖u‖dec‖θ‖cb ≤ ‖u‖dec.

Since (IdC ⊗ u)(t) =∑θ(ξj)⊗ aj , this yields∥∥∥∑ θ(ξj)π(aj)

∥∥∥ ≤ ∥∥∥∑ θ(ξj)⊗ aj∥∥∥π(A)′⊗maxA

≤ ‖u‖dec.

This proves the claim, and hence the equality δ(y) = ‖u‖dec. Then, the first assertion follows fromRemark 6.17 (applied with E = B).

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Remark 10.28. The following question seems interesting: fix ε > 0, let k be the rank of u, can weobtain the preceding factorization with n ≤ f(k, ε) for some function f ? In other words can wecontrol n by a function f depending only on k and ε?

10.4 Notes and Remarks

§10.1 is due to Kirchberg. In the separable case, exact C∗-algebras are just C∗-subalgebras ofnuclear ones, but the latter fact was proved (by Kirchberg) as the crowning achievement of a longseries of his own previous works [159, 160, 155]. We present in Theorems 10.3 and 10.18 only asimpler (but quite important) step that led him to that result. The fact that (iii) in Theorem 10.3characterizes exactness is related to what Kirchberg called locfin(A) in [159]. These ideas makesense equally well for operator spaces, but in that generality one needs to introduce a constant ofexactness, that replaces the constant 1 appearing in (10.2), see [208, ch. 17]. See [162] (or [3]) for aproof of the full result, namely that a separable exact C∗-algebra A embeds in O2, and that if A isnuclear the embedding A ⊂ O2 can be obtained together with a completely contractive projectiononto A. In §10.2 the main points are due to Choi-Effros (Theorem 10.17) and Kirchberg (Theorem10.18). The use of the δ-norm allows us to deduce them from results on general linear maps as inour previous book [208]. Theorem 10.27 is due to Junge and Le Merdy [137]. The latter statementexplains transparently why the CPAP implies a reinforced approximation property by c.p. mapsthat uniformly factorize through Mn.

11 Traces and ultraproducts

11.1 Traces

We start by some preliminaries on non-commutative (or more appropriately not necessarily com-mutative) measure spaces.

Let M be a von Neumann algebra. Let M+ denote the positive part of M . We recall that atrace on M is a map τ : M+ → [0,∞] satisfying

• (i) τ(x+ y) = τ(x) + τ(y), ∀ x, y ∈M+;

• (ii) τ(cx) = cτ(x), ∀ c ∈ [0,∞), x ∈M+;

• (iii) τ(a∗a) = τ(aa∗), ∀ a ∈M (this is the “tracial” condition).

τ is said to be faithful if τ(x) = 0 implies x = 0, normal if

(11.1) supi τ(xi) = τ(supi xi)

for any bounded increasing net (xi) in M+. Note that since (xi) is bounded there is x in M+ suchthat, for any h in H, 〈xih, h〉 ↑ 〈xh, h〉, which implies that xi tends to x weak* (see Remark 26.11)and hence x ∈M+. The operator x being obviously the least upper bound of (xi), it is natural todenote it by supi xi.By considering the net of finite partial sums

∑i∈γ Pi (γ ⊂ I), we see that (11.1) implies that

τ(∑

Pi) =∑

τ(Pi)

for any mutually orthogonal family of (self-adjoint) projections (Pi)i∈I in M , which is analogousto the σ-additivity of measures.

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When dealing with von Neumann algebras, it is customary to refer to self-adjoint projectionssimply as “projections”. Since the confusion this abuse may create is very unlikely, we adopt thisconvention: in a von Neumann algebra, the term projection will always mean a self-adjoint one.

The trace τ is called semifinite if for any nonzero x ∈M+ there is y ∈M+, such that 0 6= y ≤ xand τ(y) <∞, and finite if τ(1) <∞ (1 denoting the identity of M). In the finite case, τ(x) <∞for any x ≥ 0.

Remark 11.1. If τ is semifinite there is a family of mutually orthogonal projections (pi)i∈I in Mwith τ(pi) < ∞ for all i ∈ I such that

∑i∈I pi = 1. Indeed, let (pi)i∈I be a maximal such family

(except for the latter condition). If r = 1−∑

i∈I pi 6= 0 there is y ∈M nonzero such that 0 ≤ y ≤ rwith τ(y) <∞. By spectral theory we have y ≥ λq for some λ > 0 and some projection 0 6= q ∈M .Then q contradicts the maximality of (pi)i∈I . Therefore r = 0.

A von Neumann algebra M is called finite if the family formed of the finite normal tracesseparates the points of M . Clearly this happens if M admits a single faithful normal finite trace,but a finite M may fail to have any faithful finite trace, for instance M = `∞(Γ) with Γ uncountable.However, on a separable Hilbert space (i.e. if M is weak*-separable) the converse is also true; thenM is finite if and only if it admits a faithful normal finite trace.

The definition of semifiniteness is simpler: A von Neumann algebra M is called semifinite if Madmits a faithful normal semifinite trace.

For example, B(H) is semifinite for any H. The trace τ : B(H)+ → [0,∞] is the usual onedefined for T ∈ B(H)+ by τ(T ) =

∑〈ei, T (ei)〉 where (ei) is any orthonormal basis of H. The

abundance of finite rank (and hence finite trace) operators guarantees semifiniteness. B(H) is finiteif and only if dim(H) <∞.

Remark 11.2 (Classification of von Neumann algebras in types I, II, III). It is traditional to classifyvon Neumann algebras in three main classes called type I, type II and type III. A von Neumannalgebra is called of type I if it is isomorphic to a direct sum of the form (⊕

∑i∈I Ci⊗B(Hi))∞ where

(Hi)i∈I is a family of (mutually non-isomorphic) Hilbert spaces and (Ci)i∈I a family of commutativevon Neumann algebras. It is easy to check that these are semifinite.

A von Neumann algebra M is called of type II if it is semifinite and if there is no non zeroprojection p for which pMp is commutative (equivalently no p 6= 0 for which pMp is type I). Anysemifinite M can be decomposed as M = MI ⊕MII with MI (resp. MII) of type I (resp. II).

Then M is called of type III if there is no non zero projection p ∈M for which pMp is semifinite.Any von Neumann algebra M can be decomposed as

M = MI ⊕MII ⊕MIII

with MI (resp. MII , rresp MIII) of type I (resp. II, rresp III).See e.g. [241, ch. V] for a complete discussion.

In these notes we will be mainly concerned by finite or semifinite von Neumann algebras, (i.e.algebras of type I or II). However, in several instances we will have to consider the case of type IIIvon Neumann algebras. Then an important structure theorem due to Takesaki will come to ourrescue. We will use it as a “black box” and refer for the proof to Takesaki’s book [243, Th.XII.1.1p. 364 and Th. X.2.3] or to the original paper [240, Th. 4.5 and §8]. See also [146, §13.3]. A veryconcise description can be found in [257]. See also [111] for useful information on the same theme.

Theorem 11.3 (Takesaki’s duality theorem). Let M be a von Neumann algebra. There is asemifinite von Neumann algebra M with the following two properties:

(i) There is an embedding M ⊂ M of M as a von Neumann subalgebra and a c.p. projectionP :M→M with ‖P‖ = 1.

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(ii) There is an embedding M ⊂ M of M as a von Neumann subalgebra and a c.p. projectionQ : M →M with ‖Q‖ = 1.

In its usual form, Takesaki’s duality theorem asserts that Theorem 11.3 holds for any vonNeumann algebra of type III. Schematically, in the latter caseM is the crossed product of M witha one parameter automorphism group called the modular group. The algebra M appears as thefixed point algebra with respect to a certain one parameter (dual) automorphism group acting onM. Then a well known averaging argument based on the amenability of R (the parameter group)shows that there is a contractive projection P fromM to M . To obtain (ii) a similar constructionis applied to M (whence Q by the same averaging argument) but the resulting “double” crossedproduct is now isomorphic to M , so we obtain in this way the type III case. Since the semifinitecase is trivial, one can use the decomposition M = MI ⊕MII ⊕MIII and apply Takesaki’s dualitytheorem for the type III case to produce a semifiniteMIII associated to MIII as in Theorem 11.3.Then we obtain the general case by simply setting M = MI ⊕MII ⊕MIII .

Remark 11.4. In general the projection appearing in (i) or (ii) is not normal (see [249]), reflectingthe fact that an invariant mean is not a measure. However, just like invariant means are pointwiselimits of true measures there is a net of normal contractive c.p. maps Pi : M → M that tendpointwise to P for the weak* topology on M , and similarly for Q (see [257, p. 45] for more details).

Remark 11.5 (Lp-spaces for semifinite traces). Let τ be a semifinite faithful normal trace on a vonNeumann algebra M . Then A = x ∈ M | τ(|x|) < ∞ is a weak* dense ∗-subalgebra of M .When τ is finite, of course A = M . For any 1 ≤ p <∞ one usually defines the space Lp(τ) as thecompletion of A equipped with the norm defined for any x ∈ A by ‖x‖p = (τ(|x|p))1/p.For example, when M = B(H) with the usual trace the space L2(τ) (resp. L1(τ)) can be identifiedwith the Hilbert-Schmidt class (resp. the trace class). When 1 ≤ p < ∞ we obtain the so-calledSchatten p-class.By Remark 11.1 in any semifinite M there is a directed increasing net (pγ) of projections with finitetraces tending weak* to 1 (just take pγ =

∑i∈γ pi for any finite subset γ ⊂ I). Let Mγ = pγMpγ

and let τγ denote the restriction of τ to Mγ , which is clearly a finite faithful normal trace on Mγ .For any γ ≤ δ in the net we have Mγ ⊂Mδ and there is a natural isometric embedding

Lp(Mγ , τγ) ⊂ Lp(Mδ, τδ).

Then it is not hard to check that the space Lp(τ) that we just defined can be identified with thecompletion of the union ∪γLp(Mγ , τγ) with its natural norm. In this way the construction of thespaces Lp(τ) can be reduced to the finite trace case.Since we will be using the semifinite case only much later in §22 we prefer to concentrate for nowon the slightly less technical case of finite traces with p = 2 or p = 1.

11.2 Tracial probability spaces and the space L1(τ)

In general, a functional f on an algebra A is called tracial if f(xy) = f(yx) for any x, y ∈ A.

Definition 11.6. By a “tracial probability space”, we mean a von Neumann algebra M equippedwith a trace τ : M+ → R+ that is normalized (i.e. such that τ(1) = 1) faithful and normal.Then (see Remark 11.7) τ then extends to a normal (tracial) state on M , so that

∀x, y ∈M τ(xy) = τ(yx).

Remark 11.7. Indeed, let Msa denote the set of self-adjoint elements in M . Since any x in Msa

can be written x = x1 − x2 with x1, x2 ∈ M+, τ uniquely extends to an R-linear form on Msa by

164

setting τ(x) = τ(x1)− τ(x2). Since τ is additive on M+, this definition is unambiguous. Note thatsince τ is nonnegative on M+, this extension of τ preserves order, i.e. x ≤ y implies τ(x) ≤ τ(y).Then, by complexification, τ uniquely extends to a positive C-linear form (and hence a state) onM . The latter state is normal by Theorem 26.44. Moreover, by polarization, the tracial propertyof τ on M+ implies τ(y∗x) = τ(xy∗) for any x, y ∈M , or equivalently (replace y by y∗) τ is tracial.

If M is commutative, M can be identified with L∞(Ω,A , µ) for some abstract probability space(Ω,A , µ), then projections are indicator functions of sets in A and τ corresponds to µ on (Ω,A ),i.e. we have for all f in L∞(Ω,A , µ)

τ(f) =

∫f(ω) dµ(ω).

For this reason, (M, τ) is usually called a “non-commutative” probability space, but since we wantto include the commutative case, we prefer the term tracial probability space.

The simplest example is of course the algebra Mn of all n×n matrices equipped with the usualnormalized trace, τ(x) = n−1tr(x). As we will see in the next section, infinite tensor products andultraproducts of matrix algebras of varying sizes give us examples of tracial probability spaces. Anydiscrete group G also provides us with another fundamental example namely (MG, τG) (see §3.5).

Let (M, τ) be a tracial probability space, and let M∗ be the predual of M . By definition (see§26.16), this is the subspace of M∗ formed of all the normal (i.e. weak*-continuous) elements ofM∗. It is well known that M∗ can be identified with the (“non-commutative”) L1-space associatedto (M, τ). Let us now explain how the latter is defined and how this identification works. For anyx in M , we set

‖x‖1 = τ(|x|).

Proposition 11.8. For all x in M we have |τ(x)| ≤ τ(|x|). More generally

(11.2) τ(|x|) = sup|τ(xy)| | y ∈M ‖y‖ ≤ 1 = sup|τ(yx)| | y ∈M ‖y‖ ≤ 1.

In particular, this shows that ‖ ‖1 is a norm on M .

Proof. Recall that as any positive functional, τ satisfies the Cauchy-Schwarz inequality, that is|τ(ba)| ≤ τ(bb∗)τ(a∗a) for all a, b ∈M . Assume ‖y‖ ≤ 1. Let x = u|x| be the polar decompositionof x. Let a = u|x|1/2 and b = |x|1/2y. Note a∗a = |x| and bb∗ ≤ |x|. By Cauchy-Schwarz we have

|τ(xy)| = |τ(ab)| = |τ(ba)| ≤ (τ(a∗a)τ(bb∗))1/2 ≤ τ(|x|).

In particular, taking y = 1, we obtain |τ(x)| ≤ τ(|x|). Finally, |x| = u∗x yields the first equality in(11.2). Since τ is tracial, the second one is clear.

The space L1(M, τ) or simply L1(τ) is defined as the completion of M with respect to the norm‖ ‖1. Note that, by definition, we have an inclusion with dense range

M ⊂ L1(M, τ).

For any x ∈M , let fx ∈M∗ be the linear form defined on M by

(11.3) fx(y) = τ(xy).

Clearly, fx is normal (indeed, since τ is normal, this follows from Remark 26.37. Thus fx ∈M∗.

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Remark 11.9. Let Λ ⊂ M∗ be the linear subspace defined by Λ = fx | x ∈ M. We claim that Λis dense in M∗. To see this it suffices to show that it is σ(M∗,M)-dense, or equivalently that anyϕ ∈ (M∗)

∗ = M that vanishes on Λ must vanish on the whole of M∗. Equivalently, we must showthat if y ∈ M (corresponding to ϕ) is such that fx(y) = 0 for any x ∈ M then y = 0. Indeed thisis clear: the choice of x = y∗ gives us τ(y∗y) = 0, and hence y = 0.

Remark 11.10. More generally, let V ⊂M be any σ(M,M∗)-dense linear subspace. Then the sameargument as for Remark 11.9 shows that V is dense in L1(τ).

Theorem 11.11. The map that takes x ∈ M to fx extends to an isometric isomorphism fromL1(τ) onto M∗. In short we have

L1(τ) 'M∗ (isometrically).

Proof. By (11.2), ‖fx‖M∗ = τ(|x|) = ‖x‖1. Therefore, since Λ is dense in M∗, the latter can beviewed as the completion of (Λ, ‖.‖1).

Consequently, since M = (M∗)∗, we also have

L1(τ)∗ 'M (isometrically),

or more explicitly for the record:

Corollary 11.12. The map that takes y ∈ M to the functional ϕy ∈ L1(τ)∗ defined by ϕy(x) =τ(xy) is an isometric isomorphism from M onto L1(τ)∗. Thus we have

(11.4) ∀y ∈M ‖y‖ = sup|τ(xy)| | x ∈M, τ(|x|) ≤ 1.

11.3 The space L2(τ)

Let (M, τ) be a tracial probability space. We will denote by L2(τ) the Hilbert space associated to(M, τ) by the GNS construction (see §26.13). More precisely, the space L2(τ) is the completion ofM with respect to the norm x 7→ ‖x‖2 defined on M by

‖x‖2 = (τ(|x|2))1/2,

or equivalently ‖x‖2 = (τ(x∗x))1/2. Note that the traciality of τ ensures that

‖x‖2 = (τ(x∗x))1/2 = (τ(xx∗))1/2 = ‖x∗‖2.

The latter norm is derived from the scalar product

(11.5) 〈y, x〉 = τ(y∗x) (x, y ∈M),

which, as already mentioned, satisfies the Cauchy–Schwarz inequality

(11.6) |〈y, x〉| ≤ τ(x∗x)1/2τ(y∗y)1/2 = ‖x‖2‖y‖2.

Note that this shows in particular that x → ‖x‖2 is subadditive, and since τ is assumed faithful(i.e. such that τ(x∗x) = 0⇔ x = 0), we see that ‖ · ‖2 is indeed a norm on M .

Remark 11.13. Let (M, τ) be a tracial probability space. If x ∈ BM is such that τ(x∗x) = 1 thenx is unitary (and conversely of course). Indeed, τ(1− x∗x) = 0 and 1− x∗x ≥ 0 imply x∗x = 1 bythe faithfulness of τ . Similarly xx∗ = 1 so that x ∈ U(M).

166

Remark 11.14. Recall that the “modulus” we constantly use is defined for all x ∈ M by |x| =(x∗x)1/2. It is worthwhile to emphasize that the triangle inequality |x + y| ≤ |x| + |y| is false forthis modulus. Here is an example: Let x = ( 0 1

0 1 ). Then x−x∗ is unitary so |x−x∗| = I. However, asimple calculation shows that |x| =

√2 ( 0 0

0 1 ) and |x∗| = 2−1/2 ( 1 11 1 ) so that 〈(|x|+|x∗|)e1, e1〉 = 2−1/2

while 〈|x− x∗|e1, e1〉 = 1. Therefore |x− x∗| 6≤ |x|+ |x∗|.Note however the following useful substitute from [1]: for any pair x, y in M there are isometries

U, V in M such that |x+ y| ≤ U |x|U∗ + V |y|V ∗.From the GNS construction, we recall that for any a, x in M we have a∗a ≤ ‖a‖21 and hence

x∗a∗ax ≤ ‖a‖2x∗x which implies ‖ax‖2 ≤ ‖a‖‖x‖2. Therefore, the left multiplication by a extendsby density to a bounded operator

L(a) : L2(τ)→ L2(τ)

such that ‖L(a)‖ ≤ ‖a‖ and L : M → B(L2(τ)) is a ∗-homomorphism.Let j : M → L2(τ) be the inclusion map. Note that j has dense range by definition of L2(τ)

and the unit vector ξ = j(1) is cyclic for L i.e. L(M)ξ = L2(τ) (and L(M)ξ = j(M)). Note that

(11.7) ∀a ∈M τ(a) = 〈1, a〉 = 〈ξ, L(a)ξ〉.

Up to now we did not make use of the trace property. We will now use it to show that rightmultiplications are also bounded on L2(τ): For any a, x in L2(τ) we have

τ((xa)∗(xa)) = τ((xa)(xa)∗) = τ(xaa∗x∗) ≤ ‖a‖2τ(xx∗) = ‖a‖2τ(x∗x),

and hence ‖xa‖2 ≤ ‖a‖‖x‖2. Therefore the right multiplication by a extends to a bounded operatoron L2(τ) denoted by R(a) : L2(τ)→ L2(τ). The mapping

R : Mop → B(L2(τ))

is a ∗-homomorphism on the opposite algebra Mop, i.e. the same ∗-algebra but with the “opposite”product (defined by a · b = ba).

In fact, what precedes holds more generally for the GNS construction associated to any tracialstate on a C∗-algebra. But since τ is faithful and normal, L and R are isometric normal ∗-homomorphisms (see §26.13 and Remark 26.42), which embed M (resp. Mop) as a von Neumannsubalgebra of B(L2(τ)). The fact that L(M) (resp. R(Mop)) is a von Neumann subalgebra ofB(L2(τ)) can be deduced e.g. from Kaplansky’s density Theorem 26.47 (because BL(M) = L(BM )is weak* compact and similarly for R). The uniqueness of the predual then guarantees that L (resp.R) is a weak*-homeomorphism from M to L(M) (resp. R(Mop)).

Remark 11.15. Consider x ∈M . Then x ∈ BM if and only if

∀a, b ∈M |τ(a∗xb)| ≤ ‖a‖2‖b‖2.

Indeed, this boils down to ‖x‖ = ‖L(x)‖ (or ‖x‖ = ‖R(x)‖).Moreover, for any x ∈M

(11.8) x ≥ 0⇔ τ(a∗xa) ≥ 0 ∀a ∈M,

or equivalently

(11.9) x ≥ 0⇔ τ(yx) ≥ 0 ∀y ∈M+.

Indeed, (11.8) is the same as 〈a, L(x)a〉 ≥ 0 for any a ∈ L2(τ).

167

Another way to see that L(M) (resp. R(Mop)) is a von Neumann subalgebra of B(L2(τ)) is viathe next statement:

Proposition 11.16. We have L(M) = R(M)′ and R(M) = L(M)′, and hence L(M) = L(M)′′

and R(Mop) = R(Mop)′′.

Proof. Let T ∈ R(M)′. Assume ‖T‖ = 1 for simplicity. For any a, b ∈ M we have |〈b, T (a)〉| ≤‖a‖2‖b‖2 and also T (a) = T (R(a)1) = R(a)T (1). Therefore |〈ba∗, T (1)〉| = |〈R(a∗)b, T (1)〉| =|〈b, R(a)T (1)〉| ≤ ‖a‖2‖b‖2. Let ϕ ∈M∗ be the linear form defined by ϕ(x) = 〈x∗, T (1)〉. Using thepolar decomposition x = U |x| = ab∗ with a = U |x|1/2 and b = |x|1/2, we find |ϕ(x)| ≤ τ(|x|) for anyx ∈M . By Corollary 11.12 (and the density of M in L1(τ)) there is y ∈ BM such that ϕ(x) = ϕy(x)for any x ∈M , and hence (since M is dense in L2(τ)) T (1) = y in L2(τ). Thus T (1) ∈M so thatwe may write T (a) = R(a)T (1) = T (1)a = L(T (1))a for any a ∈ M , so that T = L(T (1)) and weconclude R(M)′ ⊂ L(M). The converse is obvious, and similarly for L(M)′.

Remark 11.17. We will use the following classical fact about the inclusion M ⊂ L2(τ): the weak*-closure (i.e. σ(M,M∗)-closure) of a bounded convex subset of M coincides with its closure for thetopology induced on M by the norm of L2(τ). Indeed, since L2(τ) is dense in M∗, the σ(M,M∗)-closure of a bounded subset of M coincides with its closure in the weak topology of L2(τ), and by(Mazur’s) Theorem 26.9 for a convex set this is the same as the norm closure in L2(τ).

Remark 11.18. In a tracial probability space (M, τ) certain inequalities are immediate consequencesof the usual probabilistic ones. For instance for any x ∈ M and any 0 < p < q < ∞ we have forany x ∈M

(11.10) (τ(|x|p))1/p ≤ (τ(|x|q))1/q and limp→∞

(τ(|x|p))1/p = ‖x‖M .

Indeed, let N ⊂ M be the commutative von Neumann subalgebra generated by |x|. Since wemay identify (N, τ|N ) with (L∞(Ω,A , µ), µ) for some abstract probability space (Ω,A , µ) (that ofcourse depends on |x|), the preceding assertion reduces to the fact that for any f ∈ L∞(Ω,A , µ)we have ‖f‖p ≤ ‖f‖q and limp→∞ ‖f‖p = ‖f‖∞.

Proposition 11.19. Let (M, τ) and (N,ϕ) be two tracial probability spaces. Let T : L2(τ)→ L2(ϕ)be a trace preserving isometry. Assume that there is a weak*-dense unital ∗-subalgebra A ⊂M suchthat T (A) ⊂ N and T|A : A → N is a unital ∗-homomorphism. Then, when restricted to M , Tdefines a normal (injective trace preserving) ∗-homomorphism embedding M into N (as a vonNeumann subalgebra).

Proof. We claim that ‖T (a)‖ = ‖a‖ for any a in A. Indeed, since (T (a)∗T (a))m = T ((a∗a)m) andϕ(T (x)) = τ(x) (a, x ∈ A), we have by Remark 11.18

‖T (a)‖ = limm→∞

↑ (ϕ((T (a)∗T (a))m))1

2m = limm→∞

↑ (τ((a∗a)m))1

2m = ‖a‖.

Let A1 = T (A) ⊂ N . Then T defines an isometric isomorphism from (A, ‖ ‖) to (A1, ‖ ‖). We mayassume without loss of generality that A1 generates N . Thus, by Remark 11.17 and by Kaplanky’s

classical density theorem, we have BM = A ∩BML2(τ)

and similarly BN = A1 ∩BNL2(τ)

. Since Tis an isometry on L2(τ) this clearly implies that T (BM ) = BN . Moreover, since the product andthe ∗-operation are continuous on (BM , ‖ ‖L2(τ)), T : M → N is a ∗-homomorphism, which haskernel = 0 and is necessarily surjective (since T (BM ) = BN ). Lastly, since T ∗(L2(τ)) ⊂ L2(τ),we have T ∗(N∗) ⊂M∗, so T is normal. Actually T : M → N , being a ∗-isomorphism between vonNeuman algebras, is necessarily bicontinuous for the topologies σ(M,M∗), σ(N,N∗) (see Remark26.38).

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Remark 11.20. Here is a typical application of Proposition 11.19. Let (M, τ) and (N,ϕ) be twotracial probability spaces. Fix n ≥ 1. Let (x1, · · · , xn) ∈ Mn and (y1, · · · , yn) ∈ Nn be n-tuplesthat have “the same ∗-distribution” in the sense that for any polynomial (or equivalently for anymonomial) P (X1, · · · , Xn, X

∗1 , · · · , X∗n) in non-commuting variables (X1, · · · , Xn) and their adjoints

we haveτ(P (x1, · · · , xn, x∗1, · · · , x∗n)) = ϕ(P (y1, · · · , yn, y∗1, · · · , y∗n)).

Then the correspondence

T : P (x1, · · · , xn, x∗1, · · · , x∗n) 7→ P (y1, · · · , yn, y∗1, · · · , y∗n)

extends to a trace preserving ∗-isomorphism between the (unital) von Neumann subalgebras Mx

and Ny generated respectively by (x1, · · · , xn) and (y1, · · · , yn). Indeed, T clearly extends to anisometry from L2(Mx, τ|Mx

) onto L2(Ny, ϕ|Ny), and Proposition 11.19 can be applied to the unital∗-algebra Ax generated by (x1, · · · , xn) in Mx.More generally, the same can be applied for arbitrary families (xi)i∈I and (yi)i∈I . In that casewe say they have the same ∗-distribution if it is the case in the preceding sense for (xi)i∈I′ and(yi)i∈I′ for any finite subset I ′ ⊂ I. Then there is a trace preserving isomorphism between the vonNeumann algebras generated by (xi)i∈I and (yi)i∈I .

We will invoke several times the following well known fact. It may be worthwhile to emphasizethat this is special to the finite trace case. In general, for instance when M = B(H) and N ⊂ Mthere may not exist any bounded projection onto N .

Proposition 11.21 (Conditional expectations). Let (M, τ) be a tracial probability space. Thenfor any von Neumann subalgebra N ⊂ M , there is a normal c.p. projection P : M → N with‖P‖ = ‖P‖cb = 1, such that P (axb) = aP (x)b for any x ∈M and a, b ∈ N .

Proof. By definition L2(N, τ|N ) can be naturally identified to a subspace of L2(M, τ), namely the

closure N of N in L2(M, τ). Let P be the orthogonal projection from L2(M, τ) to N ' L2(N, τ|N ).Fix x ∈ L2(M, τ). Just like in the classical commutative case, P (x) is the unique x′ ∈ L2(N, τ|N )such that 〈x′, y〉 = 〈x, y〉 for any y ∈ N . Then since ayb ∈ N for any a, b ∈ N , it followsthat P (axb) = aP (x)b. By Remark 11.15, if x ∈ M (resp. x ∈ M+) then P (x) ∈ N (resp.P (x) ∈ N+) and ‖P (x)‖ ≤ ‖x‖. Of course P (x) = x if x ∈ N . Moreover, for any a, b ∈ N the formx 7→ 〈a, P (x)b〉 = τ(a∗xb) is normal (since τ is normal) and hence P is normal by Remark 26.42.Thus P is a normal positive projection from M onto N with ‖P‖ = 1. Its complete positivity isautomatic by Theorem 1.45. But in any case, if we repeat the argument with N ⊂M replaced byMn(N) ⊂Mn(M) with τ replaced by τn([xij ]) =

∑τ(xjj) we obtain that P is c.p.

Remark 11.22 (A recapitulation). We have natural inclusions M ⊂ L2(τ) and M ⊂ L1(τ), andsince ‖x‖1 ≤ ‖x‖2 for any x in M , we also have a natural inclusion L2(τ) ⊂ L1(τ) with norm = 1.

Let j : M → L2(τ) denote the natural inclusion. Then the (Banach space sense) adjointj∗ : L2(τ)∗ →M∗ actually takes values into M∗. Indeed, using the canonical identification L2(τ)∗ 'L2(τ) we find that j∗ takes x ∈M ⊂ L2(τ) ' L2(τ)∗ to the linear form fx∗ where fx is as in (11.3),which is in M∗. Then it is easy to verify that the composition

T = j∗j : M → L2(τ) ' L2(τ)∗ →M∗

is the antilinear map that takes x ∈ M to fx∗ ∈ M∗, or more rigorously the C-linear map thattakes x ∈M to fx∗ ∈M∗.

Recall the canonical identification M 'Mop that takes x ∈M to x∗. Note M∗ ' (Mop)∗. Usingthis we find that T can be identified to the mapping T0 : M → (Mop)∗ defined by T0(x) = fx.

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11.4 An example from free probability: semicircular and circular systems

In Voiculescu’s free probability theory, stochastic independence of random variables is replaced byfreeness of non-commutative analogues of random variables. We refer the reader to [255] for anaccount of this beautiful theory. Here we only want to introduce the basic tracial probability spacethat is the free analogue of Rn equipped with the standard Gaussian measure. The free analogueof a family of standard independent real (resp. complex) Gaussian variables is a free semi-circular(resp. circular) family. They satisfy a similar distributional invariance under the orthogonal (resp.unitary) group.

Such families generate a tracial probability space that can be realized on the “full” Fock space,as follows. Let H be a (complex) Hilbert space. We denote by F(H)(or simply by F) the “full”Fock space associated to H, that is to say we set H0 = C, Hn = H⊗n (Hilbertian tensor product)and finally

F = ⊕n≥0Hn.

We consider from now on Hn as a subspace of F . For every h ∈ F , we denote by `(h) : F → Fthe operator defined by:

`(h)x = h⊗ x.

More precisely, if x = λ1 ∈ H0 = C1, we have `(h)x = λh and if x = x1 ⊗ x2...⊗ xn ∈ Hn we have`(h)x = h⊗ x1 ⊗ x2...⊗ xn. We will denote by Ω the unit element in H0 = C1. The von Neumannalgebra B(F) is equipped with the vector state ϕ defined by

ϕ(T ) = 〈Ω, TΩ〉,

called the vacuum state.Let (es)s∈S be an orthonormal basis of H. Since ϕ(`(h)`(h)∗) = 0 and ϕ(`(h)∗`(h)) = ‖h‖2, we seethat ϕ is not tracial on B(F). However, it can be checked (a possible exercise for the reader) thatit is tracial when restricted to the von Neumann algebra M generated by the operators `(es)+ `(es)

∗

(s ∈ S),i.e. we have ϕ(xy) = ϕ(yx) for all x, y in this ∗-subalgebra.The pair (M,ϕ) is an example of a tracial probability space. Let

(11.11) Ws = `(es) + `(es)∗ .

Then the family (Ws)s∈S is the prototypical example of a free semi-circular system, sometimes alsocalled a “free-Gaussian” family. The term semi-circular is used for any family with the same jointmoments as (Ws)s∈S . Such a family enjoys properties very much analogous to those of a standardindependent Gaussian family. Indeed, let [aij ] (i, j ∈ S) be an “orthogonal matrix”, by which wemean that aij ∈ R (i, j ∈ S) and that the associated R-linear mapping is an isometric isomorphismon `2(S). Then if we let W a

i =∑

j aijWj , we have

(11.12) ϕ(P (W ai )) = ϕ(P (Wi))

for any polynomial P in non-commuting variables (Xi)i∈S . In other words the families (Ws)s∈Sand (W a

s )s∈S have the same joint moments. This is analogous to the rotational invariance of theusual Gaussian distributions on RS when (say) S is finite.In particular, for any fixed s0 ∈ S the variables Ws0 and W a

s0 have the same moments, so thatϕ((W a

s0)2m) = ϕ((Ws0)2m) for any m ≥ 0, and letting m→∞ after taking the 2m-th root we find‖W a

s0‖ = ‖Ws0‖. Since we can adjust a so that its s0-th row matches any element (αs)s∈S in theunit sphere of `2(S,R) we obtain

∀(αs)s∈S ∈ `2(S,R) ‖∑

s∈SαsWs‖ = cR(

∑α2s)

1/2,

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where cR is the common value of ‖Ws0‖. Thus, the real Banach space R-linearly generated by(Ws)s∈I is isometric to a real Hilbert space. By (11.11) we have cR ≤ 2, and with a little morework (see e.g. [255]) one shows that

cR = 2.

For any a ∈ BB(H), let F (a) ∈ BB(F(H)) denote the linear operator that fixes the vacuum vector Ω

and acts like a⊗n

on Hn (so called “second quantization” of a). To check (11.12) the simplest wayis to observe that for all h ∈ H and all a ∈ B(H) we have

F (a)`(h) = `(ah)F (a).

If a is unitary then F (a) is unitary and we have

F (a)`(h)F (a)∗ = `(ah) and F (a)`(h)∗F (a)∗ = `(ah)∗.

Therefore, F (a)(`(h) + `(h)∗)F (a)∗ = `(ah) + `(ah)∗, and hence when the coefficients aij are allreal and ta denotes the transposed of a we find

F (ta)WsF (ta)∗ = W as ,

and hence for any polynomial P

F (ta)P (Ws)F (ta)∗ = P (W as ).

Since all the F (a)’s and their adjoints preserve Ω, (11.12) follows.Using the basic ideas of spectral theory and free probability one can show that if |S| = n (resp.

S = N) then there is a trace preserving isomorphism from (M,ϕ) to (MFn , τFn) (resp. (MF∞ , τF∞)).We now pass to the complex case. We need to assume that the index set is partitioned into

two copies of the same set S, so we replace H by H = H ⊕ H and assume given a (partitioned)

orthonormal basis es | s ∈ S∪fs | s ∈ S of H. We then define Ws = `(es) + `(fs)∗ ∈ B(F(H)).

Let M ⊂ B(F(H)) be the von Neumann algebra generated by (Ws)s∈S . Let ϕ be the vacuum

state on B(F(H)). Then again (M, ϕ) is a tracial probability space and the family (Ws)s∈S isthe prototypical example of a so-called free circular system. It is the free analogue of an i.i.d.family of complex valued Gaussian variables with covariance equal to the 2× 2 identity matrix. Asthe latter, the family (Ws)s∈S is unitarily invariant. Indeed, let [aij ] (i, j ∈ S) be the matrix of a

unitary operator on `2(S). Then if we let W ai =

∑j aijWj , we have

(11.13) ϕ(P (W ai , W

ai

∗)) = ϕ(P (Wi, W

∗i ))

for any polynomial P (Xi, X∗i ) in non-commuting variables (Xi)i∈S and (X∗i )i∈S . In other words the

families (Ws, W∗s ) and (W a

s , Was

∗) have the same joint moments. This is analogous to the unitary

invariance of the usual standard Gaussian measure on CS when S is finite. The proof of (11.13) issimilar to that of (11.12) (hint: consider F (α) for α = a⊕ a acting on H).As before, we have

∀(αs)s∈S ∈ `2(S,C) ‖∑

s∈SαsWs‖ = cC(

∑|αs|2)1/2,

where cC is the common value of ‖Wi‖, and it can be shown (see [255]) that cC = 2.

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11.5 Ultraproducts

Let (M(i), τi) | i ∈ I be a family of tracial probability spaces. Let U be an ultrafilter on theindex set I (see Remark 26.6). Then for any bounded family of numbers (xi)i∈I , the limit along Uis well defined. We denote it by

limU xi.

LetB = x ∈

∏i∈I

M(i) | supi∈I ‖x(i)‖M(i) <∞,

equipped with the norm‖x‖B = supi∈I ‖x(i)‖M(i).

As already mentioned, we adopt in these notes the notation

(11.14) B =(⊕∑

i∈IM(i)

)∞.

We define a functional fU ∈ B∗ by setting for all t = (ti)i∈I in B

fU (t) = limU τi(ti).

Clearly fU is a tracial state on B. Let HU be the Hilbert space associated to the tracial state fUin the GNS construction applied to B. We denote by

L : B → B(HU ) and R : Bop → B(HU )

the representations of B corresponding to left and right multiplication by an element of B. (Werecall that Bop is the same C∗-algebra as B but with reverse multiplication.)More precisely, let

pU (x) = limU τi(x∗ixi)

1/2.

Clearly pU is a Hilbertian seminorm on B. Let

IU = ker(pU ) = x ∈ B | limU τi(x∗ixi) = 0.

Then IU is a closed 2-sided ideal, and HU is defined as the completion of B/IU equipped with theHilbertian norm associated to pU .For any t = (ti)i in B we denote by t the equivalence class of t in B/IU .

Then L and R are defined by L(x)t =_xt and R(x)t =

_tx. Clearly since fU is tracial, these are

contractive representations of B and Bop on HU .Let us record the following simple fact (see §26.4 for background on ultrafilter limits).

Lemma 11.23. For any t = (ti)i in B we have

‖t‖HU = limU ‖ti‖L2(τi).

Moreover, for any ε > 0 there is s = (si)i in B such that s− t ∈ IU and

supi∈I ‖si‖L2(τi) < ‖t‖HU + ε = ‖s‖HU + ε.

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Proof. It is immediate that for any t, s ∈ B such that t − s ∈ IU we have limU ‖ti‖L2(τi) =limU ‖si‖L2(τi). By definition of the limit ` = limU ‖ti‖L2(τi) for any ε > 0 the set

Jε = i ∈ I | |‖ti‖L2(τi) − `| < ε

is such that limU 1Jε = 1 and limU 1I\Jε = 0. See §26.4 for details if necessary. Thus we may simplyset si = ti for any i ∈ Jε and si = 0 otherwise. Then we have limU ‖ti− si‖L2(τi) ≤ c limU 1I\Jε = 0,where c = supi∈I ‖ti − si‖L2(τi).

Actually, it turns out that for many considerations the Hilbert space HU is “too small”. Weneed to embed it in the larger Hilbert space HU that is the ultraproduct of the family (Hi) definedby Hi = L2(τi), as defined in §26.5.

We claim that we have a natural isometric inclusion

jU : HU → HU .

Indeed, to any x ∈ B we can associate (xi) in the space X =(⊕∑

i∈I Hi

)∞ and we have by

definition (see §26.5)pU (x) = limU ‖xi‖L2(τi) = ‖(xi)U‖HU .

Thus the correspondence x 7→ jU (x) = (xi)U extends to an isometric (linear of course) embeddingjU : HU → HU .

Remark 11.24. For any t = (ti)i in B, the mapping (xi) 7→ (tixi) is clearly in B(X) and takesker(pU ) to itself. Thus it defines a mapping π`(t) : HU → HU . The mapping t 7→ π`(t) is clearlya ∗-homomorphism from B/IU to B(HU ). It is easy to check that if we view HU as embedded inHU via jU , then HU ⊂ HU is an invariant subspace under π`(t) and we have

L(t) = π`(t)|HU .

See Remarks 11.28 and 11.29 for a description of HU as the subset of HU formed of the elementsadmitting a “uniformly square integrable” representative. The reader who is already familiar withthe latter can skip the next Lemma, which is but a pedestrian reformulation of the same fact.

We will invoke the following simple observation.

Lemma 11.25. For any β ∈ HU and ε > 0, there is a family (βi) ∈ X such that

(11.15) jU (β) = (βi)U ,

(11.16) sup ‖βi‖2 < ‖β‖HU + ε,

(11.17) ∀t = (ti) ∈ B ‖L(t)β‖HU = limU ‖tiβi‖2 = ‖π`(t)((βi)U )‖HU ,

and moreover, for any family of projections (pi) (pi ∈M(i)) such that limU τi(pi) = 0, we have

(11.18) limU ‖piβi‖2 = limU ‖βipi‖2 = 0.

Proof. Let β(m) be a sequence in B/IU such that β =∑β(m) and

∑‖β(m)‖HU < ‖β‖HU + ε/2.

Let q : B → B/IU denote the quotient map. Then by Lemma 11.23 we may assume that, for eachm, we have β(m) = q(b(m)) for some sequence b(m) = (bi(m))i∈I ∈ B such that

(11.19) ∀m ≥ 0 supi‖bi(m)‖2 ≤ ‖β(m)‖HU + 2−m(ε/4).

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Then, if we set

(11.20) βi =∑

mbi(m) ∈ L2(τi),

assuming 0 < ε < 1, (11.16) clearly holds. From β =∑

m β(m) (absolutely convergent series inHU ), since jU : HU → HU is isometric we deduce

jU (β) =∑

mjU (β(m)).

Note jU (β(m)) = (bi(m))U . Therefore, we have (absolutely convergent series in HU )

jU (β) =∑

m(bi(m))U .

But we have also by (11.20) (absolutely convergent series in HU )

(βi)U =∑

m(bi(m))U .

Therefore,we obtain (11.15). Also (11.17) is immediate by Remark 11.24.To check (11.18), note that for each fixed k

‖piβi‖2 ≤ ‖pi(∑

m≤kbi(m))‖2 + ‖pi(

∑m>k

bi(m))‖2

≤ ‖pi‖2‖‖∑

m≤kb(m)‖M(i) +

∑m>k

supi ‖bi(m)‖2

and since supi ‖∑

m≤k bi(m)‖M(i) ≤ ‖∑

m≤k b(m)‖B <∞ and limU ‖pi‖2 = 0 we have by (11.19)

limU ‖piβi‖2 ≤∑

m>k‖β(m)‖HU +

∑m>k

2−m(ε/4)

and letting k →∞ we obtain one part of (11.18). The other part follows similarly.

The next result seems to go back to McDuff’s 1969 early work. Note that the kernel IU is notweak* closed in B (and the quotient map is not normal), so the fact that the quotient B/IU isnevertheless (∗-isomorphic to) a von Neumann algebra is a priori somewhat surprising.

Theorem 11.26. The functional fU : B → C vanishes on IU . The associated functional

τU : B/IU → C

is a faithful tracial state on B/IU such that, if q : B → B/IU denotes the quotient map, we have

(11.21) τU (q(t)) = fU (t) ∀ t ∈ B.

The kernels of L and R coincide with the set IU . After passing to the quotient, L and R defineisometric representations

LU : B/IU → B(HU ) and RU : Bop/IU → B(HU )

with commuting ranges. Lastly, the commutants satisfy

[LU (B/IU )]′ = RU (Bop/IU ) and [RU (Bop/IU )]′ = LU (B/IU ).

In particular, LU (B/IU ) and RU (Bop/IU ) are (mutually commuting) von Neumann subalgebras ofB(HU ).

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Proof. By Cauchy-Schwarz, we have |fU (x)| ≤ (fU (x∗x))1/2(fU (1))1/2 = (fU (x∗x))1/2, and IU =x ∈ B | fU (x∗x) = 0. Therefore, fU vanishes on IU . Passing to the quotient, we obtain afunctional τU unambiguously defined by (11.21), which is clearly a tracial state on B/IU .Let 1i be the unit of M(i) and let ξ = (1i)i∈I ∈ B. We have

fU (t) = 〈ξ, L(t)ξ〉 = 〈ξ, R(t)ξ〉.

Let t ∈ B. If L(t) = 0, then L(t∗t) = 0 which by the preceding line implies fU (t∗t) = 0 hencet ∈ IU . Conversely, if t ∈ IU then x∗t∗tx ∈ IU for any x in B, whence fU (x∗t∗tx) = 0 which means_tx = 0 for all x in B, or equivalently L(t) = 0.

A similar argument applies for R, so we obtain that ker(L) = ker(R) = IU . Then, after passingto the quotient by IU , L and R define the isometric representations LU and RU with the samerespective ranges as L and R. Therefore LU and RU still have commuting ranges.

Finally, let T ∈ B(HU ) be an operator commuting with LU (B/IU ), i.e. T ∈ LU (B/IU )′. Wewill show that T must be in the range of RU . Let

β = T (ξ) ∈ HU .

We will show that there is b = (bi) in B such that β = b and that

T = R(b) = RU (b).

Indeed, we have for any t = (ti) in B

(11.22) TL(t)ξ = L(t)T ξ = L(t)β

hence

(11.23) ‖L(t)β‖HU ≤ ‖T‖ ‖L(t)ξ‖HU = ‖T‖ ‖t‖HU .

By Lemma 11.25 there is a family (βi) with βi ∈ L2(τi) satisfying (11.18) and such that sup ‖βi‖2 <∞, (βi)U = jU (β) and

∀t = (ti) ∈ B ‖L(t)β‖HU = limU ‖tiβi‖2.This, together with (11.23) implies that for any t in B

(11.24) limU τi(βiβ∗i t∗i ti) ≤ ‖T‖2 limU τi(t

∗i ti).

Let βi = hivi be the polar decomposition of βi in L2(τi) with hi ∈ L2(τi), hi ≥ 0, vi partialisometry in M(i) and hi = |β∗i | (see Remark 11.29 for clarification). Fix ε > 0. Let pi be thespectral projection of hi associated to ]‖T‖+ε,∞[. Note that βiβ

∗i pi = h2

i pi ≥ (‖T‖+ε)2pi. Hence(11.24) implies (with ti = pi)

(‖T‖+ ε)2 limU τi(pi) ≤ limU τi(βiβ∗i pi) ≤ ‖T‖2 limU τi(pi).

This forces limU τi(pi) = 0, and hence by (11.18) limU τi(βiβ∗i pi) = 0.

Therefore, if we set finally bi = (1−pi)hivi we find ‖bi‖ ≤ ‖(1−pi)hi‖ ≤ ‖T‖+ε and ‖βi−bi‖2L2(τi)≤

‖pihivi‖2L2(τi)≤ τi(βiβ∗i pi), hence limU ‖βi − bi‖L2(τi) = 0.

Let b = (bi). Then (βi)U = (bi)U , and b ∈ B with ‖b‖B ≤ ‖T‖+ ε. Then going back to (11.22) weobtain finally

TL(t)ξ = L(t)β = L(t)b = (_tibi) = RU (b)L(t)ξ.

This shows that T = RU (b), which completes the proof that LU (B/IU )′ = RU (Bop/IU ). The sameargument clearly yields RU (Bop/IU )′ = LU (B/IU ), and hence LU (B/IU )′′ = LU (B/IU ), whichproves that LU (B/IU ) is a von Neumann algebra.

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Definition 11.27. Let MU = LU (B/IU ). The tracial probability space

(MU , τU )

is called the ultraproduct of the family (M(i), τi) with respect to U .

Remark 11.28. Let HU be the usual ultraproduct of the Hilbert spaces Hi = L2(τi). Then, HUcan be identified to the closure in HU of the subspace of all elements of the form (bi)U withsupi ‖bi‖M(i) < ∞. Alternatively HU ⊂ HU can also be described as the subspace correspondingto the “uniformly square integrable” families. More precisely, let β = (βi)U be an element of HU ,with supi ‖βi‖Hi <∞. Then β belongs to HU (meaning rather jU (HU )) if and only if

limc→∞

limU τi(βiβ∗i 1βiβ∗i >c) = 0

or if and only iflimc→∞

limU τi(β∗i βi1β∗i βi>c) = 0,

where we have denoted (abusively) by 1h>c the spectral projection of the Hermitian operator hfor the interval (c,∞).

Remark 11.29. In what precedes, we invoked the polar decomposition in L2(τi), using its structureas a bimodule over M(i). In general this involves dealing with unbounded operators. But actually,we will apply the preceding result only in the case when M(i) is a (finite dimensional) matrix algebrafor which the polar decomposition is entirely elementary and classical, since L2(τi) coincides withM(i) itself.

In general, a unitary in a quotient C∗-algebra A/I does not lift to a unitary in A. However itis so when A/I is isomorphic to a von Neumann algebra, in particular for B/IU .

Lemma 11.30. Let qU : B →MU be the quotient map (given by qU = LUq). For any unitary u inMU , there is a unitary u = (ui)i∈I in B such that qU (u) = u.

Proof. Since MU is a von Neumann algebra (see Theorem 11.26), we can write u = exp ix for someself-adjoint x ∈ MU (see Remark 26.39). Since qU maps B onto MU and is a ∗-homomorphism,there is x ∈ B with x∗ = x such that qU (x) = x. Then u = exp ix is a unitary in B lifting u.

Remark 11.31. Any (self-adjoint) projection Q ∈ MU admits a lifting (Qi) ∈ B such that Qi isa (self-adjoint) projection for any i. Indeed, let (xi) ∈ BB be a lifting of Q (see Lemma 26.33).Replacing xi by its real part, we may assume all the xi’s self-adjoint. Observe that for any λ inthe spectrum of xi, we have |λ| ≤ 1 and d(λ, 0, 1) ≤ 2|λ− λ2|. Using this it is easy to show thatthere is a (self-adjoint) projection Qi (in the commutative von Neumann algebra generated by xi)such that |xi −Qi| ≤ 2|xi − x2

i |. Since Q = Q2 we have (xi − x2i ) ∈ IU and hence (xi −Qi) ∈ IU .

Thus we conclude as announced Q = q((Qi)).

Remark 11.32. We will be mostly interested with the case when the algebras M(i) are finite dimen-sional. The main and simplest case is when M(i) = MN(i) (matrices of size N(i)×N(i)) equipped

with the normalized trace τi(x) = N(i)−1tr(x). In that case we refer to MU as an ultraproduct ofmatricial tracial probability spaces.If we merely assume that all the M(i)’s are finite dimensional, the resulting MU can anyway beembedded in an ultraproduct of the preceding matricial kind. Indeed, each finite dimensional(M, τ) can be identified in a trace preserving way with Mn(1) ⊕ · · · ⊕ Mn(k) equipped with the

trace τ(x1 ⊕ · · · ⊕ xk) = w1n(1)−1tr(x1) + · · ·+ wkn(k)−1tr(xk) where the positive weights satisfy

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w1 + · · · + wk = 1. If these weights are all rational, say wj = pj/N with p1 + · · · + pk = N , thenwe can embed (M, τ) into (MN , τN ) (here τN is the normalized trace on MN ) by a block diagonalembedding repeating each factor Mn(j) with multiplicity pj . In the general case, when the weightsare arbitrary real numbers, we can approximate them by rationals, and form an ultraproduct as-sociated to these elementary numerical approximations. This shows that any finite dimensional(M, τ) (and hence any ultraproduct of such) can be embedded in an ultraproduct of matricialtracial probability spaces. See Corollary 12.7 for another way to justify this.

Remark 11.33 (On Dixmier’s approximation theorem). A von Neumann algebra M is called a factorif M ∩M ′ = CI. For example, MG is a factor if (and only if) all the nontrivial conjugacy classes ofG (i.e. the sets gtg−1 | g ∈ G with t 6= 1) are infinite. Moreover, if all the M(i)’s are factors, theultraproduct MU of the tracial probability spaces (M(i), τi) is also a factor. We leave the proofs asexercises for the reader.Let (M, τ) be a tracial probability space. If M is a factor, a classical theorem asserts that τ isthe unique tracial state on M . This is an immediate corollary of a more general result due toDixmier (see e.g. [146, p. 523] for a proof): for any x ∈ M the norm closure of the convex hull ofuxu−1 | u ∈ U(M) intersects the center Z = M ∩M ′ (when M is a factor, the latter intersectionis reduced to τ(x)1). This implies that two tracial states on M that coincide on Z must beidentical. In particular, given any other tracial probability space (N,ϕ), any unital embeddingπ : M → N automatically preserves the trace if M is a factor.

Remark 11.34. If (M, τ) is a tracial probability space then any ∗-homomorphism π : A1⊗A2 →Mis continuous with respect to the minimal norm, and hence continuously extends to A1 ⊗min A2.We only sketch the proof. Replacing M by π(A1 ⊗ A2)′′, we may assume that π(A1 ⊗ A2)′′ =M ⊂ B(L2(τ)). By Remark 26.27 it suffices to show that the tracial state A1 ⊗A2 3 x 7→ τ(π(x))continuously extends to A1⊗min A2. It is easy to show that the extreme points of the set of tracialstates on a C∗-algebra A are all factorial states, i.e. states f for which πf (A)′′ is a factor. Indeed,if the center Z of πf (A)′′ is nontrivial we can find a nonzero projection p ∈ PZ such that 0 6= p 6= 1and then τ(·) = τ(p)[τ(p)−1τ(p·)] + τ(1 − p)[τ(1 − p)−1τ((1 − p)·)] shows that τ is not extreme.Using this for A = A1 ⊗max A2, we may assume that M is a factor. We may assume π = π1 · π2 asin (4.4). Let Mj = πj(Aj)

′′. Since π1, π2 have commuting ranges, the center of Mj is included inthat of M . Therefore, if M is a factor, both M1,M2 are factors, and τj = τ|Mj

is the tracial state ofMj . Applying Dixmier’s approximation theorem (see Remark 11.33) to each of M1 and M2 we findthat τ(x1x2) = τ1(x1)τ2(x2) for all (x1, x2) ∈M1 ×M2. Then we have τ π = (τ1 π1)⊗ (τ2 π2),and hence |τ π(x)| ≤ ‖x‖min for all x ∈ A1 ⊗A2, which completes the proof.

Remark 11.35 (On ascending unions of factors). Let (M, τ) be a tracial probability space. LetM(i) ⊂ M be a family of von Neumann subalgebras directed by inclusion and let N ⊂ M be theweak* closure (or equivalently the bicommutant by Theorem 26.46) of their union. If each M(i) isa factor then N is also a factor. To check this assertion, let τi = τ|M(i) and Ei = L2(M(i), τi). Weview Ei and L2(N, τN ) as subspaces of L2(τ). Consider the orthogonal projection Pi : L2(τ)→ Ei.Clearly Pi(x) → x for all x ∈ L2(N, τ). By Proposition 11.21, Pi is a conditional expectation,so that Pi(axb) = aPi(x)b for all a, b ∈ M(i) and all x ∈ M . Since N ′ ⊂ M(i)′, this impliesthat Pi(M ∩ N ′) ⊂ M(i)′. Therefore Pi(N ∩ N ′) ⊂ M(i) ∩M(i)′ and the assertion follows sincePi(x)→ x in L2(N, τ) for any x ∈ N and we assume M(i) ∩M(i)′ = C1 for all i ∈ I.

Remark 11.36 (Reduction to factors). The notion of free product M ∗N of two finite von Neumannalgebras goes back to Ching [43] and Voiculescu (see [255]). Equivalently, they defined the freeproduct (M ∗N, τ ∗ϕ) of two tracial probability spaces (M, τ), (N,ϕ). Moreover, the constructionis done so that the canonical embeddings M → M ∗N and N → M ∗N are trace preserving. In[43] Ching proved that if M,N both admit an orthonormal basis formed of unitaries for their L2

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spaces (and dim(M) ≥ 2,dim(N) ≥ 3) then M ∗ N is automatically a factor. In [219, Th. 4.1]Sorin Popa proves a very general result of this type from which it follows that if (M, τ) is anytracial probability space, and if for instance (N,ϕ) = (MF2 , τF2) (namely the so-called “free groupfactor”) then M ∗N is automatically a factor. More precisely, the relative commutant N ′∩(M ∗N)is trivial. Actually, Popa proves this whenever (N,ϕ) is a non-atomic tracial probability space.In particular, this shows that any tracial probability space (M, τ) embeds in a trace preserving wayinto one that is a factor and is separable if M is separable.

We note in passing that it is an interesting and fundamental open question whether for anytracial probability space (M, τ) (on a separable Hilbert space and “atomless”, that is without non-trivial minimal projections) there is an orthonormal basis of L2(τ) formed of unitaries. Of coursethis holds whenever (M, τ) = (MG, τG) with G a discrete group.

For more in depth information on finite von Neumann algebras we strongly recommend [4] tothe reader.

Remark 11.37 (GNS representations on B(H) and ultraproducts). Let (Pn) be a sequence of mutu-ally orthogonal (self-adjoint) projections in B(H) with rk(Pn) = n for all n. Let U be a nontrivialultrafilter on N and let fU be the state on B(H) defined by fU (x) = limU n

−1tr(PnxPn). Then,for some infinite dimensional Hilbert space KU , we have πfU (B(H))′′ ' B(KU )⊗MU where MU isthe ultraproduct of the tracial probability spaces (Mn, τn) (with τn(·) = n−1tr(·)). This is due toAnderson and Bunce, see [6, Th. 5].

11.6 Factorization through B(H) and ultraproducts

We will describe a simple criterion that guarantees that a ∗-homomorphism π : A → M from aC∗-algebra to a von Neumann algebra factorizes completely positively through B(H). To adapt itsuse to various situations we consider more generally the restriction of π to a unital linear subspaceE ⊂ A spanned by unitaries.

Theorem 11.38. Let A be a unital C∗-algebra, (M, τ) a tracial probability space. Let π : A→Mbe a unital ∗-homomorphism. Let S ⊂ U(A) be a subset with 1 ∈ S and let E = span(S) ⊂ A.Assume that

(11.25) ∀n ≥ 1,∀x1, · · · , xn ∈ E∑‖π(xj)‖22 ≤ ‖

∑xj ⊗ xj‖min.

Then there is a state f on M ⊗min M such that

(11.26) ∀x, y ∈ E τ(π(y)∗π(x)) = f(y ⊗ x).

More precisely, there is an embedding A ⊂ B(H), a family of finite rank operators (hi)i∈I on Hwith tr(h∗ihi) = 1 and an ultrafilter U on I such that

(11.27) ∀x, y ∈ E τ(π(y)∗π(x)) = limU tr(h∗i y∗hix).

Conversely (11.27) implies (11.25).

Remark 11.39. The proof will show that any family (hi)i∈I of Hilbert-Schmidt operators on Hsatisfying (11.27) automatically also satisfies an approximate commutation condition as follows:

(11.28) ∀x ∈ E limU tr|xhi − hix|2 = 0.

This is but a simple consequence of the equality case in Cauchy-Schwarz.

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The rest of this section is devoted to the proof of Theorem 11.38 and the reinforced version inTheorem 11.42.

Proof. We will assume (without loss of generality) that A ⊂ B(H), with infinite multiplicity. Moreprecisely, this means that, starting from an embedding A ⊂ B(H1), we replace H1 by H = `2⊗2H1

and embed B(H1) into B(H) (diagonally) by T 7→ Id`2 ⊗ T. This gives us a new faithful ∗-homomorphism ρ : A→ B(H). For simplicity we view ρ as an embedding i.e. we set ρ(a) = a forall a ∈ A.

We identify H ⊗2 H with the Hilbert-Schmidt class S2(H). Let Bmin denote the unit ball of(A⊗min A)∗. Note that for any T ∈ A⊗min A we have

‖T‖min = sup<(f(T )) | f ∈ Bmin.

By our assumption we have∑‖π(xj)‖2L2(τ) ≤ ‖

∑xj ⊗ xj‖min = sup

f∈Bmin

∑<(f(xj ⊗ xj)).

We now apply Lemma 26.16. Since Bmin is convex and weak* compact, this gives us in the limit afunctional f in Bmin such that

(11.29) ∀x ∈ E ‖π(x)‖2L2(τ) ≤ <(f(x⊗ x)).

Taking x ∈ S we find 1 ≤ <(f(x⊗ x)) ≤ 1 and hence <(f(x⊗ x)) = 1 for any x ∈ S. In particular,1 = <(f(1⊗ 1)), and hence the real part of f is a state, which implies (see §26.23) that f itself is astate on A⊗min A. By Proposition 4.25 there is a net (hi)i∈I in the unit sphere of S2(H) such that

∀x ∈ E f(x⊗ x) = lim tr(x∗hixh∗i ).

Note (by the trace property) tr(x∗hixh∗i ) = tr(h∗ix

∗hix) and hence tr(x∗hixh∗i ) = 〈xhi, hix〉 where

the last inner product is relative to S2(H). Thus, for any (unitary) x ∈ S we have

1 = <f(x⊗ x) = lim<〈xhi, hix〉,

and also ‖hix‖S2(H) = ‖xhi‖S2(H) = 1. Therefore for any x ∈ S and hence for any x ∈ E

(11.30) lim ‖xhi − hix‖S2(H) = 0.

This implies for any x ∈ E

(11.31) f(x⊗ x) = lim tr(h∗ix∗hix) = lim tr(h∗ix

∗xhi) ≥ 0.

This allows us to rewrite (11.29) more simply as

(11.32) ∀x ∈ E ‖π(x)‖2L2(τ) ≤ f(x⊗ x).

We claim that equality holds in (11.32), i.e.

(11.33) ∀x ∈ E ‖π(x)‖2L2(τ) = f(x⊗ x).

It clearly suffices to show equality for all x ∈ span(S1) for any finite subset S1 ⊂ S, so we assumex ∈ span(U1, · · · , Ur) with U1, · · · , Ur ⊂ S. Consider the matrices defined for 1 ≤ i, j ≤ r byaij = 〈π(Ui), π(Uj)〉L2(τ) and bij = f(Ui ⊗ Uj). We then have the following situation: we have

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two (nonnegative) matrices a, b ∈ Mr such that a ≤ b by (11.32) and also ajj = bjj for all j, andthis clearly implies that a = b (because for c = b − a ≥ 0, tr(c) = 0 ⇒ c = 0). This proves ourclaim (11.33). By the polarization identity of sesquilinear forms, the latter and (11.31) yield (sincetr(h∗i y

∗xhi) = tr(hih∗i y∗x) by the trace property)

(11.34) ∀x, y ∈ E τ(π(y∗x)) = f(y ⊗ x) = lim tr(h∗i y∗hix) = lim tr(hih

∗i y∗x).

Since the finite rank operators are dense in S2(H) we may assume by perturbation that the hi’sare all of finite rank. Lastly, passing to an ultrafilter U refining the net (see §26.4) we may as wellassume that the preceding limits are all with respect to U . This completes the proof, since (recallProposition 2.11) the converse direction is obvious.

Remark 11.40 (Complement to the proof). Note for further use that for any unitary x ∈ B(H)

‖xhi − hix‖S2(H) = ‖x∗(xhi − hix)x∗‖S2(H) = ‖hix∗ − x∗hi‖S2(H) = ‖xh∗i − h∗ix‖S2(H),

and moreover (derivation rule) xyhi − hixy = x(yhi − hiy) + (xhi − hix)y. Therefore (11.30) stillholds for any x in the group GS ⊂ U(A) generated by S ∪ S−1. Applying the derivation rule againwe see that limU ‖xhih∗i − hih∗ix‖S1(H) = 0. Thus we have

(11.35) ∀x ∈ GS limU ‖xhih∗i − hih∗ix‖S1(H) = 0.

By perturbation, we may clearly assume that the finite rank operators hih∗i have rational eigenval-

ues.

Remark 11.41. In [39], a state ϕ on A ⊂ B(H) is called an “amenable trace” if it can be extendedto a state ϕ on B(H) such that ϕ(U∗bU) = ϕ(b) for any U ∈ U(A) and any b ∈ B(H). Inthe situation of the preceding proof, assuming E dense in A, let ϕ(x) = τ(π(x)) (x ∈ A), andϕ(b) = limU tr(h∗i bhi) (b ∈ B(H)). By (11.30) we have ϕ(b) = ϕ(U∗bU) for any U ∈ S and by(11.34) ϕ|E = ϕ|E . Therefore, if E is dense in A, (11.25) implies that ϕ is an amenable trace on A.Conversely, if ϕ is an amenable trace then (11.25) holds for E = A. This is easy to check using thePowers-Størmer inequality (11.38) which comes next. The notion of “amenable trace” generalizesto C∗-algebras that of “hypertrace” that will be discussed for von Neumann algebras in §11.7.

Theorem 11.42. The conclusion of Theorem 11.38 can be strengthened as follows:There are a Hilbert space H, an embedding σ : A → B(H), a family of finite rank projections(Ri)i∈I on H and an ultrafilter U on I such that

(11.36) ∀y, x ∈ E τ(π(y)∗π(x)) = limU (tr(Ri))−1tr(Riσ(y)∗Riσ(x)),

and

(11.37) ∀y, x ∈ E limU∣∣(tr(Ri))−1tr(Riσ(y)∗σ(x))− (tr(Ri))

−1tr(Riσ(y)∗Riσ(x))∣∣ = 0.

Note that (11.36) is similar to (11.27) with hi replaced by h′i = tr(Ri)−1/2Ri.

To complete the proof we will need the Powers-Størmer inequality and Lemma 11.45.

Remark 11.43. Actually we will prove that (11.37) holds for all y in the linear span of the groupgenerated by S in U(A) and for all x ∈ A.

We need some technical preliminary to be able to complete the proof.

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Lemma 11.44 (Powers-Størmer inequality). Let s, t ≥ 0 be trace class operators on H. Then

‖s1/2 − t1/2‖22 ≤ ‖s− t‖1.

In particular, for any unitary U ∈ B(H)

(11.38) ‖U∗t1/2U − t1/2‖22 ≤ ‖U∗tU − t‖1.

Proof. Let ek be an orthonormal basis of H consisting of eigenvectors of s1/2 − t1/2 and let λkbe the corresponding (real) eigenvalues. Note for later use that for any self-adjoint T we have−|T | ≤ T ≤ |T | and hence for any x ∈ H

(11.39) |〈x, Tx〉| ≤ 〈x, |T |x〉.

Note that since ±(s1/2 − t1/2) ≤ s1/2 + t1/2 it follows that |〈x, (s1/2 − t1/2)x〉| ≤ 〈x, (s1/2 + t1/2)x〉for all x ∈ H, and hence

|λk| ≤ 〈ek, (s1/2 + t1/2)ek〉.

Thus we have

‖s1/2 − t1/2‖22 = tr(|s1/2 − t1/2|2) =∑〈ek, |s1/2 − t1/2|2ek〉 =

∑|λk|2

≤∑|λk|〈ek, (s1/2 + t1/2)ek〉.

Let εk be the sign of λk. Then |λk|〈ek, (s1/2 + t1/2)ek〉 is the same as both

εk〈(s1/2 − t1/2)ek, (s1/2 + t1/2)ek〉 and εk〈ek, (s1/2 + t1/2)(s1/2 − t1/2)ek〉.

Therefore ∑|λk|〈ek, (s1/2 + t1/2)ek〉

=∑

2−1εk

(〈(s1/2 − t1/2)ek, (s

1/2 + t1/2)ek〉+ 〈ek, (s1/2 + t1/2)(s1/2 − t1/2)ek〉)

≤ 2−1∑|〈ek, (s1/2 − t1/2)(s1/2 + t1/2) + (s1/2 + t1/2)(s1/2 − t1/2)ek〉|

=∑|〈ek, (s− t)ek〉| ≤

∑〈ek, |s− t|ek〉 = tr|s− t| = ‖s− t‖1,

where for the last ≤ we used (11.39).

The next lemma is taken from [39, Lemma 6.2.5]. It is the most difficult step.

Lemma 11.45. For any x ∈ B(H) we denote

x′ = x⊗ Id`2 ∈ B(H ⊗2 `2).

Let h ∈ B(H) be a finite rank operator with tr(hh∗) = 1. Let t = hh∗. We assume that t = hh∗

has rational eigenvalues. Then there are an integer r and a projection R of rank r such that:

(11.40) ∀U ∈ U(H) |tr(t)− r−1tr(RU ′∗RU ′)| ≤ 2‖U∗tU − t‖1/21 .

and more generally:

(11.41) ∀U, V ∈ U(H) |tr(U∗V t)− r−1tr(RU ′∗RV ′)| ≤ 2‖U∗tU − t‖1/41 ‖V∗tV − t‖1/41 .

181

Proof. Let t = p1r Q1 + p2

r Q2 +· · ·+ pkr Qk be the spectral decomposition of t where the eigenvalues

pjr

are in increasing order and the projections Qj are mutually orthogonal. Note that∑

j pjtr(Qj) =∑j pjrk(Qj) = rtr(t) = r. Let K = `2. Let P1 ≤ P2 ≤ · · · ≤ Pk be projections on K such that

tr(Pj) = rk(Pj) = pj for any j. We then define a projection R on H ⊗K by

R =∑k

j=1Qj ⊗ Pj .

Note tr(R) = rk(R) = r. Then we observe

(11.42) ∀x ∈ B(H) tr(xt) = tr(xhh∗) = r−1tr(x′R).

Indeed, r−1tr(x′R) = r−1∑

tr(xQj)tr(Pj) = tr(xr−1∑Qjpj) = tr(xt) = tr(xhh∗).

We will first show that (11.40) implies (11.41). This uses an idea (an operator valued Cauchy-Schwarz inequality) similar to the one used to prove Theorem 5.1 about the multiplicative domainof a c.p. map (here the relevant map is x 7→ Rx′R). Let

(11.43) ∀x, y ∈ B(H) F (x, y) = r−1tr(x′∗y′R)− r−1tr(Rx′∗Ry′).

We claim that F (x, x) ≥ 0. Indeed, R ≤ I implies

Rx′∗Rx′R = (x′R)∗R(x′R) ≤ (x′R)∗I(x′R) = Rx′∗x′R,

and since R2 = R we have tr(Rx′∗Rx′) = tr(Rx′∗Rx′R) and tr(x′∗x′R) = tr(Rx′∗x′R), from whichthe claim follows. Then by Cauchy-Schwarz we have

|F (x, y)| ≤ F (x, x)1/2F (y, y)1/2.

In particular, |F (U, V )| ≤ F (U,U)1/2F (V, V )1/2. By (11.42) we have tr(U∗V t) = r−1tr(U ′∗V ′R).This shows that (11.40) ⇒ (11.41).

We now turn to the more delicate verification of (11.40), for which we follow [39]. Since thiselementary fact is implicitly used several times in the proof we remind the reader that tr(xy) =tr(x1/2yx1/2) ≥ 0 whenever x ≥ 0 and y ≥ 0 are (say) Hilbert-Schmidt.In the present situation since tr(t) = 1 the Powers-Størmer inequality (11.38) gives us

(11.44) 2(1− tr(t1/2U∗t1/2U)) = ‖t1/2 − U∗t1/2U‖22 ≤ ‖U∗tU − t‖1.

A simple verification shows that

r−1tr(RU ′∗RU ′) =∑

m,`

min(pm, p`)

rtr(QmU

∗Q`U).

Plugging in the elementary inequality

∀p, p′ ≥ 0 min(p, p′) =1

2(p+ p′ − |p− p′|) ≥ (pp′)1/2 − 1

2|p− p′|,

we find

(11.45) r−1tr(RU ′∗RU ′) ≥ β − γ,

whereβ = tr(t1/2U∗t1/2U) and γ = (2r)−1

∑m,`|pm − p`|tr(QmU∗Q`U).

182

Using |pm − p`| = |p1/2m − p1/2

` |(p1/2m + p

1/2` ), we find by Cauchy-Schwarz

γ ≤ (2r)−1(∑

m,`(p1/2m − p1/2

` )2tr(QmU∗Q`U)

)1/2 (∑m,`

(p1/2m + p

1/2` )2tr(QmU

∗Q`U))1/2

and after expanding (p1/2m − p1/2

` )2 and (p1/2m + p

1/2` )2 we find

γ ≤ (2r)−1(

2r − 2rtr(t1/2U∗t1/2U))1/2 (

2r + 2rtr(t1/2U∗t1/2U))1/2

.

Since the last term is ≤ (4r)1/2 we obtain

γ ≤(

2− 2tr(t1/2U∗t1/2U))1/2

,

and hence by (11.44)

γ ≤ ‖U∗tU − t‖1/21 .

By (11.44) again 1−β = 1−tr(t1/2U∗t1/2U) ≤ 2−1‖U∗tU−t‖1 ≤ 1. A fortiori 1−β ≤ ‖U∗tU−t‖1/21 .Thus, recalling (11.43) and (11.45), we obtain

F (U,U) ≤ 1− β + γ ≤ 2‖U∗tU − t‖1/21 .

This proves (11.40).

Proof of Theorem 11.42. Let H = H ⊗2 `2 and σ(a) = a′ for a ∈ A. We will exploit Remark 11.40and (11.35). Let (hi) be as in Theorem 11.38. By perturbation we may assume that hih

∗i has

rational eigenvalues. By Lemma 11.45 we can find a net of projections Ri of rank n(i) = tr(Ri)on H so that (11.40) and (11.41) are satisfied for t = hih

∗i . Taking U = 1 in (11.41) (or invoking

(11.42)) we find

(11.46) ∀x ∈ U(B(H)) n(i)−1tr(Rix′) = tr(xhih

∗i ).

Then (11.41) and (11.35) imply for any U ∈ GS and V ∈ U(B(H))

limU |tr(U∗V hih∗i )− n(i)−1tr(RiU′∗RiV

′)| = 0,

and hence by (11.46)

limU |n(i)−1tr(RiU′∗V ′)− n(i)−1tr(RiU

′∗RiV′)| = 0,

which implies (11.37) and Remark 11.43. Recalling (11.34) we find

∀U, V ∈ S τ(π(U−1V )) = limU n(i)−1tr(RiU′∗V ′) = limU n(i)−1tr(RiU

′∗RiV′),

which implies (11.36) since E is spanned by S.

Corollary 11.46. In the situation of Theorem 11.38, assume that E generates A (as a C∗-algebra)and that π(S) ⊂ U(M) is a subgroup generating M (i.e. such that π(S)′′ = M). Then, for someH, π factors through B(H) via unital c.p. maps. More precisely, there is a family (Mn(i))i∈I ofmatrix algebras and an ultrafilter U on I, so that π admits a factorization of the form

(11.47) Au−−→ B

qU−−−→MUv−−→M

where B = (⊕∑

i∈IMn(i))∞, qU : B → MU is the quotient ∗-homomorphism and u : A → B aswell as v : MU →M are unital c.p. maps (so that ‖u‖cb = ‖v‖cb = 1).Moreover, qUu is a ∗-homomorphism, v is normal and defines a contraction from L2(τU ) to L2(τ).Lastly, M embeds in MU .

183

Proof. With the notation from the conclusion of Theorem 11.42, let Hi ⊂ H be the range of Riand let n(i) = dimHi. For any x ∈ E, let ui(x) = Riσ(x)|Hi ∈ B(Hi). Choosing an orthonormalbasis of the range of Ri, we may view ui(x) as an element of Mn(i). Then let u(x) = (ui(x)) ∈ B.Clearly, this defines a unital c.p. map u : A→ B. By (11.36) we have

(11.48) ∀x, y ∈ S τU (qUu(y)∗qUu(x)) = τ(π(y)∗π(x)).

By sesquilinearity this remains valid for all x, y ∈ E and hence

(11.49) ∀x, y ∈ E ‖qUu(x)− qUu(y)‖L2(τU ) = ‖π(x)− π(y)‖L2(τ).

and also

(11.50) ∀x ∈ E τU (qUu(x)) = τ(π(x)).

In particular, τU (qUu(x)∗qUu(x)) = 1 for any x ∈ S, which shows by Remark 11.13 that qUu(x) ∈U(MU ). By Proposition 9.7 (or because S is in its multiplicative domain), the linear map qUu : A→MU is a ∗-homomorphism on A. Let A ⊂ M be the linear span of π(S) (i.e. A = π(E)). By ourassumption A is a weak*-dense subalgebra of M . By (11.49) the correspondence π(x) 7→ qUu(x) isa well defined linear map from A to MU and it is a ∗-homomorphism (since qUu is one). By Remark11.20, the “∗-distribution equality” (11.50) implies that the von Neumann subalgebra NU generatedin MU by qUu(x) | x ∈ S is isomorphic to M = π(S)′′, via the correspondence T : M → MUdefined on A by T (π(x)) = qUu(x). This gives us a ∗-isomorphism T : M → NU ⊂MU , and henceM embeds in MU . But, by Proposition 11.21, we also have a conditional expectation P : MU → NU ,thus setting v = T−1P , we obtain the desired factorization. Note that (recall (11.49)) T also extendsto an isometry from L2(τ) to L2(τU ), so we could invoke Proposition 11.19 instead of Remark 11.20in the preceding argument. We have clearly ‖v : L2(τU ) → L2(τ)‖ = 1 (with the obvious abuse ofnotation), and v : MU →M is normal. Lastly, since B is injective, IdB factors through B(H) (forsome H) via unital c.p. maps (see §1.5), which proves the first assertion.

Remark 11.47. To see that the factorization in (11.47) implies the embedding M ⊂ MU , let ρ =qUu : A → MU , and let ρ : A∗∗ → MU be the normal ∗-homomorphism extending ρ. Since v isnormal, π = vρ implies π = vρ. But (see (26.37)) we know A∗∗ 'M ⊕ ker(π), so that with respectto the associated embedding M ⊂ A∗∗ we have IdM = π|M . Thus with IdM = vρ|M we obtain theembedding of M in MU .

M // A∗∗

π

%%

ρ

""A?

OO

π

22ρ //MU

v //M

Corollary 11.48. In the situation of Corollary 11.46, (or simply when E = A in Theorem 11.38)assume given a unital embedding A ⊂ A1 into another C∗-algebra A1. Then any ∗-homomorphismπ : A→M satisfying (11.25) extends to a unital c.p. mapping π1 : A1 →M still satisfying (11.25).

Proof. By the extension Theorem 1.39, u : A→ B admits a u.c.p. extension u1 : A1 → B. Then letπ1 = vqUu1. Note that, by the converse direction in Theorem 11.38, qU itself satisfies the propertydescribed by (11.25). Since ‖v : L2(τU )→ L2(τ)‖ = 1 and ‖u1⊗u1 : A1⊗minA1 → B⊗minB‖ ≤ 1,the map π1 must also satisfy (11.25).

184

The extension property considered in Corollary 11.48 will be refined in a later chapter notablyin Theorem 23.29.

We now relate injectivity to Theorem 11.38.

Corollary 11.49. Let (M, τ) be a tracial probability space. The following are equivalent:

(i) M is injective.

(ii) For any finite subsets (xj) and (yj) in M we have

|∑

τ(y∗jxj)| ≤ ‖∑

yj ⊗ xj‖min.

(iii) For any finite subset (xj) in M we have∑τ(x∗jxj) ≤ ‖

∑xj ⊗ xj‖min.

Proof. Assume (i). Let L,R be as in Proposition 11.16, so that L(M) ' M is injective andL(M)′ = R(Mop). By (i) ⇒ (ii) in Theorem 8.14 applied to L(M), for any n and any (yj), (xj) inMn, since

∑R(y∗j )L(xj)1 =

∑xjy∗j , we have

|τ(∑

y∗jxj)| ≤ ‖∑

R(y∗j )L(xj)‖B(L2(τ)) ≤ ‖∑

R(y∗j )⊗ L(xj)‖R(Mop)⊗minL(M),

also ‖∑R(y∗j )⊗ L(xj)‖R(Mop)⊗minL(M) = ‖

∑y∗j ⊗ xj‖Mop⊗minM and by (2.12)

‖∑

y∗j ⊗ xj‖Mop⊗minM = ‖∑

yj ⊗ xj‖M⊗minM.

This shows (i) ⇒ (ii). (ii) ⇒ (iii) is trivial. Assume (iii). Equivalently (iii) means that (11.25)holds with A = M , π = IdM and E = A(= M). The factorization in (11.47) implies that IdMfactors through B via u.c.p. maps. Therefore M is injective.

11.7 Hypertraces and injectivity

Let (M, τ) be a tracial probability space. The goal of this section is to present a particularly neatcharacterization of the injectivity of M , refining the last corollary when M is a factor, and itsgeneralization involving the center of M when M is not. The proof uses the notion (due to Connes)of hypertrace.

Definition 11.50. A tracial state on M ⊂ B(H) is called a hypertrace if it admits an extensionto a state f on B(H) such that

(11.51) ∀U ∈ U(M), ∀x ∈ B(H), f(UxU∗) = f(x).

Note that this is the same as ∀U ∈ U(M), ∀x ∈ B(H), f(Ux) = f(xU) or equivalently:

∀y ∈M, ∀x ∈ B(H), f(yx) = f(xy).

Hypertraces, or rather the states f satisfying (11.51), are analogous to invariant means for amenablegroups. In this regard, the reader is invited to compare (11.52) to (iv) in Theorem 3.30.

For example, if P : B(H) → M is a contractive projection, then f(x) = τ(P (x)) satisfies(11.51), because P (UxU∗) = UP (x)U∗ (x ∈ B(H), U ∈ U(M)) by Theorem 1.45. Thus τ is ahypertrace if M is injective. The converse is also true:

185

Proposition 11.51. If τ is a hypertrace then M is injective.

Proof. Let L2(f) be the GNS Hilbert space associated to f and let πf : B(H)→ B(L2(f)) be theGNS representation with unit vector ξf ∈ L2(f) such that f(x) = 〈ξf , πf (x)ξf 〉. Since τ(x∗x) =

f(x∗x) for any x ∈ M , we have an isometric embedding L2(τ) ' πf (M)ξf ⊂ L2(f). We view

M ⊂ L2(τ) and denote by ψ : πf (M)ξf → L2(τ) the isometric isomorphism that takes πf (a)ξfto a for any a ∈ M . Let Q : L2(f) → πf (M)ξf be the orthogonal projection. We define P :B(H) → L2(τ) by P (x) = ψQ(πf (x)ξf ) for any x ∈ B(H). Then P (x) = x for any x ∈ M . Weclaim that P (x) ∈ M and ‖P (x)‖ ≤ ‖x‖ for any x ∈ B(H), which proves that M is injective.This will be checked by the same classical argument that was used to prove Proposition 11.21. Tocheck the claim it suffices to show by Remark 11.15 that |τ(y∗2P (x)y1)| ≤ ‖x‖‖y1‖L2(τ)‖y2‖L2(τ)

for any y1, y2 ∈ M . Note τ(y∗2P (x)y1) = τ(y1y∗2P (x)) = 〈y2y

∗1, P (x)〉L2(τ). Since ψ is isometric we

have 〈y2y∗1, P (x)〉L2(τ) = 〈πf (y2y

∗1)ξf , Q(πf (x)ξf )〉L2(f) and hence by the definition of Q and the

hypertrace property of f

〈y2y∗1, P (x)〉L2(τ) = 〈πf (y2y

∗1)ξf , πf (x)ξf 〉L2(f) = f(y1y

∗2x) = f(y∗2xy1) = 〈πf (y2)ξf , πf (x)πf (y1)ξf 〉,

which yields |τ(y∗2P (x)y1)| ≤ ‖x‖‖πf (y2)ξf‖L2(f)‖πf (y1)ξf‖L2(f) = ‖x‖‖y1‖L2(τ)‖y2‖L2(τ). Thus Pis a contractive projection onto M .

Theorem 11.52. Let (M, τ) be a tracial probability space. Let Z = M ∩M ′ the center of M .

(i) If M is a factor (i.e. Z = C1) then M is injective if and only if

(11.52) ∀n ≥ 1,∀Uj ∈ U(M) n = ‖∑n

1Uj ⊗ Uj‖min.

(ii) In general, M is injective if and only if for any nonzero projection q ∈ PZ we have

(11.53) ∀n ≥ 1,∀Uj ∈ U(M) n = ‖∑n

1qUj ⊗ qUj‖min.

Proof. We first show that (11.53) holds if M is injective. Since we may replace M by qM which isstill injective (with unit q and trace x 7→ τ(q)−1τ(qx)) it suffices to show this for q = 1. The lattercase follows from (iii) in Corollary 11.49. This settles both “only if” parts.(i) Assume (11.52). Then for any n and any U0, U1, · · · , Un−1 in U(M) with U0 = 1, for any ε > 0by (2.9) there is a unit vector h ∈ S2(H) such that

n− ε2/4n < ‖∑n

1UjhU

∗j ‖2,

and hence by (26.3) supj ‖h−UjhU∗j ‖2 ≤ ε. Thus there is a net (hi) of unit vectors in S2(H) suchthat

(11.54) ∀U ∈ U(M) ‖hi − UhiU∗‖2 → 0.

Let U be a nontrivial ultrafilter refining this net. We set f(x) = limU tr(xh∗ihi). Then (11.54)implies (11.51). A fortiori, f|M is a tracial state on M . Since (finite) factors have a unique tracialstate (see Remark 11.33), we must have f|M = τ , so that τ is a hypertrace, and M is injective byProposition 11.51.(ii) Assume (11.53). We will use the classical fact that if two tracial states coincide on Z, thenthey are equal (see Remark 11.33). Consider the set I formed by all the disjoint partitions of 1Mas a finite sum 1M =

∑m1 qk of mutually orthogonal projections in PZ . The set I is ordered by its

186

natural order: i ≤ i′ means that each projection that is part of i is a sum of some of the projectionsin i′. Since Z is commutative, the set I is directed with respect to this order.We claim that for any i ∈ I, i = (q1, · · · , qm), there is a state fi on B(H) satisfying (11.51) suchthat

(11.55) ∀(λk) ∈ Cm fi(∑m

1λkqk) =

∑λkτ(qk).

We may view (fi) as a net indexed by a directed ordered set. Let U be an ultrafilter refining thisnet and let f = limU fi (pointwise on B(H)). Taking the claim as granted for the moment, let usconclude. Clearly f still satisfies (11.51) and also for any i = (q1, · · · , qm) we have f(

∑m1 λkqk) =∑

λkτ(qk) for any (λk) ∈ Cm (because for any j ≥ i (11.55) remains true if we replace fi byfj). This shows that f|Z and τ|Z coincide on the linear span of PZ , and hence since the latter isnorm-dense in Z (here Z is isomorphic to some L∞-space) we conclude that f|Z = τ |Z. But sincef|M is a tracial state, by the preceding classical fact f|M = τ . Thus τ is a hypertrace and again Mis injective by Proposition 11.51.To prove the claim, consider i = (q1, · · · , qm). Fix 1 ≤ k ≤ m and apply the preceding argument for(i) to the von Neumann algebra qkM ⊂ B(qkH) (with unit qk) instead of M ⊂ B(H). This givesus a state fk on B(qkH) satisfying (11.51) (with respect to qkH instead of H and qkM instead ofM). Let fi(x) =

∑fk(PqkHx|qkH)τ(qk) for any x ∈ B(H). Then fi is a state on B(H) satisfying

(11.51) and (11.55). This proves the claim.

Remark 11.53. Actually, any von Neumann algebra M satisfying (11.53) must be finite. Indeed,the preceding proof shows that if (11.53) holds then qM admits a tracial state for any q ∈ PZ .From this it is easy to deduce by structural arguments that M must be finite. If M is σ-finite thisimplies that M admits a faithful normal finite trace τ (see §23.1).

Remark 11.54. Let (M, τ) be a tracial probability space, with M ⊂ B(H). Then τ is a hypertraceif and only if there is a net (hi) of unit vectors in S2(H) such that ‖UhiU∗ − hi‖S2(H) → 0 (orequivalently ‖Uhi − hiU‖S2(H) → 0) for any U ∈ U(M) and such that τ(x) = lim tr(xh∗ihi) for anyx ∈M .Indeed, if τ is a hypertrace, let f be as in Definition 11.50. Let (ti) be a net of unit vectors inS1(H) = B(H)∗ with ti ≥ 0 such that f(x) = lim tr(xti) for any x ∈ B(H) (we may assume ti ≥ 0because ‖ti‖S1(H) ≤ 1 and tr(ti)→ 1 together imply that there is t′i ≥ 0 such that ‖ti−t′i‖S1(H) → 0).Then (11.51) implies that UtiU

∗ − ti → 0 for σ(B(H)∗, B(H)). By Mazur’s Theorem 26.9 after

passing to a different net we may as well assume that ‖UtiU∗− ti‖S1(H) → 0. Let hi = t1/2i . By the

Powers-Størmer inequality (11.38) we have ‖UhiU∗ − hi‖S2(H) → 0, while since f|M = τ we haveτ(x) = lim tr(xh2

i ) for any x ∈ M . No wonder if this argument rings a bell: we used an analogousone to prove (i) ⇒ (ii) in Theorem 3.30. This proves the “only if” part. The converse is immediate(just set f(x) = limU tr(xh∗ihi) for any x ∈ B(H)).Note that the existence of such a net (hi) implies that π = IdM satisfies (11.27) and hence Propo-sition 11.51 can be deduced alternatively from Corollary 11.46 as in the proof of Corollary 11.49.

11.8 The factorization property for discrete groups

We already introduced the factorization property in Definition 7.36. We now give several equivalentproperties.

Theorem 11.55. The following properties of discrete group G are equivalent:

(i) The unitary representation (s, t) 7→ λG(s)ρG(t) on G × G, extends to a (continuous) repre-sentation on C∗(G)⊗min C

∗(G). In other words G has the factorization property.

187

(ii) The linear functional f defined on C[G]⊗ C[G] by f(x ⊗ y) =∑

t∈G x(t)y(t) extends to alinear form of norm 1 on C∗(G)⊗min C

∗(G).

(iii) For any finite sequence x1, · · · , xn in C∗(G) we have

(11.56)∑‖λG(xj)‖2L2(τG) ≤ ‖

∑xj ⊗ xj‖C∗(G)⊗minC∗(G)

.

(iv) The natural ∗-homomorphism QG : C∗(G) → MG (such that QG(UG(t)) = λG(t) for allt ∈ G) factorizes via unital c.p. maps through B(H) for some H.

Proof. Assume (i). Let f be as in (ii). Let x, y ∈ C[G]. We set λG(x) =∑

s∈G x(s)λG(s) andρG(y) =

∑t∈G y(t)ρG(t). Then f(x⊗ y) = 〈δe, λG(x)ρG(y)δe〉. This shows (i) ⇒ (ii).

Assume (ii). The linear mapping taking∑x(t)UG(t) to

∑x(t)UG(t) (∀x ∈ C[G]) extends to a

C-linear isomorphism Φ : C∗(G)→ C∗(G). Therefore (ii) implies

|∑

j

∑txj(t)yj(t)| ≤ ‖

∑xj ⊗ yj‖C∗(G)⊗minC∗(G)

,

and hence taking xj = yj we obtain (iii).Assume (iii). We apply Corollary 11.46 to the case A = C∗(G) with S = UG(t) | t ∈ G (i.e.essentially S = G) and π = QG. This implies (iv).Assume (iv). We first show that (iv) ⇒ (iii). Unfortunately, we need to invoke an inequalitysatisfied by B(H) that is proved only later on these notes, namely (22.15). The latter implies thatif a ∗-homomorphism π : C∗(G)→MG satisfies the factorization in (iv) then for any finite set (xj)in C∗(G)

‖∑

π(xj)⊗ π(xj)‖MG⊗maxMG≤ ‖

∑xj ⊗ xj‖C∗(G)⊗minC∗(G)

.

Since ∑‖π(xj)‖2L2(τG) ≤ ‖

∑π(xj)⊗ π(xj)‖MG⊗maxMG

we obtain (iii).Assume (iii). Let A = C∗(G), E = span[UG(t) | t ∈ G and π = QG as before. By Theorem 11.38choosing a suitable embedding A ⊂ B(H) there is a net (hi) in the unit ball of S2(H) such that forany x, y ∈ E, say x =

∑x(t)UG(t), y =

∑y(t)UG(t) we have∑

x(t)y(t) = τG(π(x∗y)) = limU tr(x∗hiyh∗i ).

By Proposition 2.11 this implies |∑

j

∑t xj(t)yj(t)| = limU |tr(

∑x∗jhiyjh

∗i )| ≤ ‖

∑xj ⊗ yj‖A⊗minA

.

Thus, using again the C-linear isomorphism Φ : C∗(G)→ C∗(G), (ii) follows. To conclude we show(ii)⇒ (i). This is a routine argument based on the observation that the representation appearing in(i) is a GNS representation associated to the state appearing in (ii). Let T =

∑xj⊗yj ∈ A⊗A with

‖T‖min ≤ 1. Then 1⊗1−T ∗T ∈ (A⊗min A)+. Let κ : A⊗A→ B(`2(G)) be the ∗-homomorphismassociated to (s, t) 7→ λG(s)ρG(t). Note for any z ∈ A⊗A

f(z) = 〈δe, κ(z)δe〉.

For any z ∈ A⊗A we have z∗(1⊗ 1− T ∗T )z ∈ (A⊗min A)+ and hence assuming (ii), we have

‖κ(T )κ(z)δe‖2`2(G) = f(z∗T ∗Tz) ≤ f(z∗z) = ‖κ(z)δe‖2`2(G),

which shows ‖κ(T )‖B(`2(G)) ≤ 1. This completes the proof that (ii) ⇒ (i).

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Corollary 11.56. If G has the factorization property, in particular if G is amenable (see Remark7.37) then MG embeds in a trace preserving way in an ultraproduct of matrix algebras.

Proof. This follows by applying Corollary 11.46 to the case when A = C∗(G), M = MG = λG(G)′′,S = G viewed as a subset of U(A) and π = QG.

Remark 11.57. We will show in Corollary 12.23 that all free groups have the factorization property.

11.9 Notes and Remarks

The construction of non-commutative measure theory outlined in §11.2 was motivated by quantummechanics, and hence goes far back. The results of §11.2 are all classical facts. As for the originof non-commutative Lp-spaces one usually attaches the names of Dixmier, Kunze, Segal and morerecently Nelson, see [216] for more information on this topic. The construction of ultraproductsgoes back to McDuff [176]. See [4] for a much more complete treatment of the ramifications of thisimportant topic. The reader interested in ultraproducts in the non-tracial case is referred to [241, p.115] and the more recent papers [8, 9]. Theorem 11.38 is a relatively easy fact reformulating ideasthat can be traced back to Kirchberg’s [155] in the style of a Pietsch factorization for 2-summingmaps as in [205, §5]. The unital assumption (which guarantees that π(S) ⊂ U(M)) is the key toobtain an equality as in (11.27). The proof of Theorem 11.42 is more delicate. The main pointcomes from Brown and Ozawa’s book [39, Lemma 6.2.5]. Its relevance to the situation consideredin §11.6 was pointed out by Ozawa in [191]. The notion of hypertrace, together with Proposition11.51 and Theorem 11.52 for factors are all due to Connes [61]. The generalization in Theorem11.52 (ii) comes from Haagerup’s [104] on which §11.7 is based.The factorization property was introduced by Kirchberg in [155]. It is particularly interesting inconnection with property (T) groups as in Theorem 17.5. The equivalence of the properties inTheorem 11.55 were surely known to Kirchberg [157]. For its proof we used some simplificationsdue to Ozawa [189]. See also [39, p. 219].

12 The Connes embedding problem

We now turn to the first of a series of problems that will turn out to be eventually all equivalent.Since it was formulated as a question (or, say, a problem) we do not refer to it as a conjecture.

12.1 Connes’s question

In the classical von Neumann algebra terminology, a “II1-factor” is an infinite dimensional tracialprobability space (M, τ) with trivial center, i.e. such that M ∩M ′ = CI. In his famous paper [61],Connes observed that in addition to the case when G is amenable, in the somewhat “opposite”case when G = Fn (2 ≤ n ≤ ∞), the II1-factor (MG, τG) also embeds in a trace preserving wayin (MU , τU ) for some U . Since the latter case was at the time the principal “bad apple” in theclassification theory of factors, it was natural for Connes to wonder whether in fact the sameembedding held for any discrete group G and any II1-factor. But since it can be shown, using freeproducts (see Remark 11.36) that any tracial probability space embeds (in a trace preserving way)in a II1-factor, we can rephrase the problem more generally as follows.Connes’s question: Is it true that any tracial probability space (M, τ) embeds in a trace preservingway in an ultraproduct of matricial tracial probability spaces (MU , τU ) for some U ?

A priori, the embedding in the preceding question must preserve much of the algebraic structure.But actually, much less is needed for the same conclusion: as we will show in Theorem 12.3, a mere

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isometric assumption (as opposed to a completely isometric one) implies a strong ∗-isomorphicconclusion. Not surprisingly, one of the key ingredients to prove this comes from the isometrictheory (or the Jordan theory) of C∗-algebras, namely the following result, going back to Kadison[142] and Størmer [234]. See [123] for an excellent detailed account of this theory.

Theorem 12.1. Let (M, τ) and (N,ϕ) be two tracial probability spaces, i.e. two von Neumannalgebras equipped with faithful, normal, normalized traces. Let T : L2(τ)→ L2(ϕ) be an isometrysuch that T (1) = 1 and T ∗(1) = 1. In other words, T is unital and trace preserving. Assume thatT (BM ) ⊂ BN (i.e. T defines a mapping of norm 1 from M to N). Then T : M → N decomposesas a direct sum of a ∗-homomorphism and a ∗-anti-homomorphism. More precisely, there is anorthogonal decomposition I = P + Q in N (P ⊥ Q) with P,Q ∈ T (M)′ ∩ N and an associateddecomposition

∀x ∈M T (x) = PT (x)P +QT (x)Q

such that x 7→ PT (x)P is a ∗-homomorphism and x 7→ QT (x)Q is a ∗-anti-homomorphism.

Proof. By assumption T : L2(τ) → L2(ϕ) is isometric but we assume ‖T : M → N‖ = 1. Fromnow on, we view T as a mapping from M to N . Note that the latter mapping is normal since itsadjoint takes L2(ϕ) to L2(τ), and hence by density (since L2(ϕ) is norm dense in N = L1(ϕ) ) ittakes L1(ϕ) to L1(τ), or equivalently N∗ to M∗. Recapitulating, T : M → N is unital, normal,and preserves the trace. We will show that T takes unitaries to unitaries. Indeed, for any unitaryu ∈M , we have ‖T (u)‖N ≤ 1 and ‖T (u)‖22 = ‖u‖2 = 1, so T (u) is unitary by Remark 11.13.

We claim this implies that T is a Jordan ∗-morphism, i.e. that for any hermitian h, its imageT (h) is hermitian and we have T (h2) = T (h)2. Indeed, since T (eith) = 1 + itT (h) + o(t) and‖T (eith)‖ ≤ 1, T (h) must be hermitian, and actually since T (eith) is unitary, if we develop furtherfor small t ∈ R

1 = (T (eith))∗T (eith) = 1 + t2(T (h)2 − T (h2)) + o(t2),

we find that necessarily T (h)2 = T (h2). From the theory of Jordan representations as already usedin Theorem 5.6 (see [123, p. 163] or also [147, p. 588-589]), we know that there is an orthogonaldecomposition I = P +Q with P,Q ∈ T (M)′ ∩N , P ⊥ Q that gives us a decomposition

∀x ∈M T (x) = PT (x)P +QT (x)Q

such that x 7→ PT (x)P is a ∗-homomorphism and x 7→ QT (x)Q is a ∗-anti-homomorphism. (Ofcourse these are a priori non-unital.)

Proposition 12.2. Let (M, τ) and (N,ϕ) be two tracial probability spaces and let W ⊂ U(M) bea weak*-dense subset of the unitary group of M , i.e. such that U(M) is the closure of W in L2(τ).The following are equivalent:

(i) There is a unital trace preserving (C-linear) isometry T : L2(τ) → L2(ϕ) such that ‖T :M → N‖ ≤ 1.

(ii) There is a function r : 1 ∪W → U(N) such that r(1) = 1 and

(12.1) ∀u1, u2 ∈ 1 ∪W τ(u∗1u2) = ϕ(r(u1)∗r(u2)).

(ii)’ There is a function r : W → BN such that

(12.2) ∀u1, u2 ∈ W τ(u∗1u2) = ϕ(r(u1)∗r(u2)).

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Proof. Assume (i). The first step of the proof of Theorem 12.1 shows that T (U(M)) ⊂ U(N).Then (i) ⇒ (ii) is obvious: we just let r = T|1∪W and observe that

∀u1, u2 ∈ U(M) ∀µ ∈ C ‖u1 + µu2‖2L2(τ) = 1 + |µ|2 + 2<(µτ(u∗1u2)).

Since ‖u1 + µu2‖2L2(τ) = ‖r(u1) + µr(u2)‖2L2(ϕ), it follows that r = T|1∪W satisfies (ii).

Assume (ii). Assuming for a moment that such a T exists, let T : span[1 ∪ W] → N be thelinear mapping extending r. For any x ∈ span[1 ∪W], we have by (12.1)

‖T (x)‖2L2(ϕ) = ‖x‖2L2(τ),

and this shows that T is unambiguously well defined. Note that T preserves the trace (take u1 = 1in (12.1)), and T (1) = r(1) = 1. By our density assumption, the unitaries in W are dense for theL2(τ)-norm in the set U(M) of unitaries of M (which linearly spans M), and hence span[W] isdense in L2(τ). Therefore T extends to a unital trace preserving isometry, still denoted by T , fromL2(τ) to L2(ϕ), such that T (W) ⊂ U(N). Moreover, since the set of unitary elements is closed inL2(ϕ) (see e.g. Remark 11.13), and W is assumed dense in U(M), we have T (U(M)) ⊂ U(N). Bythe Russo-Dye Theorem 26.18, the convex hull of U(M) is norm-dense (and a fortiori L2(τ)-dense)in the unit ball of M , and the unit ball of N is closed in L2(ϕ). Therefore ‖T : M → N‖ = 1.This shows that (i) holds, and hence (i) ⇔ (ii).(ii) ⇒ (ii)’ is trivial. Conversely, assume (ii)’. Then ϕ(r(u)∗r(u)) = 1 for any u ∈ W. The factthat r(W) ⊂ U(N) is automatic by Remark 11.13. To take care of the condition r(1) = 1, wewill change W and r. We pick a fixed u0 ∈ W and we set W ′ = u−1

0 W = u−10 u | u ∈ W and

r′(u−10 u) = r(u0)−1r(u). Then 1 ∈ W ′, r′(1) = 1, (W ′, r′) satisfy (12.1) and W ′ is still dense in

U(M). Therefore, by the (already proved) implication (ii) ⇒ (i) applied to W ′ and r′, we see that(i) holds, and we already proved that (i) implies (ii).

The following criterion for M to embed in some MU is due to Kirchberg.

Theorem 12.3 (Kirchberg’s criterion). Let (M, τ) be a tracial probability space. Let W ⊂ U(M)be a weak*-dense unital subset of the unitary group of M , i.e. we assume that U(M) is the closureof W in L2(τ). The following are equivalent:

(i) There is a (unital) trace preserving embedding of M in an ultraproduct of matrix algebras.

(ii) For any ε > 0, any n and any u1, · · · , un ∈ W there is an integer N and unitary N × Nmatrices v1, · · · , vn such that

(12.3) ∀i, j = 1, · · · , n |τ(u∗iuj)− τN (v∗i vj)| ≤ ε,

and

(12.4) ∀j = 1, · · · , n |τ(uj)− τN (vj)| ≤ ε.

(iii) For any ε > 0, any n and any u1, · · · , un ∈ W there is an integer N and N × N matricesx1, · · · , xn in the unit ball of MN , equipped with its normalized trace τN , such that

(12.5) ∀i, j = 1, · · · , n |τ(u∗iuj)− τN (x∗ixj)| ≤ ε.

Moreover, if W ⊂ U(M) is countable, (ii) ⇒ (i) holds for an ultraproduct based on a sequence ofmatrix algebras.

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Proof. (i)⇒ (ii) is essentially obvious withW = U(M) (recall Lemma 11.30). (ii)⇒ (iii) is trivial.We give the rest of the proof assuming that W is countable. The general case can be treatedsimilarly.Assume (iii). To show that (i) holds we will use Proposition 12.2 and Theorem 12.1.Let u1, u2, · · · be an enumeration of W. Let I be the set of pairs (n, ε) with n ∈ N and ε > 0.We view it as a directed set for the order defined by (n, ε) ≤ (n′, ε′) if n ≤ n′ and ε′ ≤ ε. Wemay restrict ε to be in a countable sequence decreasing to 0 so that I be countable. Let U be anultrafilter refining the resulting net (see Remark 26.6). For each i = (n, ε), we can find N(i) and(x1(i), · · · , xn(i)) in the unit ball of MN(i) such that (12.5) holds. We then set

∀k ≤ n vk(i) = xk(i),

∀k > n vk(i) = 1 (say).

The values for k > n will turn out to be irrelevant. Let M(i) = MN(i) equipped with its normalizedtrace. Let B and MU be as before. We may define Vk ∈ BB by setting

Vk = (vk(i))i∈I .

Clearly, by (12.5) , for any k, ` ∈ N

limU τi(Vk(i)∗V`(i)) = τ(u∗ku`).

In other words, if we denote by vUk ∈ BMU the equivalence class of Vk modulo IU , we have

(12.6) τU (vUk∗vU` ) = τ(u∗ku`).

Thus if we define r : W → BMU by r(uk) = vUk , then (12.2) holds for ϕ = τU . By Proposition 12.2and Theorem 12.1 there is a unital trace preserving isometry T : M → MU and an orthogonaldecomposition I = P +Q with P,Q ∈ T (M)′ ∩MU that induces a decomposition

∀x ∈M T (x) = PT (x)P +QT (x)Q

such that x 7→ PT (x)P is a ∗-homomorphism and x 7→ QT (x)Q is a ∗-anti-homomorphism.We claim that there is a unital trace preserving ∗-anti-isomorphism κ : QMUQ → QMUQ. Usingthis claim we can obtain a bona fide unital and trace preserving ∗-homomorphism T ′ embedding Min MU by setting T ′(x) = PT (x)P +κ(QT (x)Q), whence (i). To check the claim, first observe thatMU is clearly ∗-anti-isomorphic to itself by a trace preserving map y 7→ ty (associated to matrixtransposition). More precisely, for any y = qU ((yi)) ∈ MU we set ty = qU ((tyi)). Note that thelatter map gives us a ∗-anti-isomorphism from QMUQ (with unit Q) to tQMU

tQ (with unit tQ).Let (Qi) ∈ B be a representative of Q such that Qi is a projection in M(i) = MN(i) for all i ∈ I(see Remark 11.31). Since Qi and tQi have the same trace, their ranges have the same dimension,so there is a unitary matrix Υi such that Qi = Υi

tQiΥ∗i . Then if Υ ∈MU is the unitary associated

to (Υi) the mapping κ defined by κ(y) = ΥtyΥ∗ is the desired ∗-anti-isomorphism.

Remark 12.4. By (i)⇔ (iii) in Theorem 12.3 to show that a tracial probability space (M, τ) embedsin a trace preserving way in an ultraproduct of matrix algebras, it suffices to check that this holdsfor any finitely generated (and a fortiori) weak* separable von Neumann subalgebra of M .

Remark 12.5 (Separable factors suffice). Recall that, by definition, a von Neumann algebra M is afactor if its centre is trivial. By Remark 11.36, to answer positively Connes’s question for all finitevon Neumann algebras, it actually suffices to answer it for finite “factors” on a separable Hilbert

192

space (which incidentally is the original question raised in [61]). Indeed, by Remark 11.36 anyweak* separable tracial probability space embeds in a trace preserving way into one that is a factor.Moreover, by Remark 12.4 the weak* separable case of the Connes embedding problem implies thegeneral one. Actually, any embedding of a factor M as a C∗-subalgebra of an ultraproduct MU ofmatrix algebras is automatically trace preserving (and hence normal) since M has a unique tracialstate (see Remark 11.33). Thus for the Connes embedding problem it suffices to show that anyweak* separable finite factor embeds as a C∗-subalgebra of a matricial MU .

Using Lemma 11.30, one easily deduces the following fact (which can also be proved by observingthat an ultraproduct of ultraproducts is again an ultraproduct).

Corollary 12.6. Let (M(i), τi)i∈I be a family of tracial probability spaces. Assume that each oneof them embeds in a trace preserving way into an ultraproduct of matrix algebras. Then the sameis true for their ultraproduct (MU , τU ) relative to any ultrafilter U on I.

For future reference it may be worthwhile to formulate the following obvious consequence:

Corollary 12.7. Let (M, τ) and (N,ϕ) be two tracial probability spaces. Assume that there is atrace preserving embedding of M in an ultraproduct of matrix algebras. For the same to hold for(N,ϕ) the following condition is sufficient: For any ε > 0, any n and any u0, · · ·un ∈ U(N) thereare v0, v1, · · · vn ∈ BM such that

(12.7) ∀i, j = 0, · · · , n |ϕ(u∗iuj)− τ(v∗i vj)| ≤ ε.

Proof. We may replace M by MU . Then N satisfies (iii) in Theorem 12.3 with W = U(N).

The next variant is more involved. Here we use Lemma 11.45 to further refine the precedingcriterion.

Theorem 12.8. The conditions (i)-(iii) in Theorem 12.3 are equivalent to the following ones:

(iv) For any ε > 0, any n and any u1, · · ·un ∈ W, there are an integer N , matrices x1, · · ·xn inthe unit ball of MN and η ∈MN with τN (η∗η) = 1 such that

(12.8) ∀i, j = 1, · · · , n |τ(u∗iuj)− τN (x∗i ηxjη∗)| < ε.

(iv)’ For any ε > 0, any n and any u1, · · ·un ∈ W, there are x1, · · ·xn in the unit ball of B(`2)and a Hilbert-Schmidt operator h ∈ B(`2) with tr(h∗h) = 1 such that

(12.9) ∀i, j = 1, · · · , n |τ(u∗iuj)− tr(x∗ihxjh∗)| < ε.

(v) For any ε > 0, any n and any u1, · · ·un ∈ W there are an integer N , N ×N unitary matricesv1, · · · vn and η ∈MN with τN (η∗η) = 1 such that

(12.10) ∀i, j = 1, · · · , n |τ(u∗iuj)− τN (v∗i ηvjη∗)| < ε.

Proof. (iii)⇒ (iv) is obvious (with η = 1) and (iv)⇒ (iv)’ is trivial (with h = N−1/2η). Conversely,if (iv)’ holds we may assume by density of the finite rank operators in S2(`2) that there is a projectionP of finite rank N such that h = PhP . We can replace xj by PxjP and setting η = N1/2h we seethat (12.9) becomes (12.8). This shows (iv) ⇔ (iv)’.

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Assume (iv). By polar decomposition we can write xj = vj |xj | with vj unitary and also x∗j =v∗j (vj |xj |v∗j ) = v∗j |x∗j |. Then

τN (x∗i ηxjη∗) = τN (v∗i |x∗i |ηvj |xj |η∗)

Thus to show (v) it suffices to prove that

(12.11) |τN (v∗i |x∗i |ηvj |xj |η∗)− τN (v∗i ηvjη∗)| ≤ f1(ε)

where f1 depends only on ε and f1(ε) = o(ε). We will denote f2, f3, ... functions of the same kind.To prove (12.11) it suffices to show that

(12.12) ‖|xj |η∗ − η∗‖L2(τN ) ≤ f1(ε)/2 and ‖|x∗i |η − η‖L2(τN ) ≤ f1(ε)/2.

Taking i = j in (12.8) we find

(12.13) |1− 〈xjη∗, η∗xj〉L2(τN )| ≤ ε,

and hence ‖η∗xj − xjη∗‖2L2(τN ) ≤ 2ε. Therefore |1 − 〈xjη∗, xjη∗〉L2(τN )| ≤ f2(ε) = ε +√

2ε, which

means |1 − τN (ηx∗jxjη∗)| ≤ f2(ε). Since |xj |2 ≤ |xj | we have 0 ≤ τN (ηx∗jxjη

∗) ≤ τN (η|xj |η∗) ≤ 1and hence |1 − τN (η|xj |η∗)| ≤ f2(ε). The latter means |1 − 〈η∗, |xj |η∗〉L2(τN )| ≤ f2(ε), and hence‖η∗ − |xj |η∗‖2L2(τN ) ≤ 2f2(ε), which proves the first part of (12.12). We now reapply the same

argument starting from |1 − 〈x∗jη, ηx∗j 〉L2(τN )| ≤ ε instead of (12.13) and this gives us the secondpart of (12.12), so that (v) follows.It remains to show (v) ⇒ (iii). Assume (v). We have for any j

|1− 〈vjη, ηvj〉L2(τN )| = |1− τN (v∗j ηvjη∗)| ≤ ε,

and hence ‖ηvj − vjη‖2L2(τN ) ≤ 2ε. Using ηη∗ − vjηη∗v∗j = (ηvj − vjη)v∗j η∗ + vjη(v∗j η

∗ − η∗v∗j ) wefind

‖ηη∗ − vjηη∗v∗j ‖L1(τN ) ≤ 2(2ε)1/2,

and also

(12.14) |τN (v∗i (ηvj)η∗)− τN (v∗i (vjη)η∗)| ≤ (2ε)1/2.

We may assume that ηη∗ has rational eigenvalues. Let H = `N2 and let v′ = v⊗ Id`2 ∈ B(H ⊗2 `2)as in Lemma 11.45. The latter associates to t = N−1ηη∗ a projection R ∈ B(H⊗2 `2) of finite rankr, such that

|τN (v∗i vjηη∗)− r−1tr(Rv′iR

∗v′j)| ≤ f3(ε).

By (12.14)|τN (v∗i ηvjη

∗)− r−1tr(Rv′i∗Rv′j)| ≤ (2ε)1/2 + f3(ε),

and hence lastly|τ(u∗iuj)− r−1tr(Rv′i

∗Rv′j)| ≤ ε+ (2ε)1/2 + f3(ε).

Let H ⊂ H ⊗2 `2 be the range of R. We conclude that (iii) holds with r now playing the role of Nand xj = Rv′j |H viewed as an operator on H, or equivalently as an element of Mr.

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12.2 The approximately finite dimensional (i.e. “hyperfinite”) II1-factor

Although we prefer not to use this in the sequel, we wish to briefly mention here another wayto formulate the Connes question in terms of the so-called approximately finite dimensional (i.e.“hyperfinite”) II1-factor (R, τ0), which can be defined as the unique countably generated, approxi-mately finite dimensional, tracial probability space with trivial center and no atom (i.e. no nonzerominimal projection).

One can legitimately think of (R, τ0) as a non-commutative analogue of the Lebesgue interval([0, 1], dt); indeed, the latter is also the unique countably generated atomless probability space. Theuniqueness of (R, τ0) (up to isomorphism) goes back to Murray and von Neumann who proved thatany two countably generated finite factors with no atom (the so-called II1-factors) are isomorphicif each is approximately finite dimensional. For a detailed proof see [146, p. 896]. Many yearslater, solving a longstanding problem, Connes [61] proved that for such algebras injective implies(and hence is equivalent to) approximately finite dimensional. Thus (R, τ0) can be described asthe unique (up to isomorphism) injective, tracial probability space with trivial center and no atomon a separable Hilbert space. We will not prove any of these uniqueness theorems. We brieflydescribe one classical construction by which (R, τ0) can be produced, thus showing its “existence”but we take its uniqueness for granted. The quick construction we outline highlights that thereis a copy of (R, τ0) inside an ultraproduct of matricial tracial probability spaces. The Connesembedding problem can then be reformulated using the “ultrapowers” of (R, τ0). Let (R(i))i∈Ibe a family of copies of R. Then the ultraproduct of (R(i), τ0) with respect to an ultrafilter U iscalled an ultrapower of (R, τ0); we denote it by (RU , τU0 ). We will show that a tracial probabilityspace (M, τ) embeds (trace preservingly) in RU for some U if and only if it similarly embeds in anultraproduct of matricial tracial probability spaces.

Let (mi)i∈N be a sequence of integers ≥ 2. The traditional way is to define (R, τ0) as an infinitetensor product (often called “ITPFI” in the literature) of the form

(12.15)⊗i∈N

(Mmi , τmi).

By the uniqueness results just mentioned, up to isomorphism, the resulting algebra does not dependon the choice of the sequence (mi), and the simplest choice is clearly to take mi = 2 for all i, butwe will not prove this either. Let N(i) =

∏k≤imk. Let

M(i) = MN(i) ' ⊗k≤iMmk

and let τi be the normalized trace on M(i). Note that M(k + 1) ' M(k) ⊗ Mmk+1and more

generally, for any i > kM(i) 'M(k)⊗Mmk+1

⊗ · · · ⊗Mmi

so that we have a natural trace preserving embedding M(k) → M(i) taking x ∈ M(k) to x ⊗ 1 ⊗· · · ⊗ 1 ∈ M(i). Using these embeddings we may think of M(k) as a ∗-subalgebra of M(i) andform the unital ∗-algebra that is the union (i.e. formally the inductive limit) A = ∪M(i). It isconvenient to think of a typical element x of A as x = a ⊗ 1 ⊗ 1 ⊗ · · · with a ∈ M(i) for some i,followed by an infinite sequence of ⊗1’s. Then τ(x) = τi(a) defines a linear functional on A, suchthat τ(x∗x) = τ(xx∗) ≥ 0 and τ(1) = 1. The GNS construction applied to A produces a Hilbertspace H and an injective ∗-homomorphism π : A → B(H), such that τ(x) = 〈1, π(x)1〉 (x ∈ A).We then define ⊗i∈N(Mmi , τmi) as R = π(A)′′ equipped with the natural extension of τ , that isa faithful normal trace τ0 on π(A)′′. We have a natural identification H ' L2(τ0) with which πbecomes the representation L of left multiplication on L2(τ0).

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We now relate this construction to ultraproducts. Let U be a non trivial ultrafilter on N and let(MU , τU ) be the ultraproduct of (M(i), τi). We will show that ⊗i∈N(Mmi , τmi) embeds in a tracepreserving way into MU .As before, let B =

(⊕∑

i∈IM(i))∞, with quotient map qU : B → MU , let vk : M(k) → B be the

map taking a ∈ M(k) to b = (bi) ∈ B defined by bi = a ⊗ 1 ⊗ · · · ⊗ 1 for all i ≥ k and (this isactually somewhat irrelevant) bi = 0 for all i < k. Then let uk : M(k) → MU be the map takinga ∈ M(k) to qU (vk(a)). Note τU (uk(a)) = τk(a) and uk : M(k) → MU is an embedding. LetAk = uk(M(k)) ⊂ MU . With the obvious identification (corresponding to a 7→ a ⊗ 1) we haveAk ⊂ Ak+1. This gives us an embedding ψ : A ⊂ MU such that τU (ψ(x)) = τ0(x) for any x ∈ A.Clearly, this extends to an isometry T : L2(R, τ0) → L2(MU , τU ). By Proposition 11.19 that samemap defines a trace preserving embedding of (R, τ0) into (MU , τU ).

Proposition 12.9. Let (M, τ) be a tracial probability space on a separable Hilbert space. Thenthere is a trace preserving embedding of M into an ultraproduct of matrix algebras if and only ifthere is one of M into RU for some ultrafilter U .

Proof. By Corollary 12.6 there is a trace preserving embedding of RU into an ultraproduct of matrixalgebras. This settles the “if part”. For the converse, recall that (12.15) gives us a copy of R nomatter what (mi) is. Thus MN(i) embeds in R for each i, and hence any ultraproduct of (MN(i))

relative to U embeds in RU .

Remark 12.10. As most ultraproducts, the von Neumann algebra RU is defined on a non-separableHilbert space and this is unavoidable. The appearance of large cardinals suggests that issues fromlogic should play a role. Consider for instance the very natural question whether the ultrapowersRU (for varying nontrivial ultrafilters on N) are all isomorphic: a positive answer turns out to beequivalent to the continuum hypothesis [86]. See [86, 87, 88, 89] where the analogous question forultraproducts of (Mn)n≥1 is discussed as well as other similar issues.

12.3 Hyperlinear groups

Definition 12.11. A discrete group G will be called approximately linear (also called “hyperlin-ear”) if (MG, τG) embeds in a trace preserving way in an ultraproduct of matricial tracial probabilityspaces, such as (MU , τU ) with all M(i)’s matricial.

In other words, G is approximately linear (or “hyperlinear”) if for (MG, τG) the answer to theConnes question is positive. We already know (see Remark 11.56) that this holds if G is amenableor has the factorization property.

Just like “hyperfinite”, the term “hyperlinear” is hardly a good choice since it does not im-ply linear in any reasonable sense, so we prefer to use “approximately linear” which seems moreappropriate.

Theorem 12.12. The following properties of a discrete group G are equivalent:

(i) The group G is approximately linear (so-called hyperlinear).

(ii) There is a group representation π : G→ U(MU ) embedding G into the unitary group U(MU )of an ultraproduct of matricial tracial probability spaces and satisfying

(12.16) ∀t ∈ G τU (π(t)) = τG(λG(t)).

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(iii) For any finite subset S ⊂ G containing the unit e and any ε > 0 there is an integer N < ∞and a function ψ : S → UN with values in the group UN = U(MN ) of N×N -unitary matricessuch that

∀s, t ∈ S ‖ψ(s)ψ(t)− ψ(st)‖L2(τN ) < ε,

and|τN (ψ(e))− 1| < ε and ∀t ∈ S, t 6= e |τN (ψ(t))| < ε.

Proof. Assume (i). Let Φ : MG → MU the embedding. Let π(t) = Φ(λG(t)) (t ∈ G). Then (ii) isimmediate.Assume (ii). By Lemma 11.30 each π(t) has a unitary representative modulo IU . Thus, for eachi there is πi(t) ∈ U(M(i)) such that qU ((πi(t))) = π(t). Since π(st) = π(s)π(t) (s, t ∈ G), wehave limU ‖πi(s)πi(t) − πi(st)‖L2(τi) = 0 and since τ(π(e)) = 1 and τ(π(t)) = 0 if t 6= e, we havelimU |τi(πi(e)) − 1| = 0 and limU |τi(πi(t))| = 0 if t 6= e. Recall that (M(i), τi) = (MN(i), τN(i)) forsome N(i) <∞ Thus, it suffices to take ψ = πi and to choose i far enough relative to U to obtain(iii).Assume (iii). Let I be the set of pairs (S, ε) with S ⊂ G finite subset and ε > 0, with the usualordering (S, ε) ≤ (S′, ε′) if S ⊂ S′ and ε′ ≤ ε. Let U be an ultrafilter refining the net associatedto this directed set (see Remark 26.6). For any i = (S, ε), when t ∈ S we set πi(t) = ψ(t) whereψ is the function given by (iii); and when t 6∈ S (this is actually irrelevant) we set πi(t) = 1. Letπ(t) = qU ((πi(t))). Then it is easy to check that (ii) holds, but our goal is (i).Let A = span[λG(t) | t ∈ G] ⊂ L2(τG). Let T : A →MU be the ∗-homomorphism taking λG(t) toπ(t). Note that ‖T (a)‖2L2(τU ) = τU (π(a∗a)) and ‖a‖2L2(τG) = τG(a∗a). By (12.16), for any a ∈ A,

we have ‖T (a)‖L2(τU ) = ‖a‖L2(τG), and hence T extends to an isometric embedding from L2(τG) toL2(τU ). Therefore, by Proposition 11.19, T also extends to a normal (trace preserving) embeddingon MG into MU , showing that (i) holds. Alternatively, for (ii)⇒ (i) we could invoke Remark 11.20,observing that (ii) simply means that (λG(t))t∈G and (π(t))t∈G have the same ∗-distribution.

Remark 12.13. If we take s = t = e in ‖ψ(s)ψ(t)− ψ(st)‖L2(τN ) < ε < 1 it follows that the unitary

matrix ψ(e) satisfies |τN (ψ(e))− 1| <√

2ε, so the condition |τN (ψ(e))− 1| < ε could be omitted.

12.4 Residually finite groups and Sofic groups

We denote by SN the group of permutations of the set withN elements, i.e. the so-called “symmetricgroup”. Note the embedding SN ⊂ UN obtained by identifying a permutation σ with the N ×Nunitary matrix u defined by u(ej) = eσ(j). Note that

τN (u) = N−1|j ∈ 1, · · · , N | σ(j) = j|

is the proportion of the number of fixed points of σ.

Definition 12.14. A discrete group G is called “sofic” if it satisfies the condition (iii) in Theorem12.12 with a function ψ taking values in permutation matrices.Equivalently, and more explicitly, G is sofic if for any finite subset S ⊂ G with eG ∈ S and anyε > 0 there is an integer N <∞ and a function ψ : S → SN such that

∀s, t ∈ S |j ∈ 1, · · · , N | (ψ(s)ψ(t))(j) 6= ψ(st)(j)| ≤ εN,

and

|j ∈ 1, · · · , N | ψ(e)(j) 6= j| ≤ εN and ∀t ∈ S \ e |j ∈ 1, · · · , N | ψ(t)(j) = j| ≤ εN.

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Remark 12.15 (About examples of sofic groups). Using Følner sequences (see Remark 3.33), weshow next that amenable groups are sofic, but also as we will soon show (see Lemma 12.19) allfree groups are sofic, which leads one to the important open question whether every group is sofic.This seems to be the group theoretic analogue of the Connes problem. The notion was introducedby Gromov and the term “sofic” was coined by B. Weiss (sofi means finite in hebrew). We refer toa series of papers by Elek and Szabo (for instance [81]) for more information on sofic groups.

Lemma 12.16. Amenable groups are sofic.

Proof. By Remark 3.33 there is a net (Bi) formed of finite subsets of our amenable G such that

∀t ∈ G lim |Bi \ t−1Bi||Bi|−1 = 0.

Let ψi(t) be a permutation of Bi that is equal to x 7→ tx for any x ∈ Bi ∩ t−1Bi and that isextended to Bi \ t−1Bi in such a way that ψ(t) : Bi → Bi is bijective. Then for any unital finite setS ⊂ G and ε > 0 when i is far enough in the net we will have |Bi \ (st)−1Bi| < (ε/3)|Bi| for any(s, t) ∈ S × S (and hence also for (s, eG) and (eG, t)). It is then easy to check that ψ = ψi satisfiesthe conditions required in Definition 12.14 for G to be sofic.

Definition 12.17. A group G is called residually finite if there exists a collection of finite groups(Γi) and homomorphisms ϕi : G→ Γi separating the points of G, i.e. for any finite subset S ⊂ Gthere is an i for which the restriction of ϕi to S is injective. Without loss of generality, we mayassume that Γi = G/Ni where each Ni ⊂ G is a normal subgroup with finite index and ϕi is thecanonical quotient map. Thus, G is residually finite if and only if it admits a family of normalsubgroups with finite index (Ni), directed by (downward) inclusion and such that

⋂i∈I Ni = eG.

Proposition 12.18. Any residually finite group is sofic and any sofic group is approximately linear(so-called hyperlinear).

Proof. If G is residually finite, let S ⊂ G be a finite subset. There is a finite group Γ and a grouphomomorphism ψ : G→ Γ that is injective on S. Let N = |Γ|. We may view Γ as acting on itselfby translation (and hence any t 6= e acts without fixed points), so that Γ ⊂ SN . Then, viewing ψ asacting into SN , we obtain the properties in Definition 12.14 with ε = 0. Therefore G is clearly sofic.The implication sofic⇒ approximately linear (so-called hyperlinear) is obvious given the definitionof sofic and (iii) ⇒ (i) in Theorem 12.12.

The following fact is classical.

Lemma 12.19. Free groups are residually finite (and a fortiori sofic).

Proof. Let G = FI . Let gi | i ∈ I be the (free) generators. Let C ⊂ G be a finite subset. Itsuffices to produce a (group) homomorphism h : G → Γ into a finite group Γ such that, for anyc in C, we have h(c) 6= eΓ if c 6= e, where eΓ denotes the unit in Γ and e the unit in G (i.e. the“empty word”).We may assume that C ⊂ G′ where G′ is the subgroup generated by a finite subset gi | i ∈ Jof the generators. Let k = max|c| | c ∈ C (here |c| denotes the length of the reduced wordassociated to c, i.e. the number of elements in gi, g−1

i | i ∈ I used to express c in reduced form).We then set

S = t ∈ G′ | |t| ≤ k.

We will take for Γ the (finite) group of all permutations of the (finite) set S. For any i in J , weintroduce

Si = t ∈ S | git ∈ S.

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Then clearly Si ⊂ S and giSi ⊂ S. Hence (since |Si| = |giSi| and S is finite) there is a permutationσi : S → S such that σi(s) = gis for any s in Si. Then if s, t ∈ S and if git = s (or equivalentlyt = g−1

i s) we have σi(t) = s (or equivalently t = σ−1i (s)). Thus it is easy to check that if a reduced

word t = gε1i1 gε2i2. . . gεmim (m ≤ k εi = ±1) lies in S (note that, by definition of S, e and all the

subwords of t also lie in S) we have

σε1i1 σε2i2. . . σεmim (e) = t.

Therefore, if we define h : G → Γ as the unique homomorphism such that h(gi) = σi ∀i ∈ J andσ(gi) = eΓ ∀i /∈ J , we find, for t as before, h(t) = σε1i1 σ

ε2i2. . . σεmim and h(t)(e) = t, in particular we

have h(t) 6= eΓ whenever t ∈ S and t 6= e. Since C ⊂ S, we obtain the announced result.

Remark 12.20. By a famous result due to Malcev [177], finitely generated linear groups are resid-ually finite. Using this, Lemma 12.19 could be deduced from Choi’s Theorem 9.18.

Consequently, we obtain the following important fact, which was mentioned in passing byConnes in [61, p. 105] as motivation for his question discussed in §12.

Theorem 12.21 ([256]). There is a trace preserving embedding of the von Neumann algebra ofthe free groups Fn or F∞ into an ultraproduct of matrix algebras (in other words free groups areapproximately linear).

Corollary 12.22. When G is a free group, there is a factorization of the canonical ∗-homomorphismQG : C∗(G) → MG of the form QG : C∗(G)

w−→B v−→MG where w is a ∗-homomorphism, v is c.p.with ‖v‖cb ≤ 1, and B is a von Neumann algebra with the WEP (actually B is injective).

Proof. By Lemma 12.19 and Proposition 12.18, G is approximately linear (so-called hyperlinear).We have MG ⊂ MU with a c.p. projection (the conditional expectation) P : MU → MG. Theunitary representation π : G→ U(MU ) appearing in property (ii) in Theorem 12.12 admits a liftingto a unitary representation π : G → U(B) where B =

(⊕∑

i∈IM(i))∞ as in (11.14). Indeed, by

the freeness of G, it suffices for this to be able to lift the images under π of each free generator,and this is guaranteed by Lemma 11.30. Then π extends to a ∗-homomorphism C∗(G)

w−→B and,denoting as before by q : B →MU the quotient map, we can take v = Pq. Note that B is injectiveand hence has the WEP (see Proposition 1.48 and Corollary 9.26).

By Theorem 11.55, we deduce from Corollary 12.22:

Corollary 12.23. Free groups have the factorization property described in §11.8.

12.5 Random matrix models

The term “matrix model” is frequently used with respect to a tracial probability space (M, τ). Thisis an alternative way to discuss the embedding in the Connes question. More precisely, assume thatM is generated by a family of elements (xs)s∈S (indexed by some set S), consider a sequence of

matricial sizes (nN )N≥1 and families (x(N)s )s∈S in MnN such that for any polynomial P (Xs, X

∗s ) in

the non-commuting formal variables (Xs)s∈S we have

limN→∞ τnN (P (x(N)s , x(N)

s

∗)) = τ(P (xs, x

∗s)).

We then say that (x(N)s )s∈S is a matrix model for (xs)s∈S , or (somewhat abusively) for (M, τ).

Thus to say that (M, τ) admits a matrix model is but another way of saying that it embeds in atrace preserving way in an ultraproduct of matricial tracial probability spaces. However, the matrix

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model terminology is better adapted to the theory of random matrices. The latter provides veryinteresting and fruitful examples of matrix models, in connection with Voiculescu’s free probabilitytheory (see [255]). For instance, the convergence of both the unitary and Gaussian models in thefollowing statements, in which nN = N and S = 1, 2, · · · , is due to Voiculescu:

Theorem 12.24 (Random unitary matrix model). Let (U(N)s )s≥1 be an independent, identically

distributed (i.i.d. in short) sequence of random matrices uniformly distributed over the unitarygroup UN . We assume (for convenience) all random elements defined on a probability space (Ω,P).Let (M, τ) = (MF∞ , τF∞). Consider the sequence xs = λF∞(gs) (s ≥ 1) in M . Then for almost all

ω ∈ Ω, the sequence (U(N)s (ω))s≥1 is a matrix model for (xs)s≥1 (with respect to N →∞).

Remark 12.25 (Random permutation matrix model). The same result is valid if we assume that

(U(N)s )s≥1 is an i.i.d. family uniformly distributed over the subgroup of UN formed of matrices of

permutation of size N ×N . This is due to Nica (see [179]).

Fix a nontrivial ultrafilter U on N. Let MU be the ultraproduct of the family (MN , τN ) withrespect to U . Let B = (⊕

∑N≥1MN )∞. Let qU : B → MU be the quotient map. Fix ω such that

(U(N)s (ω))N≥1 is a matrix model for (λF∞(gs)). We know by Remark 11.20 that the correspondence

λF∞(gs) 7→ qU ((U(N)s (ω))N≥1) extends to a trace preserving ∗-homomorphism JUω : MF∞ → MU

embedding MF∞ as a von Neumann subalgebra of MU . Moreover, by Proposition 11.21 there is ac.p. contractive projection PUω from MU onto JUω (MF∞), whence the following statement.

Corollary 12.26. For almost all ω, the preceding map JUω : MF∞ →MU is a trace preserving (vonNeumann sense) embedding and there is a c.p. contractive projection PUω from MU onto JUω (MF∞).

The case of Gaussian random matrices is central. To state the result in that case one needs thenotion of a free semi-circular sequence (xs), and that of a circular one (ys), for which we refer to§11.4. It is known that the von Neumann algebra generated by either (xs) or (ys) is isomorphic tothe von Neumann algebra M = MF∞ .

Let (X(N)s )s≥1 (resp. (Y

(N)s )s≥1) be an independent, identically distributed (i.i.d. in short) sequence

of random matrices each with the same distribution as a Gaussian model X(N) (resp. Y (N)). Bydefinition its entries X(N)(i, j) (for 1 ≤ i ≤ j ≤ N) are all independent mean zero (real valued)Gaussian variables with E|X(N)(i, j)|2 = 1/N for all i < j and E|X(N)(j, j)|2 = 2/N for all j;the other entries are determined by X(N)(i, j) = X(N)(j, i) so that X(N) is a symmetric randommatrix. This is known as the GOE random matrix model.By definition the entries Y (N)(i, j) (for 1 ≤ i, j ≤ N) are i.i.d. complex valued Gaussian variablessuch that <(Y (N)(i, j)) and =(Y (N)(i, j)) are independent (real valued) Gaussian with mean zeroand such that

E|<(Y (N)(i, j))|2 = E|=(Y (N)(i, j))|2 = (2N)−1 = (1/2)E|Y (N)(i, j)|2.

Theorem 12.27 (Gaussian random matrix model). Let N ≥ 1 be any matrix size. Let (xs)s≥1

(resp. (ys)s≥1) be a free semi-circular (resp. circular) sequence in M . Then for almost all ω ∈ Ω,

the sequence (X(N)s (ω))s≥1 (resp. (Y

(N)s (ω))s≥1 ) is a matrix model for (xs)s≥1 (resp. (ys)s≥1).

We refer the reader to [255, 7] for the proofs.

12.6 Characterization of nuclear von Neumann algebras

It is easy to see from the definition that for any family (Hi)i∈I of Hilbert spaces the von Neumannalgebra

M = (⊕∑

i∈IB(Hi))∞

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is injective. In the particular case I = N with Hn n-dimensional, the von Neumann algebra

B = (⊕∑

n≥1Mn)∞

is injective (and a fortiori WEP). However, it is not nuclear. This was proved by S. Wassermann:

Lemma 12.28 ([256]). The von Neumann algebra B is not nuclear.

Proof. Let G = Fn with 1 < n ≤ ∞. By Theorem 12.21, MG embeds in an ultraproduct MUof matrix algebras and by Proposition 11.21 there is a contractive c.p. projection (conditionalexpectation) P : MU → MG. A priori MU is a quotient of (⊕i∈I

∑B(Hi))∞ for some family of

finite dimensional Hilbert spaces. Since G is countable and residually finite, from the proof ofTheorem 12.21, we may as well assume that MU is a quotient of B. Now the nuclearity of B wouldimply by Theorem 8.15 the injectivity of MU , and hence also of MG (recall Corollary 1.47). Sincewe know by Theorem 3.30 (and the remarks after it) that MG is not injective (because G is notamenable) we conclude that B is not nuclear.

More precisely, we have:

Theorem 12.29 ([256]). Let M be a von Neumann algebra. The following are equivalent:

(i) M is nuclear.

(ii) M does not contain a copy of B as a von Neumann subalgebra.

(iii) There is a finite set I, integers n(i) ≥ 1 (i ∈ I) and commutative von Neumann algebras Cisuch that M is isomorphic to (⊕

∑i∈I Ci ⊗min Mn(i))∞.

Sketch of proof. Since B is injective, (i) ⇒ (ii) follows from the preceding Lemma (recall Remark9.3). Assume (iii). By Remarks 4.9 and 4.10, M is nuclear, so (iii) ⇒ (i).The remaining implication (ii) ⇒ (iii) lies deeper. Its fully detailed proof requires classical resultsfrom the structural theory of von Neumann algebras that would take us too far off to cover in thesenotes. We merely outline the argument for the convenience of the reader. We have a decompositionM ' MI ⊕MII ⊕MIII into 3 (possibly vanishing) parts called respectively of type I, II, III. Bygeneral results, it can be shown that if either MII 6= 0 (resp. MIII 6= 0) then B embedsin MII (resp. MIII). Thus we may assume M = MI , i.e. that M is of type I. Moreover,assuming (ii) we know that B(`2) does not embed in M = MI . By the classification of type I vonNeumann algebras, M is isomorphic to a direct sum (⊕n≥1

∑Cn ⊗min Mn)∞ where each Cn is a

commutative von Neumann algebra (i.e. Cn = L∞(Ωn, µn) for some measure space (Ωn, µn) andCn ⊗min Mn = L∞(Ωn, µn;Mn)). Let I = n ≥ 1 | Cn 6= 0. The assumption (ii) implies that I isa finite set. Thus we obtain (iii).

12.7 Notes and Remarks

As already mentioned in the text, the ideas for Theorem 12.1 go back to Kadison [142] and Størmer[234] (see [123]). Proposition 12.2 is an elementary fact formulated for the convenience of ourpresentation. Kirchberg’s criterion in Theorem 12.3 is much more substantial and Theorem 12.8is even more so. The latter refined criterion is essentially due to Ozawa [191, Th. 29]. It will becrucially used to prove Theorem 14.7. Concerning injective factors and in particular the uniquenessof the injective factor R, the fundamental reference is Connes’s paper [61]. The notion of sofic grouphas recently become quite popular. The names of Gromov and Weiss are associated with it. Initialwork by Elek and Szabo [81] has been influential. Theorem 12.21 and its corollary are due to S.

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Wassermann [256] and independently to Connes [61]. The results in §12.5 are due to Voiculescu(see [255]), except for the random permutation model due to Nica [179]. The results of §12.6 aredue to S. Wassermann [256].

13 Kirchberg’s conjecture

We now turn to the second problem of our series, which actually was explicitly formulated as aconjecture by Kirchberg.

13.1 LLP ⇒ WEP ?

At the end of his landmark paper [155] Eberhard Kirchberg formulated several conjectures aboutthe properties WEP and LLP. Essentially, he asked whether they are equivalent. However, theimplication WEP⇒ LLP was soon disproved by Marius Junge and the author in [141], where itwas proved that the prototypical WEP C∗-algebra, namely B = B(`2) fails the LLP. We return tothis in more detail in §18.1.

This left open the remaining conjecture, namely the implication LLP⇒WEP. Given that C isthe prototypical example of a C∗-algebra with LLP, one can reformulate the conjecture like this:Kirchberg’s Conjecture: The C∗-algebra C has the Weak Expectation Property (WEP).

Because of its equivalence with Connes’s problem, this is now widely considered as one of themost important open problems in Operator Algebra theory (if not the most important one).

At this point, it is worthwhile to list several equivalent forms of Kirchberg’s conjecture.

Proposition 13.1. The following conjectures are all equivalent:

(i) C is WEP.

(i)’ The pair (C ,C ) is a nuclear pair.

(ii) C ⊗max C is residually finite dimensional.

(iii) C ⊗max C has a faithful tracial state.

(iv) For any free group F, C∗(F) has the WEP.

(v) Any unital C∗-algebra is isomorphic to a quotient of a WEP C∗-algebra. (We call theseQWEP.)

(vi) Any von Neumann algebra is QWEP.

(vii) LLP⇒WEP.

Proof. (i)⇔(i)’ is tautological in view of our definition of WEP. We first consider (i)-(iii).(i)⇒ (ii) follows from Theorem 9.18 and Remark 9.19.(ii)⇒ (iii) follows from Remark 9.17 since, of course, each Mn has a faithful tracial state.(iii)⇒ (i)’ follows from Remark 11.34 applied to the GNS representation of the faithful tracial state,which is isometric on C ⊗max C . Whence (i)-(iii) are equivalent.(i)’⇒ (iv) follows from (9.4) (applied with B = C ). (iv)⇒(v) is clear since, by Proposition 3.39,any unital C∗-algebra is a quotient of C∗(F) for some F, and (v)⇒ (vi) is trivial.Let us show (vi)⇒ (vii). Assume (vi). Let C be a C∗-algebra with the LLP. By (vi), C∗∗ isQWEP. By (i) ⇒ (ii) in Theorem 9.67, the linear map iC : C → C∗∗ is WEP. Since the latter ismax-injective, it follows that IdC and hence C itself is WEP, so that (vii) holds.Lastly we have (vii)⇒(i) since C has the LLP (by Theorem 9.6).

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Remark 13.2 (One-for-all...). By Corollary 9.69 if C is QWEP, then it has the WEP and byProposition 13.1 every C∗-algebra is QWEP. Motivated by this, we will say that a unital C∗-algebra A is “one-for-all” if the property:

(viii) A is QWEPimplies that all C∗-algebras are QWEP (in other words implies the Kirchberg conjecture).

If A has a one-for-all quotient, then A itself is one-for-all. Moreover, by Corollary 9.70 if theidentity of a one-for-all C∗-algebra B factors with unital c.p. maps through A, then A is alsoone-for-all (more generally see Remark 13.2).

The obvious example of one-for-all is C∗(F) when F is any non-Abelian free group. Moregenerally any C∗-algebra that admits C∗(F2) as a quotient is also an example, but it turns outthere are more noteworthy examples. For instance the universal unital C∗-algebra of a contractiondescribed in Remark 2.27 is one-for-all. Clearly C∗u〈C〉 is generated (as a unital C∗-algebra) by thesingle polynomial P (X) = X, and any singly generated unital C∗-algebra is a quotient of C∗u〈C〉.It follows that any C∗-algebra that is generated by a pair of (a priori non-commuting) hermitiancontractions x1, x2 is a quotient of C∗u〈C〉 since it is generated by (x1 + ix2)/2. In particular, if Cjis generated by xj then the full (unital) free product C1 ∗C2 is a quotient of C∗u〈C〉. It is a simpleexercise to check that the C∗-algebra C∗(G) of a finite Abelian group G can be generated by asingle hermitian element. Thus (see (4.13)) if G1, G2 are finite Abelian groups then C∗(G1 ∗G2) isa quotient of C∗u〈C〉. This shows e.g. that C∗(Z2 ∗ Z3) is a quotient of C∗u〈C〉, but it is well known(see Remark 3.40) that F∞ is a subgroup of Z2 ∗Z3, and hence by Proposition 3.5 there is a unitalc.p. factorization of the identity of C through C∗(Z2 ∗ Z3). Therefore, we can now conclude: ifC∗u〈C〉 is QWEP then so is C∗(Z2 ∗ Z3), and by Corollary 9.70 so is C . This shows that C∗u〈C〉 isone-for-all. We remind the reader that Z2 ∗ Z2 being amenable, we do need Z3 here, see Remark3.40.

The same argument shows that the unital free product A1 ∗A2 of two unital C∗-algebras is one-for-all if (say) A1 (resp. A2) admits C∗(Zn) (resp. C∗(Zm)) as a quotient with n ≥ 2 and m ≥ 3.More generally, by Boca’s Theorem 2.24, the same holds if we have a unital c.p. factorization ofthe identity of C∗(Zn) (resp. C∗(Zm)) through A1 (resp. A2). For instance, C∗(Zn) admits sucha factorization through Mn (because C∗(Zn) ' `n∞ and the latter can be identified with diagonalmatrices in Mn, see (3.17)). This shows that Mn ∗ C∗(Z) is one-for-all if n ≥ 2. It is easy to showthat Mn ∗ C∗(Z) = Mn(Bn) for some unital C∗-algebra Bn called the Brown algebra (see Remark6.31). Clearly B is QWEP if and only if Mn(B) is QWEP. Therefore Bn is one-for-all. It canbe shown that Bn is the unital C∗-algebra generated by the entries of a universal unitary blockmatrix in Mn(B(H)). In passing we observe that Mn ∗C∗(Z) and hence Bn has the LLP since, byTheorem 9.44, the LLP is stable by free products.

The free product of Cuntz algebras On ∗Om is one-for-all if n,m ≥ 1.Another interesting example is B ⊗max B. Indeed, let G = F∞. By Theorem 4.11 the identity

of C factorizes with unital c.p. maps through MG⊗maxMG. More precisely, the latter factorizationis of the form

(13.1) C → C ⊗max C →MG ⊗max MG → C .

But since by Corollary 12.22 the natural map C → MG factorizes with unital c.p. maps throughsome B(H), it follows that in (13.1) the mapping C ⊗max C →MG⊗maxMG factorizes with unitalc.p. maps through B(H)⊗max B(H). It follows that the identity of C factorizes similarly throughB(H)⊗max B(H). In fact (see the proof of Corollary 12.22) we can take here H = `2 so we obtainthat C factorizes with unital c.p. maps through B ⊗max B. Thus by Corollary 9.70 if B ⊗max Bis QWEP so is C . In other words B ⊗max B is one-for-all.

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13.2 Connection with Grothendieck’s theorem

Curiously, there seems to be a connection between Grothendieck’s theorem (in short GT) (orGrothendieck’s inequality) and the Kirchberg conjecture. Indeed, the latter problem can be phrasedas the identity of two norms on `1⊗ `1, while GT implies that these two norms are equivalent, andtheir ratio is bounded by the Grothendieck constant KG.

Here is the simplest formulation of the classical GT.

Theorem 13.3 (GT). Let [aij ] be an n × n scalar matrix (n ≥ 1). Assume that for any n-tuplesof scalars (αi), (βj) we have

(13.2)∣∣∣∑ aijαiβj

∣∣∣ ≤ supi |αi| supj |βj |.

Then for any Hilbert space H and any n-tuples (xi), (yj) in H we have

(13.3)∣∣∣∑ aij〈xi, yj〉

∣∣∣ ≤ K sup ‖xi‖ sup ‖yj‖,

where K is a numerical constant. The best K (valid for all H and all n) is denoted by KG.

In this statement the scalars can be either real or complex, but that affects the constant KG,so we must distinguish its value in the real case KR

G and in the complex case KCG. To this day, its

exact value is still unknown although it is known that 1 < KCG < KR

G ≤ 1.782, see [210] for moreinformation. In our context (in connection with spectral theory) it is natural to restrict to the caseof complex scalars.

Remark 13.4. If we restrict to positive semidefinite matrices [aij ], then the best constant in (13.3)is known to be exactly 4/π in the complex case (and π/2 in the real case). This is called the “little”GT, see [210, §5] for details.

Let (ei) denote the canonical basis of `1. Let t =∑aijei⊗ ej ∈ `1⊗ `1 be a tensor in the linear

span of ei ⊗ ej. We denote

(13.4) ‖t‖H′ = sup∣∣∣∑ aij〈xi, yj〉

∣∣∣where the supremum is over all Hilbert spaces H and all xi, yj in the unit ball of H.We identify `n1 ⊂ `1 with the linear span of ei | 1 ≤ i ≤ n in `1, so we may consider that t 7→ ‖t‖H′is a norm on `n1 ⊗ `n1 for each integer n ≥ 1.The classical “injective” Banach space tensor norm for an element t =

∑aijei⊗ej ∈ `n1⊗`n1 ⊂ `1⊗`1

is given by the following formula.

(13.5) ‖t‖∨ = sup

∣∣∣∑ aijαiβj

∣∣∣ ∣∣∣∣ αi, βj ∈ C, supi |αi| ≤ 1, supj |βj | ≤ 1

.

With this notation, GT in the form (13.3) can be restated as follows: there is a constant K suchthat for any n and any t in `n1 ⊗ `n1 we have

(13.6) ‖t‖H′ ≤ K‖t‖∨.

We will also need another norm introduced by Grothendieck as follows. We abusively denote againby ei | 1 ≤ i ≤ n the canonical basis of `n∞. Note that we have isometrically both `n∞ = (`n1 )∗ and`n1 = (`n∞)∗. Let t′ =

∑a′ijei ⊗ ej ∈ `n∞ ⊗ `n∞ (a′ij ∈ C). We define

(13.7) ‖t′‖H = infsupi ‖xi‖ supj ‖yj‖

204

where the infimum runs over all Hilbert spaces H and all xi, yj in H such that a′ij = 〈xi, yj〉 for alli, j = 1, . . . , n. It is an easy exercise to check directly that this is a norm but actually this followsfrom Proposition 1.57.We denote by `n1 ⊗H′ `n1 (resp. `n∞ ⊗H `n∞) the space `n1 ⊗ `n1 (resp. `n∞ ⊗ `n∞) equipped with theH ′-norm (resp. H-norm). By definition of ‖t‖H′ , we have clearly

(13.8) ‖t‖H′ = sup∣∣∣∑ aija

′ij

∣∣∣ | ‖t′‖H ≤ 1

= sup|〈t, t′〉| | ‖t′‖H ≤ 1,

so we have isometrically

`n1 ⊗H′ `n1 = (`n∞ ⊗H `n∞)∗ and `n∞ ⊗H `n∞ = (`n1 ⊗H′ `n1 )∗.

In operator theory terms, the norm ‖a‖H′ can be rewritten as follows. For any n and anyaij ∈ C (1 ≤ i, j ≤ n) we have

(13.9) ‖∑

aijei ⊗ ej‖H′ = sup

∥∥∥∑ aijuivj

∥∥∥B(H)

where the sup runs over all Hilbert spaces H and all n-tuples (ui), (vj) in the unit ball of B(H).Indeed, it is an easy exercise (left to the reader) to check that (13.4) and (13.9) are equal.Equivalently (by the Russo-Dye Theorem 26.18)

(13.10) ‖∑

aijei ⊗ ej‖H′ = sup

∥∥∥∑ aijUiVj

∥∥∥B(H)

where the sup runs over all Hilbert spaces H and all n-tuples (Ui), (Vj) of unitaries in B(H). Inboth cases it suffices to consider H = `2.

Remark 13.5. Consider the free product F∞∗F∞ and its associated (full) C∗-algebra C∗(F∞∗F∞) =

C ∗ C . Let (U(1)j ) (resp. (U

(2)j )) denote the free unitary generators of the first (second) of the two

“factors” of the free product. Then it is easy to check that (13.10) is equivalent to:

(13.11)∥∥∥∑ aijei ⊗ ej

∥∥∥H′

=∥∥∥∑ aijU

(1)i U

(2)j

∥∥∥C ∗C

.

To explain the connection with GT, we first give several equivalent reformulations of the Kirch-berg conjecture. As before, we set C = C∗(F∞). Let (Uj)j≥1 denote the unitaries in C thatcorrespond to the free generators of F∞. For convenience of notation we set U0 = 1 (i.e. the unitin C ). We recall that the closed linear span E ⊂ C of Uj | j ≥ 0 is isometric to `1 (see Remark3.14) and that E generates C as a C∗-algebra.

Fix n ≥ 1. Consider a family of matrices aij | i, j ≥ 0 with aij ∈MN for all i, j ≥ 0 such that|(i, j) | aij 6= 0| <∞. We denote

a =∑

aij ⊗ Ui ⊗ Uj ∈MN ⊗ E ⊗ E.

We denote, again for C = C∗(F∞):

‖a‖min =∥∥∥∑ aij ⊗ Ui ⊗ Uj

∥∥∥MN (C⊗minC )

and ‖a‖max =∥∥∥∑ aij ⊗ Ui ⊗ Uj

∥∥∥MN (C⊗maxC )

.

Then, on one hand, going back to the definitions, it is easy to check that

(13.12) ‖a‖max = sup

∥∥∥∑ aij ⊗ uivj∥∥∥MN (B(H))

205

where the supremum runs over all H and all possible unitaries ui, vj on the same Hilbert space Hsuch that uivj = vjui for all i, j. On the other hand, using the known fact that C embeds into adirect sum of matrix algebras (see Theorem 9.18), one can check that

(13.13) ‖a‖min = sup

∥∥∥∑ aij ⊗ uivj∥∥∥MN (B(H))

| dim(H) <∞

where the sup is as in (13.12) except that we restrict it to all finite dimensional Hilbert spaces H.Indeed, since dim(H) <∞⇒ B(H)⊗minB(H) = B(H)⊗maxB(H), the sup in (13.13) is the sameas the supremum over all finite dimensional H’s and all unitaries ui, vj on H of

‖∑

aij ⊗ (ui ⊗ vj)‖MN (B(H)⊗minB(H))

and by Theorem 9.18, the latter is the same as ‖a‖min.We may ignore the restriction u0 = v0 = 1 because we can always replace (ui, vi) by (u−1

0 ui, viv−10 )

without changing either (13.12) or (13.13).The following was observed in [204].

Proposition 13.6. Let C = C∗(F∞). The following assertions are equivalent:

(i) C ⊗min C = C ⊗max C (i.e. Kirchberg’s conjecture is correct).

(ii) For any N ≥ 1 and any aij | i, j ≥ 0 ⊂ MN as previously the norms (13.12) and (13.13)coincide i.e. ‖a‖min = ‖a‖max.

(iii) The identity ‖a‖min = ‖a‖max holds for all N ≥ 1 but merely for all families aij in MN

supported in the union of 0 × 0, 1, 2 and 0, 1, 2 × 0.

Note that (iii) reduces the Kirchberg conjecture to a statement about an operator space ofdimension 5 ! But it requires to control the whole operator space structure of this 5-dimensionalspace, so the size N of the 5 matrix coefficients is unbounded in (iii).

It was conceivable that the equality ‖a‖max = ‖a‖min might hold when aij ∈ C i.e. in thecase N = 1 in the previous (ii). But recently in [191, Th. 29] Ozawa proved that this is actuallyequivalent to the Kirchberg conjecture. More precisely, he showed that the conditions in Proposition13.6 are equivalent to:

(iv) For any aij | i, j ≥ 0 ⊂ C as previously the norms (13.12) and (13.13) coincide, i.e. wehave

‖a‖min = ‖a‖max, for all a in the C-linear span of Ui ⊗ Uj.Note that Ozawa’s argument requires infinitely many generators in (iv).This result will be restated and proved later on as Theorem 14.7. It should be compared with thenext two statements.

Theorem 13.7 (Tsirelson [252]). We consider the case N = 1 and aij ∈ R for all i, j. Then

‖a‖max = ‖a‖min = ‖a‖H′ ,

where we have denoted by ‖a‖H′ the norm appearing either in (13.9), (13.10) or (13.11). Moreover,these norms are all equal to

(13.14) sup ‖∑

aijuivj‖

where the sup runs over all d ≥ 1 and all self-adjoint unitary d × d matrices ui, vj such thatuivj = vjui for all i, j.

206

Proof. Recall (see (13.4)) that ‖a‖H′ ≤ 1 if and only if for any unit vectors xi, yj in a Hilbert spacewe have ∣∣∣∑ aij〈xi, yj〉

∣∣∣ ≤ 1.

Note that, since aij ∈ R, whether we work with real or complex Hilbert spaces does not affect thiscondition. The resulting H ′-norm is the same. We have trivially

‖a‖min ≤ ‖a‖max ≤ ‖a‖H′ ,

so it suffices to check ‖a‖H′ ≤ ‖a‖min. Consider unit vectors xi, yj in a real Hilbert space H. Wemay assume that aij is supported in [1, . . . , n]× [1, . . . , n] and that dim(H) = n. From classicalfacts on “spin systems” (Pauli matrices, Clifford algebras and so on), we claim that there are self-adjoint unitary matrices ui, vj (of size 2n) such that uivj = vjui for all i, j and a (vector) state

f such that f(uivj) = 〈xi, yj〉 ∈ R. Indeed, let H = Rn, H = Cn and let H∧k denote the k-foldantisymmetric tensor product, which admits ei1 ∧ · · · ∧ eik | i1 < · · · < ik as orthonormal basis.

Let F = C⊕H⊕H∧2⊕· · · denote the (2n-dimensional) antisymmetric Fock space associated to Hwith vacuum vector Ω (Ω ∈ F is the unit in C ⊂ F). For any x, y ∈ H = Rn, let c(x), c(y) ∈ B(F)(resp. d(x), d(y) ∈ B(F)) be the left (resp. right) creation operators defined by c(x)Ω = x (resp.d(x)Ω = x) and c(x)t = x ∧ t (resp. d(x)t = t ∧ x) for x ∈ H∧n with n > 0. Let vy = d(y) + d(y)∗

and ux = c(x)+c(x)∗. Then ux, vy are commuting selfadjoint unitaries and setting f(T ) = 〈Ω, TΩ〉,we find f(uxvy) = 〈x, y〉. So applying this to xi, yj yields the claim. Thus we obtain by (13.13)∣∣∣∑ aij〈xi, yj〉

∣∣∣ =∣∣∣f (∑ aijuxivyj

)∣∣∣ ≤ ‖a‖min,

and hence ‖a‖H′ ≤ ‖a‖min. This proves ‖a‖H′ = ‖a‖min but also ‖a‖H′ ≤ (13.14). Since, by (13.13),we have (13.14) ≤ ‖a‖min, (13.14) must be also equal to the number ‖a‖H′ = ‖a‖min = ‖a‖max.

Remark 13.8. When aij ∈ C, the preceding argument shows that

sup∣∣∣∑ aij<〈xi, yj〉

∣∣∣ | xi, yj ∈ B`2 ≤ ‖∑ aijUi ⊗ Uj‖min.

Indeed, the supremum remains the same when reduced to unit vectors xj , yi ∈ `n2 (C), and then<〈xi, yj〉 can be thought of as the scalar product in the “real” Hilbert space `2n2 (R) ' `n2 (C).

This inequality fails (for aij ∈ C) if one replaces <〈xi, yj〉 by 〈xi, yj〉 (see Proposition 13.10).However, we have

Proposition 13.9. Assume N = 1 and aij ∈ C for all i, j ≥ 0 then ‖a‖max ≤ ‖a‖H′ and ‖a‖∨ ≤‖a‖min. Therefore ‖a‖max ≤ KC

G‖a‖min.Moreover, for any positive semidefinite matrix [aij ] we have ‖a‖max ≤ (4/π)‖a‖min.

Proof. By (13.13) we have

sup

∣∣∣∑ aijsitj

∣∣∣ ∣∣∣∣ si, tj ∈ C |si| = |tj | = 1

= ‖a‖∨ ≤ ‖a‖min.

For any unit vectors x, y in H, we have∣∣∣∑ aij〈y, uivjx〉∣∣∣ =

∣∣∣∑ aij〈u∗i y, vjx〉∣∣∣ ≤ ‖a‖H′ ,

so that‖∑

aijuivj‖ ≤ ‖a‖H′

207

and actually this holds without any extra assumption (such as mutual commutation) on the ui’sand the vj’s. A fortiori, we have ‖a‖max ≤ ‖a‖H′ , and by Theorem 13.3 we conclude that‖a‖max ≤ KC

G‖a‖min. The last assertion follows from Remark 13.4.

Remark. The equalities ‖a‖H′ = ‖a‖min and ‖a‖H′ = ‖a‖max in Theorem 13.7 do not extend tothe case of matrices with complex entries. This was proved by Eric Ricard. See the subsequentProposition 13.10. In sharp contrast, if [aij ] is a 2× 2 matrix, they do extend because, by [69, 250]for 2 × 2 matrices in the complex case (13.3) happens to be valid with K = 1 (while in the realcase, for 2× 2 matrices, the best constant is

√2).

See [223, 224, 75, 138, 95, 57] for related contributions.Recall that we view `1 ⊂ C∗(F∞) completely isometrically. In this embedding the canonical

basis of `1 is identified with the canonical free unitaries Uj | j ≥ 0 generating C∗(F∞), withthe convention U0 = 1 (see Remark 3.14). Similarly, for any n = 1, 2, · · · , we may restrict toUj | 0 ≤ j ≤ n− 1 and identify `n1 ⊂ C∗(F∞) as the span of Uj | 0 ≤ j ≤ n− 1. Equivalently,we could replace F∞ by Fn−1 and view `n1 ⊂ C∗(Fn−1).

The following proposition and its proof were kindly communicated by Eric Ricard.

Proposition 13.10 (Ricard). For any n ≥ 4, the H ′-norm on `n1 ⊗ `n1 does not coincide with thenorm induced by the max-norm on C∗(F∞)⊗ C∗(F∞) (or equivalently on C∗(Fn−1)⊗ C∗(Fn−1))

Proof. The proof is inspired by some of the ideas in [115]. Let us consider a family x0, x1, ... in theunit sphere of C2 and let ajk = 〈xj , xk〉. We identify `n1 with the span of the canonical free unitaries1 = U0, U1, · · · , Un−1 generating C∗(Fn−1). Then consider the element a =

∑n−1j,k=0 ajkej⊗ ek. Note

that a is clearly in the unit ball of `n∞ ⊗H `n∞. For simplicity let A = C∗(Fn−1). Assume that themax-norm coincides with the H ′-norm on `n1 ⊗ `n1 . Then any such a defines a functional of norm atmost 1 on `n1⊗`n1 ⊂ A⊗maxA (with induced norm). Therefore (Hahn-Banach) there is f in the unitball of (A⊗max A)∗ such that f(Uj ⊗Uk) = ajk, and since f(1) = f(U0 ⊗U0) = a00 = 〈x0, x0〉 = 1,f is a state (see Remark 1.33). By the GNS representation of states (26.16) there is a Hilbert spaceH, a unit vector h ∈ H and commuting unitaries uj , v

∗k on H (associated to a unital representation

of A⊗max A) such that u0 = v0 = 1 and

(13.15) 〈xj , xk〉 = f(Uj ⊗ Uk) = 〈h, v∗kujh〉 = 〈vkh, ujh〉 (0 ≤ j, k ≤ n− 1).

Moreover the vector h can be chosen cyclic for the ∗-algebra generated by uj , v∗k (or equivalently

of course uj , vk). Since ajj = 〈xj , xj〉 = 1, we have ujh = vjh for all j, and hence h must also becyclic for each of the ∗-algebras Au and Av generated by (uj) and (vj) respectively (indeed notethat Auh = Avh implies AuAvh = Auh = Avh). But by a classical elementary fact (see Lemma26.62) this implies that h is separating for each of the commutants A′u and A′v, and since we haveAu ⊂ A′v and Av ⊂ A′u, it follows that h is separating for each of the ∗-algebras Au and Av.We claim that there is an isomorphism T : span[uj ] → span[xj ] such that T (uj) = xj and inparticular the linear span of (uj) is at most 2-dimensional. Indeed, first observe that by (13.15)(since ujh = vjh) any linear combination

∑bjxj (bj ∈ C) satisfies ‖

∑bjxj‖2 = ‖

∑bjujh‖2 ≤

‖∑bjuj‖2, and hence T : span[uj ] → span[xj ] such that T (uj) = xj is well defined. But now

any linear relation∑bjxj = 0 (bj ∈ C) implies ‖

∑bjujh‖2 = 0 and (since h is separating on Au)∑

bjuj = 0. This shows that T : span[uj ] → span[xj ] is injective, and hence an isomorphism sothe linear span of (uj) is at most 2-dimensional.Suppose that x0, x1 are linearly independent. Then so is 1, u1 (recall u0 = 1), and the other uj ’sare linear combinations of 1, u1 (so that Au is the unital commutative C∗-algebra generated by u1).So let us choose simply for x0, x1 the canonical basis of C2, and recall T−1xj = uj . Choosing (here

208

i =√−1) x2 = (1, i)2−1/2 = (x0 + ix1)2−1/2 and x3 = (1, 1)2−1/2 = (x0 +x1)2−1/2, by the linearity

of T−1 we must have T−1(x2) = (1 + iu1)2−1/2. The latter must be unitary, equal to u2, as well asT−1(x3) = (1 +u1)2−1/2 equal to u3. But this is impossible because (1 + iu1)2−1/2 unitary requiresthe spectrum of u1 included in ±1, while (1 + u1)2−1/2 unitary requires that it is included in±i. Thus we obtain a contradiction for any n ≥ 4.

This should be compared with Dykema and Juschenko’s results in [75].

13.3 Notes and Remarks

§13.1 is due to Kirchberg. Remark 13.2 on “one for all” C∗-algebras combines various observations,some of them already in [208]. The remark that the Brown algebra is such an example appears in[128]. For more remarks on a similar theme, see [91], [128] and [63].For §13.2 most references are credited in the text. The papers [153, 152, 91] contain severalequivalent formulations of the WEP as a Riesz interpolation property and in terms of matrixcompletion problems. The formulation of Grothendieck’s inequality in Theorem 13.3 was putforward by Lindenstrauss and Pe lczynski in 1968, see [210] for more (and more precise) references.

14 Equivalence of the two main questions

We will now prove the equivalence of both problems.

14.1 From Connes’s question to Kirchberg’s conjecture

Theorem 14.1. If the answer to Connes’s question is positive, then any von Neumann algebra isQWEP and consequently Kirchberg’s conjecture follows.

The proof of Theorem 14.1 is based on the three steps described in the following lemmas.

Lemma 14.2. If the answer to Connes’s question is positive then any finite von Neumann algebraM is QWEP .

Proof. Assume M ⊂MU in a trace preserving way. Since we are dealing with finite traces, there isa c.p. conditional expectation P : MU → M with ‖P‖ = 1 (see Proposition 11.21). Clearly MU isQWEP by definition. Thus the result follows from Corollary 9.61.

Lemma 14.3. If all finite von Neumann algebras are QWEP , then all semifinite von Neumannalgebras M are QWEP .

Proof. If M is semifinite we can write

M = ∪Mγweak∗

where the von Neumann (unital) subalgebras Mγ are all finite and form a net directed by inclusion.Indeed, let Pγ be an increasing net of projections with finite trace tending weak* to the identityon M (see Remark 11.1). Then the unital subalgebras Mγ = PγMPγ + C(1 − Pγ) are finite andtheir increasing union is weak*-dense in M . Since all the Mγ ’s are QWEP , we conclude that M isQWEP by Theorem 9.74.

Lemma 14.4. If all semifinite von Neumann algebras are QWEP , then all von Neumann algebrasare QWEP .

209

Proof. By Takesaki’s Theorem 11.3 any von Neumann algebra M can be embedded in a semifinitealgebra N , so we have M ⊂ N , in such a way that there is a c.p. projection P : N → M with‖P‖ = 1. Using this, the lemma follows immediately from Corollary 9.61.

Proof of Theorem 14.1. By the three preceding Lemmas, if the answer to Connes’s question ispositive then any von Neumann algebra is QWEP . By Proposition 13.1 this implies Kirchberg’sconjecture.

Remark 14.5. Let (M(i), τi) be a family of tracial probability spaces, with ultraproduct MU . If allthe M(i)’s are QWEP, then MU is also QWEP by Corollary 9.62. Moreover, any von Neumannsubalgebra M ⊂ MU is also QWEP. Indeed, since the conditional expectation is a c.p. projectionP : MU →M , this follows from Corollary 9.61.

Remark 14.6 (On the Effros-Marechal topology). In a series of remarkable papers, Haagerup withWinsløw and Ando ([121, 122, 9]) make a deep study of the Effros-Marechal topology on theset vN(H) of all von Neumann algebras M ⊂ B(H) (on a given Hilbert space H). The lattertopology is defined as the weakest topology for which the map M 7→ ‖ϕ|M‖ is continuous for everyϕ ∈ B(H)∗. In particular, they show that a von Neumann algebra is QWEP if and only if it is in theclosure (for that topology) of the set of injective factors. Thus their work shows that the Kirchbergconjecture is equivalent to the density of the latter set in vN(H), say for H = `2. Actually, it isalso equivalent to the density of the set of type I factors and, as expected, to the density of the setof finite dimensional factors (i.e. matricial factors).

14.2 From Kirchberg’s conjecture to Connes’s question

LetE = spanUj | j ≥ 0 ⊂ C

be the linear span of the generators Uj | j ≥ 1 and the unit (recall the convention U0 = 1) inC = C∗(F∞).

We will work with E ⊗ E ⊂ C ⊗ C .In E ⊗E we distinguish the cone (E ⊗E)+ formed by all the “positive definite” tensors, more

precisely

(E ⊗ E)+ = ∑r

1xk ⊗ xk ∈ E ⊗ E | xk ∈ E, r ≥ 1.

Alternatively:

(14.1) (E ⊗ E)+ = ∑

aijUi ⊗ Uj | n ≥ 1, a ∈ (Mn)+.

We call the elements of (Mn)+ “positive definite” matrices (they are often called positive semi-definite). To check (14.1) observe that (by classical linear algebra) any such matrix can be writtenas a finite sum aij =

∑rk=1 xk(i)xk(j) (xk(j) ∈ C), and conversely any matrix of the latter form is

positive definite. Then we have∑aijUi ⊗ Uj =

∑r1 xk ⊗ xk where xk =

∑j xk(j)Uj .

We define analogously

(14.2) (E ⊗ E)+ = ∑

aijUi ⊗ Uj | n ≥ 1, a ∈ (Mn)+.

We also define

(E ⊗ E)+min = t ∈ (E ⊗ E)+ | ‖t‖min ≤ 1 and (E ⊗ E)+

max = t ∈ (E ⊗ E)+ | ‖t‖max ≤ 1.

We have then

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Theorem 14.7 (Ozawa [191]). If the min and max norms coincide on E ⊗ E ⊂ C ⊗ C (orequivalently if they coincide on E ⊗E ⊂ C ⊗C ), then the Connes problem has a positive solution.

Actually with the same idea as Ozawa’s we will prove the following refinement:

Theorem 14.8. If (E ⊗ E)+min = (E ⊗ E)+

max, then the Connes problem has a positive solution.

Since a positive answer to Connes’s question implies Kirchberg’s conjecture (see §14.1), we have:

Corollary 14.9. If the min and max norms coincide on E⊗E ⊂ C⊗C (resp. on (E⊗E)+ ⊂ C⊗C )then they coincide on C ⊗ C (resp. on C ⊗ C ), meaning the Kirchberg conjecture holds.

Remark 14.10. Curiously there is currently no direct proof available for the preceding corollary.

Remark 14.11. For any C∗-algebra A we have A ' Aop. The mapping being x 7→ x∗ (see §2.3). Sothe assumption (E ⊗ E)+

min = (E ⊗ E)+max can be rewritten as

‖∑

x∗k ⊗ xk‖Aop⊗minA = ‖∑

x∗k ⊗ xk‖Aop⊗maxA.

Remark 14.12. Recall (see Remark 3.7) that when A = C∗(F∞) we have A ' A and we have aC-linear isomorphism Φ : A → A that takes UF∞(t) to UF∞(t). In particular, this isomorphismtakes Ui to Ui and C onto C . Therefore, the assumption (E⊗E)+

min = (E⊗E)+max is equivalent to

the following one:For any n and any a ∈ (Mn)+ we have

‖∑

aijUi ⊗ Uj‖C⊗minC = ‖∑

aijUi ⊗ Uj‖C⊗maxC .

The main point is as follows:

Theorem 14.13. Let (M, τ) be a tracial probability space on a separable Hilbert space. Let π :C →M be a ∗-homomorphism taking the free generators Uj to an L2(τ)-dense subset of U(M).Assume that for any finite set (xj) in span[Uj ] ⊂ C we have

(14.3)∑‖π(xj)‖22 ≤ ‖

∑xj ⊗ xj‖C⊗minC .

Then M embeds in a trace preserving way in a matricial MU for some U on N.

Proof. Let U0 = 1. We apply Theorem 11.38 with S = Uj | j ≥ 0 andW = π(S). This shows thatW satisfies the condition (iv)’ in Kirchberg’s criterion (see Theorem 12.8). By the L2(τ)-densityof W in U(M) it follows that there is a trace preserving embedding of M in an ultraproduct ofmatrix algebras, which completes the proof.

Remark 14.14 (A variant). If we assume that π(S) is an L2(τ)-dense subgroup of U(M) then wecan conclude using Corollary 11.46 instead of Kirchberg’s criterion (Theorem 12.3).

Corollary 14.15. A finite von Neumann algebra (on a separable Hilbert space) is QWEP if andonly if it embeds in a matricial MU for some U on N.

Proof. If M is QWEP, let π be as in Theorem 14.13 (or Remark 14.14). By Theorem 9.67,

‖IdC ⊗ π : C ⊗min C → C ⊗max M‖ ≤ 1,

and hence a fortiori‖π ⊗ π : C ⊗min C →M ⊗max M‖ ≤ 1.

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From this, since∑τ(π(xj)

∗π(xj)) =∑〈1, L(π(xj)

∗)R(π(xj))1〉, we deduce∑τ(π(xj)

∗π(xj)) ≤ ‖∑

π(xj)⊗ π(xj)‖M⊗maxM≤ ‖

∑xj ⊗ xj‖C⊗minC .

By Theorem 14.13 (or Remark 14.14) M embeds in some matricial MU . Conversely if M embeds insome matricial MU , the latter is obviously QWEP, and there is a unital c.p. projection (conditionalexpectation) from MU onto M . By Corollary 9.61 M inherits QWEP from MU .

We can now easily complete the proof of Theorems 14.8 and 14.7. All the difficult ingredientshave been already proved mainly in §11.6.

Proof of Theorems 14.8 and 14.7. It clearly suffices to prove Theorem 14.8. Let (M, τ) be a tracialprobability space on a separable Hilbert space. Then there is a countable subgroup W ⊂ U(M)that is L2(τ)-dense in U(M). We may assume that there is a unital bijection from S = Uj | j ≥ 0onto W. Let π : C →M be the ∗-homomorphism extending the latter bijection (this exists by thefreeness of S). Since left-hand and right-hand multiplications on L2(τ) have commuting ranges wehave for any xj ∈ C

(14.4)∑

τ(π(xj)∗π(xj)) ≤ ‖

∑π(xj)⊗ π(xj)‖M⊗maxM

≤ ‖∑

xj ⊗ xj‖C⊗maxC .

Thus, if the min and max norms coincide on (E⊗E)+, (14.3) holds, and by Theorem 14.13, M tracepreservingly embeds in some matricial MU , so that the Connes question has a positive answer.

Remark 14.16 (A slight refinement). If the min and max norms coincide on

T + T ∗ | T ∈ (E ⊗ E)+ = ∑

xj ⊗ xj +∑

x∗j ⊗ x∗j | xj ∈ E

then they coincide on C ⊗ C .Indeed, with the notation of the preceding proof, we have∑

τ(π(xj)∗π(xj)) + τ(π(xj)π(xj)

∗) ≤ ‖∑

π(xj)⊗ π(xj) +∑

π(x∗j )⊗ π(x∗j )‖M⊗maxM

≤ ‖∑

xj ⊗ xj +∑

x∗j ⊗ x∗j‖max = ‖

∑xj ⊗ xj +

∑x∗j ⊗ x

∗j‖min ≤ 2‖

∑xj ⊗ xj‖C⊗minC .

But since τ(π(xj)∗π(xj)) = τ(π(xj)π(xj)

∗) (trace property) we again obtain (14.3), and we concludeas in the preceding proof.

Remark 14.17 (A slight refinement (bis)). If the min and max norms coincide on

T + T ∗ | T ∈ (E ⊗ E)+

then they coincide on C ⊗ C .Indeed, by Remark 14.12 we have Φ((E ⊗ E)+) = (E ⊗ E)+.

14.3 Notes and Remarks

Most of the results are due to Kirchberg. One major exception is Ozawa’s Theorem 14.7 derivedfrom [191, Th. 29] which is a quite significant improvement over Proposition 13.6. The refinementfrom Theorem 14.7 to Theorem 14.8 or Theorem 14.13 is obtained by an easy adaptation of Ozawa’sideas. Corollary 14.15 is due to Kirchberg.

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15 Equivalence with finite representability conjecture

Definition 15.1. A Banach space X is finitely representable in another Banach space Y if forany ε > 0 and any finite dimensional subspace E ⊂ X there is a subspace E ⊂ Y that is (1 + ε)-isomorphic to E. In that case we write X f.r. Y .

We discuss equivalent forms of finite representability in §26.6.We denote by S1(H) the Banach space formed of all the trace class operators on H (see §26.10).The space S1(H) is isometric to the predual of B(H). Recall the latter space is also isometric tothe dual of K(H). When H = `2 we set S1 = S1(`2).

15.1 Finite representability conjecture

In the late 1970’s, the following conjecture began to circulate:Finite representability conjecture: A∗ f.r. S1 for any C∗-algebra A.Equivalently, M∗ f.r. S1 for any von Neumann algebra M .

This conjecture was popularized in the Banach space community by talks and papers notablyby Pe lczynski and Garling, but it probably can also be traced back to Haagerup.

Remark 15.2. Let S1(H) be the space of trace class operators on a Hilbert space H. Let x ∈ S1(H)with H infinite dimensional. Then there is clearly a subspace K ' `2 such that x = PKxPK .More generally, given a finite set x1, · · · , xn ∈ S1(H) there is a K such that this holds for anyx ∈ x1, · · · , xn. Therefore any finite dimensional subspace E ⊂ S1(H) embeds isometrically inS1 = S1(`2). A fortiori, S1(H) f.r. S1.

Theorem 15.3 (Kirchberg). The finite representability conjecture is equivalent to the Kirchbergconjecture. More precisely, a C∗-algebra A is QWEP if and only if A∗ f.r. S1.

Proof. We first show that if A is QWEP then A∗ f.r. S1. The converse is much more delicate.Assume that A is QWEP. By Theorem 9.72, A∗∗ is a 1-complemented subspace of B(H)∗∗, andhence A∗∗∗, and a fortiori A∗ ⊂ A∗∗∗, embeds isometrically in B(H)∗∗∗. By Proposition 26.15 (i.e.the local reflexivity principle) we know that B(H)∗∗∗ f.r. B(H)∗, and also (applying it again) thatB(H)∗ f.r. B(H)∗, while Remark 15.2 tells us B(H)∗ f.r. S1. Thus we obtain that A∗ f.r. S1.Assume A∗ f.r. S1. By Lemma 26.8 there is a set I and a (metric) surjection Q : `∞(I;B(`2))→ A∗∗

taking the closed unit ball of `∞(I;B(`2)) onto that of A∗∗. Let W = `∞(I;B(`2)) (which is WEPby, say, Proposition 9.34) and N = W ∗∗. Note that N is QWEP by Theorem 9.66. Replacing Qby ϕ = Q : W ∗∗ → A∗∗ (as defined in (26.32)) we find a normal (metric) surjection ϕ : N → A∗∗

taking the closed unit ball of N onto that of A∗∗. To conclude we call Jordan algebras to therescue via Theorem 5.7 applied to M = A∗∗, as follows. For any projection p in N , the C∗-algebraspNp and (pNp)op are QWEP by Corollary 9.70. If q is another such projection, with q ⊥ p, thenpNp⊕ (qNq)op is QWEP (recall Remark 9.16). By Theorem 5.7, the identity of M = A∗∗ factorscompletely positively through pNp ⊕ (qNq)op for some p, q. Therefore, by Corollary 9.70 A∗∗ isQWEP, and a fortiori by Theorem 9.66, A itself is QWEP.

Remark 15.4. An operator space X is finitely representable in the operator sense in another oneY if for any finite dimensional subspace E ⊂ X and any ε > 0 there is E ⊂ Y that is completely(1+ε)-isomorphic to E. We say that X is strictly locally reflexive if X∗∗ is finitely representable inthe operator sense in X. By [78] this holds whenever X∗ is completely isometric to a C∗-algebra (i.e.for any so-called non-commutative L1-space), in particular for X = B(H)∗ and X = S1. Combinedwith the first part of the preceding argument, this shows that A∗ is finitely representable in theoperator sense in S1 whenever A is QWEP.

213

In the same spirit as the finite representability conjecture, we note one more characterizationof QWEP von Neumann algebras involving only their Banach space structure:

Theorem 15.5 (Kirchberg). A von Neumann algebra N is QWEP if and only if it is isometric asa Banach space to a (Banach space sense) quotient of B(H) for some Hilbert space H.

Proof. Consider first the special case N = B(H)∗∗. In that case Proposition 26.12 tells us that Nis isometric to a quotient of some (⊕

∑i∈I Xi)∞ with Xi = B(H) for all i ∈ I. Let H = ⊕

∑i∈I Hi

with Hi = H for all i ∈ I. The projection onto the diagonal defines a metric surjection B(H) →(⊕∑

i∈I Xi)∞. Therefore (⊕∑

i∈I Xi)∞, and a fortiori N , is isometric to a quotient of B(H).If N is QWEP, by Theorem 9.72 its bidual N∗∗ is isometric to a quotient of some B(H)∗∗ andhence by what precedes to a quotient of some B(H). By Remark 26.52 the same is true for N .Conversely, assume N is isometric to a quotient of B(H). Then N∗ embeds isometrically in B(H)∗.A fortiori N∗ f.r. B(H)∗. By the local reflexivity principle (Proposition 26.15) and Remark 15.2 weknow that B(H)∗ f.r. B(H)∗ f.r. S1. A fortiori N∗ f.r. S1, and N is QWEP by Theorem 15.3.

15.2 Notes and Remarks

The results here come from Kirchberg’s [155], but Ozawa’s expository paper [189] greatly clarifiedthe picture.

16 Equivalence with Tsirelson’s problem

It is difficult to introduce Tsirelson’s problem without mentioning the unusual saga that led to it.In a remarkable paper [252] in 1980, Tsirelson observed that the famous Bell inequality, widelycelebrated in quantum mechanics, could be viewed as a particular instance of Grothendieck’s in-equality equally celebrated by Banach space theorists. Both inequalities involve matrices of theform [〈xi, yj〉] where xi, yj are in the unit ball of a Hilbert space H. Tsirelson was particularlyinterested in the case when H = L2(M, τ) on a tracial probability space (M, τ). In his discus-sion he asserted without proof that for his specific purpose (to be described) the case of a general(M, τ) could be reduced to the matricial one. Years later, in connection with the development ofQuantum Information Theory it was pointed out to him that this reduction was not clear and hehimself advertised the problem widely. The goal of this section is to show that the latter Tsirelsonproblem is actually equivalent to the Connes embedding problem. The proof of this equivalencewas completed in several steps, in the papers [95] and [138], the final step being taken by Ozawain [191].

16.1 Unitary correlation matrices

As a preliminary, it seems worthwhile to describe several reformulations of the Connes and Kirch-berg problem in terms of unitary correlation matrices.

Let Rc(d) be the set of d× d-matrices x = [xij ] of the form

(16.1) xij = 〈ξ, u∗i vjξ〉

where (ui) and (vi) are d-tuples of unitaries in B(H) such that uivj = vjui (or equivalently u∗i vj =vju∗i ) for all i, j, where H is arbitrary and ξ ∈ H is a unit vector. We then define the analogous

set Rs(d) but restricted to finite dimensional H’s.Explicitly: let Rs(d) be the set of d× d-matrices x = [xij ] of the form xij = 〈ξ, u∗i vjξ〉 where ui, vjare unitaries on H with dim(H) <∞ such that uivj = vjui for all i, j and ξ ∈ H is a unit vector.

214

Let E(d) ⊂ C be the linear span of U1, · · · , Ud. For any state f on C ⊗maxC (resp. C ⊗minC )let xf ∈Md be the matrix defined by:

xfij = f(Ui ⊗ Uj).

Clearly xf 7→ f|E(d)⊗E(d) is a linear isomorphism. Let Smax (resp. Smin) be the set of states onC ⊗max C (resp. C ⊗min C ). Then

(16.2) Rc(d) = xf | f ∈ Smax.

Indeed, if x is as in (16.1), let π : C ⊗ C → B(H) be a unital ∗-homomorphism taking Ui ⊗ Uj tou∗i vj . We have x = xf with f(·) = 〈ξ, π(·)ξ〉 in Smax. Conversely, the GNS factorization of f (see§26.13) shows that xf ∈ Rc(d) for any f ∈ Smax. Since Smax is weak* closed, (16.2) implies thatRc(d) is closed (this can also be shown using ultraproducts as in (16.9)). Analogously, we claimthat

(16.3) Rs(d) = xf | f ∈ Smin.

Indeed, this is easy to deduce from the same reasoning but using Proposition 9.20.Obviously, Rs(d) ⊂ Rc(d), and since Rc(d) is closed we have Rs(d) ⊂ Rc(d).

Proposition 16.1. The Kirchberg conjecture is equivalent to the assertion that Rs(d) is dense inRc(d) for any d.

Proof. The Kirchberg conjecture implies Smax = Smin and hence Rs(d) = Rc(d) by (16.2) and(16.3). We turn to the converse. Let [aij ] be a positive definite matrix in Md. In particular a = a∗

so that aij = aji. Let T =∑aijUi ⊗ Uj ∈ (E ⊗ E)+ ⊂ C ⊗ C . We claim that

(16.4) sup|∑

aij(xij + xji)| | x ∈ Rc(d) = ‖T + T ∗‖max

while

(16.5) sup|∑

aij(xij + xji)| | x ∈ Rs(d) ≤ ‖T + T ∗‖min.

In fact we will see that (16.5) is an equality. Taking these claims for granted, the density of Rs(d)in Rc(d) implies ‖T + T ∗‖max = ‖T + T ∗‖min. Then the Kirchberg conjecture follows by Corollary14.9 and Remark 14.17.We now verify the claim. By definition of the max-norm we have

‖T + T ∗‖max = sup‖∑

aiju∗i vj + aijuiv

∗j ‖,

where the sup runs over all (ui), (vj) as in the definition of Rc(d). Thus

‖T + T ∗‖max = sup|〈ξ,∑

aiju∗i vjξ〉+ 〈ξ,

∑aijuiv

∗j ξ〉|

where the sup runs over all unit vectors ξ. Equivalently

‖T + T ∗‖max = sup|∑

xijaij +∑

xijaij | | x ∈ Rc(d)

which (since a = a∗) is the same as (16.4).Let Au (resp. Av) denote the C∗-algebra generated by (ui) (resp. (vi)). We have by a similarargument

sup|∑

aij(xij + xji)| | x ∈ Rs(d) = sup‖∑

aiju∗i vj + aijuiv

∗j ‖

215

where the sup runs over all (ui), (vj) on a finite dimensional H as in the definition of Rs(d);moreover, since Au is nuclear if dim(H) <∞, the latter (ui), (vj) satisfy

‖∑

aiju∗i vj+aijuiv

∗j ‖ ≤ ‖

∑aijui⊗vj+aiju∗i⊗v

∗j ‖Au⊗maxAv

= ‖∑

aijui⊗vj+aiju∗i⊗v∗j ‖Au⊗minAv

,

and we have (using the ∗-homomorphisms Ui 7→ ui, Uj 7→ vj)

‖∑

aijui ⊗ vj + aiju∗i ⊗ v∗j ‖Au⊗minAv

≤ ‖∑

aijUi ⊗ Uj + aijU∗i ⊗ U∗j ‖C⊗minC .

Thus we obtain sup|∑aij(xij + xji)| | x ∈ Rs(d) ≤ ‖T + T ∗‖min, which is (16.5). Actually (we

do not need this in the sequel) equality holds in (16.5). This is easy to check using Proposition9.20.

Remark 16.2. Let R⊗(d) be the set of d× d-matrices of the form

xij = 〈ξ, (ui ⊗ vj)ξ〉

where (ui) and (vi) are d-tuples of unitaries in B(H) where H is any finite dimensional Hilbertspace and ξ ∈ H ⊗2 H is a unit vector.We claim R⊗(d) = Rs(d). Obviously R⊗(d) ⊂ Rs(d) (since ui ⊗ 1 commutes with 1 ⊗ vj). Tocheck the converse consider x ∈ Rs(d), say xij = 〈ξ, u∗i vjξ〉 with ui, vj ∈ U(B(H)) such thatuivj = vjui for all i, j and dim(H) < ∞. Let Au, Av be again the C∗-algebras generated inB(H) by u∗i and vj. Let f be the state defined on Au ⊗max Av by f(z ⊗ y) = 〈ξ, zyξ〉. Sincethe algebras are finite dimensional this is a state on the min-tensor product, and hence extendsto a state on B(H) ⊗min B(H) = B(H ⊗2 H). Since dim(H) < ∞ this implies that there is afinite family of unit vectors ξk in H ⊗2 H and wk > 0 (1 ≤ k ≤ m) with

∑wk = 1 such that

f(z ⊗ y) =∑

k wk〈ξk, (z ⊗ y)ξk〉 for all z, y ∈ B(H). Let H = `m2 ⊗2 [H ⊗2 H]. We now use themultiplicity trick to write:

(16.6) f(z ⊗ y) = 〈(∑

k

√wkek ⊗ ξk),

(∑kekk ⊗ [z ⊗ y]

)(∑

k

√wkek ⊗ ξk)〉.

In particular, we findxij = f(u∗i ⊗ vj) = 〈ξ′, (u′i ⊗ v′j)ξ′〉

with a unit vector ξ′ =∑

k

√wkek ⊗ ξk ∈ H, u′i =

∑k ekk ⊗ ui ⊗ 1 ∈ U(B(H)) and v′j =

∑k ekk ⊗

1⊗ vj ∈ U(B(H)). This shows that x ∈ R⊗(d), which completes the proof.

Remark 16.3. It is easy to see that both Rs(d) and Rc(d) are convex sets. Indeed, this can bechecked by the same multiplicity trick as in (16.6).

16.2 Correlation matrices with projection valued measures

The Tsirelson problem asks whether the analogue of Proposition 16.1 holds for projection valuedmeasures, in short PVMs. In quantum theory, a PVM with m outputs is an m-tuple of mutuallyorthogonal projections (Pj)1≤j≤m on a Hilbert space H with

∑Pj = I. The terminology reflects the

fact that if we set µP (A) =∑

j∈A Pj , then µP is indeed a projection valued measure on 1, · · · ,m.Seen from another viewpoint, we have a ∗-homomorphism πP : `m∞ → B(H) defined by

(16.7) πP (x) =∑m

j=1xjPj .

Let (Qj)1≤j≤m be another PVM on H, such that Qj commutes with Pi for all i, j, or equivalentlythe ranges of πP and πQ commute. In quantum mechanics, the commutation corresponds to the

216

independence of the corresponding experimental measurements. When we are given a state of thesystem in the form of a unit vector ξ ∈ H the probability of the event (i, j) is given by 〈ξ, PiQjξ〉and the matrix [〈ξ, PiQjξ〉] represents their correlation. It is then natural to wonder whether thesame numbers can be obtained by modeling the system on a finite dimensional Hilbert space H.The first and simplest question is whether the m×m so-called “covariance matrix” [〈ξ, PiQjξ〉] canbe approximated by m ×m-matrices of the same form but realized on a finite dimensional H. Inthe present simplest situation, the answer is yes. To justify this we choose a pedantic but hopefullyinstructive way. Let f : `m∞⊗max `

m∞ → C be the linear form defined by f(x⊗y) = 〈ξ, πP (x)πQ(y)ξ〉.

Clearly f is a state and of course `m∞ ⊗max `m∞ = `m∞ ⊗min `

m∞ can be identified with `m

2

∞ . A state of`m

2

∞ is simply an element z = (zij) with zij ≥ 0 such that∑

i

∑j zij = 1. Thus we may describe f as

f = (fij) with fij ≥ 0 such that∑

i

∑j fij = 1 given by fij = 〈ξ, PiQjξ〉. Now letH = `m

2

2 with o.n.basis eij , and let pi (resp. qj) denote the orthogonal projection onto the span of eij | 1 ≤ j ≤ m(resp. eij | 1 ≤ i ≤ m). Let ξ′ =

∑i

∑j f

1/2ij eij . Then ξ′ is a unit vector, (pi), (qj) are commuting

PVMs on H with dim(H) <∞ and

(16.8) ∀i, j 〈ξ, PiQjξ〉 = 〈ξ′, piqjξ′〉.

A fortiori our approximation problem is solved: we even obtain an equality.A more serious difficulty appears when we consider the same problem for several PVMs. We

give ourselves two d-tuples of PVMs (P 1, · · · , P d), (Q1, · · · , Qd) (each with m outputs) on the sameH and we assume that P ki commutes with Qlj for any 1 ≤ i, j ≤ m, 1 ≤ k, l ≤ d.

Let Qc(m, d) denote the set of all “covariance matrices” x = [x(k, i; l, j)] of the form

x(k, i; l, j) = 〈ξ, P ki Qljξ〉

(on an arbitrary H), and let Qs(m, d) be the same set of matrices but restricted to finite dimensionalHilbert spaces H. We first claim that Qc(m, d) is closed. Let us briefly sketch the easy ultraproductargument for this: Assume that a = [aklij ] is the limit when γ →∞ of 〈ξ(γ), P ki (γ)Qlj(γ)ξ(γ)〉 thatwe assume relative to H(γ). Let H(U) be the Hilbert space that is the ultraproduct of the H(γ)’sfor a nontrivial U , let ξ(U), P ki (U), Qlj(U) be the associated objects on H(U), then we find

(16.9) aklij = limU 〈ξ(γ), P ki (γ)Qlj(γ)ξ(γ)〉 = 〈ξ(U), P ki (U)Qlj(U)ξ(U)〉

and we conclude a ∈ Qc.Since Qs(m, d) ⊂ Qc(m, d) is obvious, it follows that

Qs(m, d) ⊂ Qc(m, d).

We can now state Tsirelson’s question.Tsirelson’s Problem: Is it true that Qs(m, d) = Qc(m, d) for all m, d ?

As we already mentioned in Remark 3.40, the free product Z2 ∗ Z2 is amenable and hence`2∞ ∗ `2∞ is nuclear. It is not difficult to deduce from this (as should be clear from the sequel) thatQs(m, d) = Qc(m, d) when m = d = 2, and of course we already checked in (16.8) that this alsoholds for any m if d = 1. For the other values the problem is open, and in fact:

Theorem 16.4 ([95, 138, 191]). The Tsirelson problem is equivalent to the Kirchberg conjecturewhether the min and max norms coincide on C ⊗ C .

To clarify the connection we need more notation. Let

A(m, d) = `m∞ ∗ · · · ∗ `m∞ (d times).

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Let Aj ⊂ A(m, d) denote the j-th copy of `m∞ in the (unital) free product A(m, d).Let E(m, d) ⊂ A(m, d) be the linear subspace formed of all the sums x1 + · · ·+xd where xj ∈ Aj

for all 1 ≤ j ≤ d. Note that being unital and self-adjoint, E(m, d) is an operator system.We start by an elementary observation.

Remark 16.5 (From PVMs to A(m, d)⊗maxA(m, d)). Consider two d-tuples of PVMs (P 1, · · · , P d),(Q1, · · · , Qd) in B(H) such that all P ki commute with all Qlj as before (1 ≤ k, l ≤ d, 1 ≤ i, j ≤ m).Let πPk : `m∞ → B(H) (resp. πQl : `m∞ → B(H)) be the associated ∗-homomorphism as in (16.7)).Let πp : A(m, d)→ B(H) (resp. πq : A(m, d)→ B(H)) be the ∗-homomorphism on the free productcanonically defined by (πPk)1≤k≤d (resp. (πQl)1≤l≤d) . Our commutation assumption implies thatπp and πq have commuting ranges. Therefore πp.πq is a ∗-homomorphism on A(m, d)⊗maxA(m, d),and hence the linear map F : A(m, d)⊗A(m, d)→ C defined by F (x⊗y) = 〈ξ, πp(x)πq(y)ξ〉 extendsto a state on A(m, d)⊗max A(m, d).

This gives us:

Proposition 16.6. Let F be a state on A(m, d)⊗max A(m, d). Let ekj denote the j-th basis vector

of `m∞ in the k-th copy of `m∞ inside A(m, d) = `m∞ ∗ · · · ∗ `m∞. Let xF (k, i; l, j) = F (eki ⊗ elj). Then

(16.10) Qc(m, d) = xF | F state on A(m, d)⊗max A(m, d).

Proof. The preceding remark shows Qc ⊂ xF . To show the converse, just consider the GNSconstruction applied to F on A(m, d)⊗max A(m, d).

The next lemma records an elementary (linear algebraic) observation.

Lemma 16.7. Let (Pj)1≤j≤m be a PVM on a Hilbert space H. Let H ⊂ H be finite dimensional.

There is a finite dimensional K0 with H ⊂ K0 ⊂ H such that for any finite dimensional K withK0 ⊂ K ⊂ H there is a PVM (P ′j)1≤j≤m on K such that PHP

′j |H = PHPj |H for all 1 ≤ j ≤ m.

Proof. Let Ej = Pj(H). Of course Pj and PEj commute and they coincide on H. Let K0 =

E1 + · · ·+ Em. Note H ⊂ K0. Assume K0 ⊂ K ⊂ H. Let

pj = PEj |K : K → Ej ⊂ K.

In order to replace (pj) by an m-tuple (P ′j) with sum = IdK , we set P ′j = pj for j < m andP ′m = pm + [IdK − (p1 + · · · + pm)]. Then for any h ∈ H we have P ′jh = pjh = PEjh = Pjh for

j < m and P ′mh ∈ Pmh+K K0 ⊂ Pmh+H⊥. This yields the desired property.

Lemma 16.8. Let (Pi)1≤i≤m and (Qj)1≤j≤m be PVM’s respectively on H1 and H2. Let H1 ⊂ H1

and H2 ⊂ H2 be finite dimensional subspaces. There are finite dimensional subspaces K1,K2 withH1 ⊂ K1 ⊂ H1 and H2 ⊂ K2 ⊂ H2 and PVM’s P ′k and Q′l (1 ≤ k, l ≤ d) respectively on K1 andK2 such that for all 1 ≤ i, j ≤ m and all 1 ≤ k, l ≤ d we have

PH1⊗2H2 [P ′ik ⊗Q′j

l]|H1⊗2H2

= PH1⊗2H2 [Pik ⊗Qlj ]|H1⊗2H2

.

Proof. We apply Lemma 16.7. This tells us that if K1 and K2 are chosen large enough there areP ′i

k, Q′jl as required.

Lemma 16.9. Let H1, H2 be finite dimensional. Let u : A(m, d) → B(H1) and v : A(m, d) →B(H2) be unital c.p. maps. There are finite dimensional Hilbert spaces K1,K2 with K1 ⊃ H1,K2 ⊃ H2 and ∗-homomorphisms π1 : A(m, d)→ B(K1), π2 : A(m, d)→ B(K2) such that

∀x ∈ E(m, d)⊗ E(m, d) (u⊗ v)(x) = PH1⊗2H2(π1 ⊗ π2)(x)|H1⊗2H2.

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Proof. By Stinespring’s theorem we know that this holds (on the whole of A(m, d)⊗A(m, d)) witharbitrary Hilbert spaces, say H1, H2 and ∗-homomorphisms π′1 : A(m, d)→ B(H1), π′2 : A(m, d)→B(H2) in place of H1, H2. Let then P k (resp. Ql) be the PVM’s associated respectively to therestrictions of π′1 (resp. π′2) to the k-th (resp. l-th) copy of `m∞ in A(m, d). Let K1,K2 and P ′i

k, Q′jl be

as in Lemma 16.8. Then the ∗-homomorphisms π1, π2 with values in B(K1) and B(K2), associatedrespectively to (P ′i

k) and (Q′jl) satisfy the required equality on E(m, d)⊗ E(m, d).

Remark 16.10. Let Q⊗(m, d) be the set of matrices x = [x(k, i; l, j)] of the form xf (k, i; l, j) =〈ξ, (P ki ⊗Qlj)ξ〉 where P k and Ql are PVM’s on a finite dimensional space H and ξ is a unit vectorin H ⊗2 H. The same reasoning as for Remark 16.6 shows that Q⊗(m, d) = Qs(m, d).

Proposition 16.11. Let f be a state on A(m, d)⊗minA(m, d). Let xf (k, i; l, j) = f(eki ⊗elj). Then

(16.11) Qs(m, d) = Q⊗(m, d) = xf | f state on A(m, d)⊗min A(m, d).

Proof. By (iii) in Theorem 4.21 f is the pointwise limit of a net (Fγ) of states of the form

F γ(x⊗ y) = ψγ(uγ(x)⊗ vγ(y))

where ψγ is a state on B(Hγ1 ⊗2H

γ2 ) for finite dimensional Hilbert spaces Hγ

1 , Hγ2 and where uγ1 , u

γ2

are unital c.p. maps from A(m, d) to B(Hγ1 ) and B(Hγ

2 ). By the same reasoning as for (16.6)(see also Remark 4.24), we may assume that ψγ is a vector state. By Lemma 16.9 there are ∗-homomorphisms πγ1 : A(m, d)→ B(Kγ

1 ) and πγ2 : A(m, d)→ B(Kγ2 ) with Kγ

1 ,Kγ2 finite dimensional

such that for all x, y ∈ E(m, d)

uγ1(x)⊗ uγ2(y) = PHγ1⊗2H

γ2(πγ1 (x)⊗ πγ2 (y))|Hγ

1⊗2Hγ2.

Applying that to x = eki and y = elj we obtain

xf (k, i; l, j) = limγ xγ(k, i; l, j) where xγ(k, i; l, j) = F γ(eki ⊗ elj) = ψγ(uγ1(eki )⊗ u

γ2(elj))

and we claim that xγ ∈ Q⊗(m, d). Indeed, if ψγ(t) = 〈ξ, tξ〉 for some unit vector ξ ∈ Hγ1 ⊗2 H

γ2

we find xγ(k, i; l, j) = 〈ξ, (P ki ⊗ Qlj)ξ〉 with P ki = πγ1 (eki ) and Qlj = πγ2 (elj). This shows that

xf ⊂ Qs. To show the converse, by the preceding Remark 16.10 it suffices to show Q⊗ ⊂ xf(since the latter set is obviously closed). This inclusion is immediate: let x ∈ Q⊗ of the formx(k, i; l, j) = 〈ξ, (P ki ⊗ Qlj)ξ〉. Then x = xf where f is the state on A(m, d) ⊗min A(m, d) definedby f(x⊗ y) = 〈ξ, (πPk(x)⊗ πQl(y))ξ〉.

Putting together Proposition 16.6 and the last one we obtain:

Proposition 16.12. The following assertions are equivalent for any fixed (m, d).

(i) Qs(m, d) = Qc(m, d).

(ii) The restrictions to E(m, d) ⊗ E(m, d) of the states of A(m, d) ⊗α A(m, d) for α = min andα = max form identical subsets of (E(m, d)⊗ E(m, d))∗.

Moreover, these assertions imply

(iii) The min and max norms of A(m, d)⊗A(m, d) coincide on the subspace T +T ∗ | T ∈ E(m, d)⊗E(m, d).

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Proof. Observe that (eki ⊗ elj) is a basis of the linear space E(m, d) ⊗ E(m, d). Therefore therestriction of a state f of A(m, d) ⊗α A(m, d) to E(m, d) ⊗ E(m, d) is entirely determined by xf .The equivalence of (i) and (ii) is then clear by (16.10) and (16.11).Let T ∈ E(m, d) ⊗ E(m, d). Clearly ‖T + T ∗‖α = sup|f(T + T ∗)| where the sup runs over allstates f on A(m, d)⊗α A(m, d). Thus (ii) ⇒ (iii).

We will now relate the last statement to C∗(Fd)⊗ C∗(Fd).

Lemma 16.13. The C∗-algebra A(m, d) has the LLP. Moreover, the identity of A(m, d) factorsvia unital c.p. maps through C∗(Fd).

Proof. The first assertion is clear since the LLP is stable by free products (see Theorem 9.44).For the second one (which actually also implies the first one) we use Boca’s Theorem 2.24. It iseasy to see that the identity of `m∞ = C∗(Z/mZ) factors via unital c.p. maps through C∗(Z) (thisfollows by applying Remark 9.54 to the natural quotient ∗-homomorphism C∗(Z) → C∗(Z/mZ)).By Theorem 2.24 the identity of A(m, d) factors through C∗(Z) ∗ · · · ∗ C∗(Z) = C∗(Z ∗ · · · ∗ Z) (dtimes) and Z ∗ · · · ∗ Z = Fd.

Lemma 16.14. Let E(d) ⊂ C∗(Fd) be the linear span of the d unitary generators and the unit.There are unital c.p. maps

vm : C∗(Fd)→ A(m, d) and wm : A(m, d)→ C∗(Fd)

such that vm(E(d)) ⊂ E(m, d) for any m and such that wmvm → IdC∗(Fd) pointwise when m→∞.

Proof. It is easy to check this for d = 1 (see Remark 26.20 and recall Remark 1.32). Then byBoca’s Theorem 2.24 the general case follows.

We can now complete the proof of the equivalence of the Tsirelson problem with the Connes-Kirchberg one.

Proof of Theorem 16.4. Assume Qs(m, d) = Qc(m, d) for all m. Recall that any pair of unitalc.p. maps gives rise to a contraction on the maximal tensor product (see Corollary 4.18). Thenby Proposition 16.12 and Lemma 16.14 the min and max norms coincide on T + T ∗ | T ∈E(d) ⊗ E(d) ⊂ C∗(Fd) ⊗ C∗(Fd). Since this holds for all d it holds for d = ∞, i.e. with E ⊂ Cin place of E(d) ⊂ C∗(Fd). By Corollary 14.9 and Remark 14.17 we conclude that the min andmax norms coincide on C ⊗ C , which means the Kirchberg conjecture holds. Conversely, if theKirchberg conjecture holds, by Proposition 13.1 we know from LLP ⇒ WEP that the min andmax norms coincide on A⊗A for any A with the LLP. Therefore, by Lemma 16.13 and Proposition16.12 again we conclude that Qs(m, d) = Qc(m, d) for all m, d.

Remark 16.15 (About POVMs). A positive operator valued measure (in short POVM) with moutputs is an m-tuple of self-adjoint operators (aj)1≤j≤m on a Hilbert space H with

∑aj = I.

Although this notion is more general than PVMs, it is easy to reduce the study of correlationmatrices of POVMs to that of PVMs. Indeed, consider two d-tuples of POVMs (a1, · · · , ad),(b1, · · · , bd) (each with m outputs) on the same H and assume that aki commutes with blj for any1 ≤ i, j ≤ m, 1 ≤ k, l ≤ d. We wish to address the same problem as before but for

x(k, i; l, j) = 〈ξ, aki bljξ〉.

There is a 1-1 correspondence between POVMs and unital positive (and hence c.p. by Remark1.32) linear maps from `m∞ to B(H). Just consider for any a = (ai)1≤i≤m the map ua : `m∞ → B(H)

220

defined by ua(ei) = ai. If two such POVMs (ai)1≤i≤m and (bj)1≤j≤m mutually commute, theranges of ua and ub also commute. By Boca’s Theorem 2.24 and by Corollary 4.19 we have a unitalc.p. map u : A(m, d) ⊗max A(m, d) → B(H) such that u(eki ⊗ elj) = aki b

lj or any 1 ≤ i, j ≤ m,

1 ≤ k, l ≤ d. By Stinespring’s dilation Theorem 1.22 we have H ⊃ H and a ∗-homomorphismπ : A(m, d)⊗maxA(m, d)→ B(H) such that u(·) = PHπ(·)H . It follows that for any ξ ∈ H we have

〈ξ, aki bljξ〉 = 〈ξ, P ki Qljξ〉

where P k and Ql are the PVMs associated to π by setting π(eki ⊗ 1) = P ki and π(1 ⊗ elj) = Qlj .This shows that the Tsirelson problem for POVMs reduces to the case of PVMs.

16.3 Strong Kirchberg conjecture

In [192] Ozawa introduced a strong version of Kirchberg’s conjecture that he proved to be equivalentto the assertion that

C ⊗min C ⊗min B = C ⊗max C ⊗max B.

It is an easy exercise to see that this is the same as the two assertions that C ⊗minC = C ⊗maxCtogether with (C ⊗max C ) ⊗min B = (C ⊗max C ) ⊗max B. In other words, this is the same as theKirchberg conjecture together with the assertion that C ⊗max C = C∗(F∞ × F∞) has the LLP.The strong Kirchberg conjecture from [192] asserts the following for any d ≥ 2:For any δ > 0 ∃ε > 0 such that given unitaries U1, · · · , Ud, V1, · · · , Vd in B(H) with dim(H) <∞,satisfying

∀i, j ‖[Ui, Vj ]‖ ≤ ε

there is H ⊃ H with dim(H) <∞ and unitaries U1, · · · , Ud, V1, · · · , Vd in B(H) satisfying

∀i, j [Ui, Vj ] = 0 and ‖Ui − PHUi|H‖ ≤ δ, ‖Vj − PH Vj |H‖ ≤ δ.

Actually, its validity for some d ≥ 2 implies the same for all d ≥ 2.The next lemma somewhat explains the role of the LLP.

Lemma 16.16. Consider the assertion of the strong Kirchberg conjecture without the requirementthat H be finite dimensional. Then this is equivalent to the LLP for C∗(Fd × Fd).

In [192], Ozawa also considers an analogous equivalent conjecture involving positive operatorvalued measures (POVMs) in the style of Tsirelson’s conjecture. At this stage we refer to [192] fordetailed proofs.

16.4 Notes and Remarks

The elementary results of §16.1 are only formulated to clarify (hopefully) those of the subsequent§16.2. See [139, 140] for more recent information on Bell’s inequalities and their connections withoperator spaces. See Dykema and Juschenko’s [75] for more results on unitary correlation matrices.In [129] Harris and Paulsen study a different kind of unitary correlation matrices, derived fromconsideration of the Brown algebra (see Remark 13.2) and prove an analogue of Theorem 16.4 intheir setting. See Slofstra’s [231, 232] (and Dykema and Paulsen’s [76]) for proofs that the setQs(m, d) is not closed in general.In a different direction, the papers [115, 116] by Haagerup and Musat relate what they call theasymptotic quantum Birkhoff property for factorizable Markov maps to the Connes embeddingproblem.

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17 Property (T) and residually finite groups. Thom’s example

Definition 17.1. A finitely generated discrete group G, with generators g1, g2, . . . , gn, is said tohave property (T) if the trivial representation is isolated in the set of all unitary representations ofG. More precisely, this means that there is a number δ > 0 such that, for any unitary representationπ the condition

(∃ξ ∈ Hπ, ‖ξ‖ = 1, supj≤n ‖π(gj)ξ − ξ‖ < δ)

suffices to conclude that π admits a nonzero invariant vector, or in other words that π contains thetrivial representation (as a subrepresentation).

It is easy to see that this property actually does not depend on the choice of the set of generators,but of course the corresponding δ does depend on that choice.

The classical example of discrete group with property (T) is G = SL3(Z). This goes back toD. Kazhdan (1967) and property (T) groups are often also called “Kazhdan groups”.

We will use the following basic fact.

Lemma 17.2. If G with generators S = g1, g2, . . . , gn has property (T) then there is a functionf : (0,∞)→ (0,∞) with limε→0 f(ε) = 0 such that, for any unitary representation π : G→ B(Hπ),if ξ ∈ Hπ is a unit vector satisfying

supj≤n ‖π(gj)ξ − ξ‖ < ε

there is a unit vector ξ′ ∈ Hπ such that

π(g)ξ′ = ξ′ ∀g ∈ G and ‖ξ − ξ′‖ < f(ε).

Proof. Let H invπ ⊂ Hπ be the subspace of π-invariant vectors, i.e. vectors x ∈ Hπ such that

π(g)x = x for all g ∈ G. Let P be the orthogonal projection from Hπ onto H invπ . Clearly, H inv

π⊥

isinvariant under π and the restriction of π to it has no nonzero invariant vector. Therefore for any

ξ ∈ H invπ⊥

with ‖ξ‖ = 1 we must have supj≤n ‖π(gj)ξ − ξ‖ ≥ δ, where δ is as in Definition 17.1.By homogeneity this implies supj≤n ‖π(gj)ξ − ξ‖ ≥ δ‖ξ‖ for all ξ and since π(gj)Pξ = Pξ we haveπ(gj)ξ − ξ = π(gj)(ξ − Pξ)− (ξ − Pξ) for all 1 ≤ j ≤ n, and hence

∀ξ ∈ Hπ supj≤n ‖π(gj)ξ − ξ‖ ≥ δ‖ξ − Pξ‖.

Therefore∀ξ ∈ Hπ d(ξ,H inv

π ) ≤ δ−1 supj≤n ‖π(gj)ξ − ξ‖.

Let ξ′ = Pξ‖Pξ‖−1. If supj≤n ‖π(gj)ξ − ξ‖ < ε and ‖ξ‖ = 1 we have ‖ξ − Pξ‖ ≤ ε/δ and hence‖Pξ‖ ≥ 1 − ε/δ. Assuming 0 < ε < δ, we obtain ‖ξ′ − ξ‖ ≤ ‖ξ′ − ξ‖Pξ‖−1‖ + ‖ξ‖Pξ‖−1 − ξ‖ ≤(ε/δ)(1− ε/δ)−1.

Property (T) can also be reformulated in terms of spectral gap, as follows. We will return tothat theme in §19.3.

Proposition 17.3. A discrete group G generated by a finite subset S ⊂ G containing the unit hasproperty (T) if and only if there is ε ∈ (0, 1) such that for any unitary representation π : G→ B(Hπ)we have

(17.1) ‖|S|−1∑

s∈Sπ(s)− PHinv

π‖ ≤ 1− ε.

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Proof. Let n = |S|. By (26.6) if

(17.2) n−1‖∑

s∈Sπ(s)P|Hinv

π⊥‖ > 1− ε

there is a unit vector ξ in H invπ⊥

such that ‖π(s)ξ − π(t)ξ‖ < 2√

2εn for any s, t ∈ S. In particularsince the unit of G is in S we have sups∈S ‖π(s)ξ − ξ‖ < 2

√2εn.

Now assume that G has property (T). Let δ be as in Definition 17.1. Then if 2√

2εn = δ, i.e. if

ε = δ2/8n, we conclude that (17.2) is impossible: otherwise the restriction of π to H invπ⊥

wouldhave an invariant unit vector and this is absurd by the very definition of H inv

π . This shows that(17.1) holds for ε = δ2/8n.Conversely, if (17.1) holds for some ε > 0 and if sups∈S ‖π(s)ξ− ξ‖ < δ for some unit vector ξ then

‖ξ − PHinvπξ‖ ≤ 1− ε+ δ.

Thus if we choose δ < ε, we find PHinvπξ 6= 0 so that the vector PHinv

πξ is a non-zero invariant vector

for π, and G has property (T).

When the group G has property (T) the structure of C∗(G) is very special, in particular it splitsas a direct sum pC∗(G)⊕ (1− p)C∗(G), as shown by the next statement.

Proposition 17.4. Let G be a finitely generated group with Property (T). Then there is a self-adjoint projection p in the center of C∗(G) such that for any ∗-homomorphism π : C∗(G)→ B(Hπ)associated to a unitary representation π : G→ B(Hπ) we have

π(p) = PHinvπ,

in particular, π(p) = 1 if π is the trivial representation and π(p) = 0 if π is any nontrivial irreducibleunitary representation.

Proof. Note that H invπ is also the set of invariant vectors of the range of π so we will denote it

simply by H invπ . Let S be a finite symmetric set of generators. Let t = |S|−1

∑s∈S UG(s), so that

π(t) = |S|−1∑

s∈S π(s). The projection p is simply the limit in C∗(G) of tm when m → ∞. Letus show that this limit exists. Note π(t)m − PHinv

π= π(t)m(1 − PHinv

π) for any m ≥ 1 and π(t)

commutes with (1 − PHinvπ

) so that π(t)m − PHinvπ

= (π(t)(1 − PHinvπ

))m = (π(t) − PHinvπ

)m. ByProperty (T) as in (17.1) there is ε ∈ (0, 1) such that ‖π(t)− PHinv

π‖ ≤ 1− ε for any π, and hence

(17.3) ‖π(t)m − PHinvπ‖ ≤ (1− ε)m

for any m ≥ 1. A fortiori, ‖π(t)m − π(t)m+1‖ ≤ 2(1 − ε)m, and since this holds for any π wehave ‖tm − tm+1‖C∗(G) ≤ 2(1 − ε)m. Therefore by the Cauchy criterion tm converges to a limitp ∈ C∗(G). Since t∗ = t we have p∗ = p, and also p2 = lim t2m = p. Moreover, by (17.3) for any πwe have π(p) = limπ(tm) = limπ(t)m = PHinv

π, and since PHinv

πobviously commutes with the range

of π, we have π(p)π(x) = π(x)π(p) for any x ∈ C∗(G). In particular, taking π = IdC∗(G) we seethat p is in the center of C∗(G).

We now connect the factorization property, introduced in Definition 7.36 and further studiedin Theorem 11.55, with property (T). For simplicity, we still denote by λG : C∗(G) → C∗λ(G) the∗-homomorphism that extends the unitary representation λG, so that λG(UG(t)) = λG(t) for t ∈ G.

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Theorem 17.5. Let G be a discrete group with property (T). Let S ⊂ G be an arbitrary generatingset with 1 ∈ S. Let E ⊂ C∗(G) be the linear span of S. Assume that

(17.4) ∀n ∀x1, ..., xn ∈ E∑

τG(|λG(xj)|2)) ≤ ‖∑

xj ⊗ xj‖min.

Then S is residually finite dimensional (RFD in short), i.e. S is separated by the set of finitedimensional unitary representations. In particular, if (17.4) holds for S = G, then G is RFD.

Before giving the proof we start by comments and consequences to motivate Theorem 17.5.

Remark 17.6. By a classical Theorem of Malcev [177], any finitely generated linear group is RF.Thus for finitely generated groups RFD implies RF. Actually, the groups G that are separated bytheir finite dimensional unitary representations are characterized as subgroups of compact groups.They are usually called “maximally almost periodic”.

By Theorem 11.55, the factorization property implies (17.4) with S = G, therefore:

Corollary 17.7. If G has property (T) and the factorization property, then G is residually finite.

Corollary 17.8. If C∗(G) has the WEP and G has property (T), then G is residually finite.

Proof. If C∗(G) has the WEP, then the min and max norms coincide on the set ∑xj ⊗ xj ⊂

C∗(G) ⊗ C∗(G). Therefore we again have (17.4) for S = G. Property T implies that G is finitelygenerated, thus by Malcev’s theorem (see Remark 17.6) G is RF.

Proof of Theorem 17.5. Let C = C∗(G). By Theorem 11.38 and Remark 11.39 for a suitableembeddding C ⊂ B(H) there is a net (hi) of Hilbert-Schmidt operators on H with tr(h∗ihi) = 1such that

∀y, x ∈ E τG(λG(y)∗λG(x)) = limi tr(y∗xh∗ihi)

and such that, denoting by g → U(g) ∈ B(H) the unitary representation of G obtained by restrict-ing to G the embedding C → B(H), we have

∀s ∈ S ‖U(s)hi − hiU(s)‖2 = ‖U(s)hiU(s)∗ − hi‖2 → 0.

It follows that (hi) is an approximately invariant unit vector for the representation U ⊗ U of Gon S2(H) = H ⊗2 H. Since G has property (T) there is, by Lemma 17.2, a net of invariant unitvectors (h′i) in S2(H) such that ‖hi − h′i‖2 → 0. Let Ti = h′∗i h

′i. We have then

∀y, x ∈ E τG(λG(y)∗λG(x)) = limi tr(y∗xTi),

tr(Ti) = 1 and U(s)Ti = TiU(s) for all s ∈ S, and hence for all g ∈ G.The latter condition implies that U(g) commutes with all the spectral projections of Ti ≥ 0, andsince Ti is compact these are finite dimensional. Thus if we write the spectral decomposition of Tias

Ti =∑

kλikP

ik

withλik > 0

∑kλiktr(P

ik) = 1

we find that πik : g → U(g)|P ik(H) is a finite dimensional unitary representation of G on P ik(H), suchthat for any y, x ∈ E

τ(λG(y)∗λG(x)) = limi

∑kλiktr(π

ik(y)∗πik(x)).

In particular for any s, t ∈ Sδs,t = limi

∑kλiktr(π

ik(s−1t)).

From which it is clear that πik separates the points of S.

224

In [245], Andreas Thom exhibited a group G with property (T), that is approximately linear(i.e. “hyperlinear”), and even sofic, but not residually finite. This is a remarkable example (orcounterexample) because:

Proposition 17.9. Let G be a group with property (T), that is approximately linear (i.e. “hyper-linear”) but not residually finite. Then C∗(G) has neither the WEP nor the LLP.

Proof. That C∗(G) fails the WEP follows from Corollary 17.8. Since G is approximately linear,we know (see Corollary 9.60 or Remark 14.5) that we can write MG as a quotient MG = A/I of aWEP C∗-algebra A. Let E ⊂ C∗(G) be any finite dimensional subspace. If C∗(G) had the LLP,the natural map u : C∗(G)→ A/I (which is the same as QG) would be locally 1-liftable. We claimthat this would imply (17.4). Indeed, there would be a map uE : E → A with ‖uE‖cb ≤ 1 suchthat quE = u|E . Let xj ∈ E and yj = uE(xj) ∈ A (1 ≤ j ≤ n). Then we would have u(xj) = q(yj)and hence∑

‖u(xj)‖2L2(τG) ≤ ‖∑

u(xj)⊗ u(xj)‖MG⊗maxMG≤ ‖

∑xj ⊗ yj‖C∗(G)⊗maxA

By Corollary 9.40 (general form of Kirchberg’s Theorem) and since yj = uE(xj) and ‖uE‖cb ≤ 1we would have

‖∑

xj ⊗ yj‖C∗(G)⊗maxA= ‖

∑xj ⊗ yj‖C∗(G)⊗minA

≤ ‖∑

xj ⊗ xj‖C∗(G)⊗minE,

and since ‖∑xj ⊗ xj‖C∗(G)⊗minE

= ‖∑xj ⊗ xj‖C∗(G)⊗minC∗(G)

, our claim (17.4) would follow. By

Theorem 17.5 this would contradict the fact that G is not residually finite, thus showing that C∗(G)fails the LLP.

Remark 17.10. Let us say that an operator u : C → B between C∗-algebras locally factors (com-pletely boundedly) through a C∗-algebra A if there is a constant c such that for any finite di-mensional subspace E ⊂ C the restriction u|E admits a factorization u|E : E

v−→A w−→B with‖v‖cb‖w‖cb ≤ c. Using the subsequent inequality (22.16) the preceding argument can be modifiedto show more generally that the inclusion u : C∗(G) → MG does not locally factor (completelyboundedly) through a WEP C∗-algebra. This negates at the same time both WEP and LLP forC∗(G), when we know that MG is QWEP and G has (T).

17.1 Notes and Remarks

Theorem 17.5 originally comes from [157]. A simpler proof appears in [17] which we recommend tothe reader for (much) more information on Property (T) (see also [127]). Proposition 17.4 is dueto Valette [253]. A. Thom’s construction of a group G as in Proposition 17.9 appears in [245] towhich we refer the reader for full details. This is the first example for which C∗(G) fails the LLP.

18 The WEP does not imply the LLP

Although S. Wassermann had proved in his 1976 paper [256] that B(H) is not nuclear (assumingdim(H) = ∞), the problem whether A ⊗min B = A ⊗max B when A = B = B(H) remained openuntil [141]. In the latter paper, several different proofs were given. Taking into account the mostrecent information from [120], we now know that

(18.1) sup

‖t‖max

‖t‖min

∣∣∣ t ∈ B(H)⊗B(H), rk(t) ≤ n≥ n

2√n− 1

.

225

This estimate is rather sharp asymptotically, since it can be shown that the supremum appearingin (18.1) is ≤

√n (see [141] or [208, p. 353]).

Remark 18.1. We need to clarify a few points regarding complex conjugation, which we alreadydiscussed in §2.3. In general, we will need to consider the conjugate A of a C∗-algebra A. Thisis the same object but with the complex multiplication changed to (λ, a) → λa, so that A is anti-isomorphic to A. Recall that for any a ∈ A, we denote by a the same element viewed as an elementof A. Recall (see Remark 2.13) that A ' Aop via the mapping a→ a∗.

The distinction between A and A is necessary in general, but not for A = B(H) since in thatcase, using H ' H, we have B(H) ' B(H) ' B(H), and in particular MN ' MN . In the case ofMN , the mapping a 7→ [aij ] is an embedding of MN into MN , taking eij to eij . As a consequence,for any matrix a in MN (A) we have

(18.2) ‖∑

eij ⊗ aij‖MN⊗minA= ‖

∑eij ⊗ aij‖MN⊗minA = ‖[aij ]‖MN (A).

By (2.13) this also implies‖[aij ]‖MN (A) = ‖[aij ]‖MN (A).

Note however that H ' H depends on the choice of a basis so the isomorphism B(H) ' B(H)is not canonical. Nevertheless, this shows that the problem whether the min and max norms arethe same is identical for B(H)⊗B(H) and for B(H)⊗B(H).

Remark 18.2. Consider a1, . . . , an in A and b1, . . . , bn in B. Using the preceding remark we have∥∥∥∑ aj ⊗ bj∥∥∥A⊗αB

=∥∥∥∑ a∗j ⊗ bj

∥∥∥Aop⊗αB

for any “reasonably” well behaved C∗-norm, in particular for α = min or max. Moreover, we have∥∥∥∑ aj ⊗ bj∥∥∥A⊗maxB

=∥∥∥∑ a∗j ⊗ bj

∥∥∥Aop⊗maxB

= sup∥∥∥∑π(a∗j )σ(bj)

∥∥∥where the supremum runs over all commuting range pairs π : A → B(H), σ : B → B(H) with σa representation and π an anti-representation on the same (arbitrary) Hilbert space H.

Remark 18.3. Let M be a C∗-algebra equipped with a tracial state τ . Then the GNS construction(see §26.13) associated to (M, τ) produces a Hilbert space H = L2(τ), a cyclic unit vector ξin H associated to 1M and commuting left-hand and right-hand actions of M induced by thecorresponding multiplications on M . As earlier, we denote them by L(a)h = a · h (L is what wedenoted πf in §26.13) and R(a)h = h · a. Then L (resp. R) is a representation of M (resp. Mop)on B(H) and the ranges of L and R commute (see the beginning of §11).We have then for any n-tuple (u1, . . . , un) of unitaries in M

(18.3)∥∥∥∑n

1uj ⊗ uj

∥∥∥M⊗maxM

=∥∥∥∑n

1u∗j ⊗ uj

∥∥∥Mop⊗maxM

= n.

Indeed, this is

(18.4) ≥∥∥∥∑n

1L(u∗j )R(uj)

∥∥∥ ≥ ∥∥∥∑n

1u∗j · ξ · uj

∥∥∥but since ξ ∈ L2(τ) is the element associated to 1M we have u∗j · ξ · uj = ξ hence (18.4) is ≥ n and≤ n is trivial by the triangle inequality.In particular, for any unitary matrices u1, . . . , un in MN we have

(18.5)∥∥∥∑n

1uj ⊗ uj

∥∥∥min

=∥∥∥∑n

1uj ⊗ uj

∥∥∥max

= n.

226

18.1 The constant C(n): WEP 6⇒ LLP

In [141], a crucial role is played by a certain constant C(n), defined as follows: C(n) is the infimumof the constants C such that for each m ≥ 1, there is Nm ≥ 1 and an n-tuple [u1(m), . . . , un(m)]of unitary Nm ×Nm matrices such that

(18.6) supm 6=m′

∥∥∥∑n

j=1uj(m)⊗ uj(m′)

∥∥∥min≤ C.

By (18.2) the preceding min-norm can be understood either in MNm ⊗min MNm′ 'MNmNm′ (with

uj(m) denoting the usual matrix with conjugate entries) or in MNm⊗minMNm′ with uj(m) ∈MNm .The connection of C(n) to B(H)⊗B(H) goes through the following statement.

Theorem 18.4 ([141]). For any n ≥ 1 and ε > 0, there is a tensor t of rank n in B⊗B such that

‖t‖max/‖t‖min ≥ n/C(n)− ε.

Remark 18.5. By Corollaries 22.13 and 22.16 we have ‖t‖max = ‖t‖min for any tensor t ∈ B(H)⊗B(H) of the form t =

∑xj ⊗ xj . This perhaps explains why Theorem 18.4 is not so easy.

We have trivially C(n) ≤ n for all n. The crucial fact to show that B(H) ⊗min B(H) 6=B(H)⊗max B(H) is that C(n) < n for at least one n > 1. The final word on this is now:

Theorem 18.6 ([120]).

(18.7) ∀n ≥ 2 C(n) = 2√n− 1

The (much easier) lower bound 2√n− 1 ≤ C(n) was proved in [206]. The complete proof of the

upper bound uses a delicate random matrix ingredient (namely Theorem 18.16) the proof of whichis beyond the scope of these notes, but we will give the proof of (18.7) modulo this ingredient inthe next section. See the next chapter for simpler proofs that C(n) < n.

Theorem 18.4 is an immediate consequence of the next Theorem 18.9. To prove the latter we willuse a compactness argument for “convergence in distribution” (or rather in moments) of n-tuplesof operators, that is described in the next two lemmas. By “distribution” we mean the collection ofall “moments” in the operators (viewed as non-commutative random variables) and their adjoints.This is the same notion as that of ∗-distribution used by Voiculescu in free probability (see [255]),but our notation is slightly different.

Let S be the set consisting of the disjoint union of the sets

Sk = [1, . . . , n]k × 1, ∗k.

For any w = ((i1, . . . , ik), (ε1, . . . , εk)) in Sk and any n-tuple x = (x1, . . . , xn) in B(H) we denote

w(x) = xε1i1 xε2i2. . . xεkik

(where xε = x if ε = 1 and xε = x∗ if ε = ∗). Let x = (x1, . . . , xn) be an n-tuple in a von Neumannalgebra M equipped with a tracial state τ . By “the distribution of x,” we mean the function

µx : S → C

defined byµx(w) = τ(w(x)).

When x = (x1, . . . , xn) is an n-tuple of unitary operators, we may as well consider that µx is afunction defined on Fn (free group with generators g1, . . . , gn) by setting µx(w) = τ(πx(w)) for any“word” w ∈ Fn, where πx : Fn →M is the unitary representation defined by π(gj) = uj .

The following is elementary and well known.

227

Lemma 18.7. Fix n ≥ 1. Let (M(m), τm) be a sequence of von Neumann algebras equipped with(tracial) states. Let x(m) = (x1(m), . . . , xn(m)) be a bounded sequence of n-tuples with x(m) ∈M(m)n. Then there is a subsequence mk such that the distributions of x(mk) converge pointwiseon S when k →∞.

To identify the limit of a sequence of distribution, it will be convenient to use ultraproducts. Let(M(m), τm) be as before (m ∈ N). Let U be a nontrivial ultrafilter on N. Let B = (⊕

∑mM(m))∞.

Let HU be the GNS Hilbert space for the state τU defined on B by ∀y = (ym) ∈ B τU (y) =limU τm(ym). Let IU = y = (ym) | limU τm(y∗mym) = 0. As explained earlier (see §11.2), since τUvanishes on IU it defines a tracial state on B/IU , that we still denote (albeit abusively) by τU . Wehave an isometric representation: a→ L(a) ∈ B(L2(τU )) of B/IU on L2(τU ) (associated to left-handmultiplication) and an isometric representation a → R(a) ∈ B(L2(τU )) of (B/IU )op (associated toright-hamd multiplication). We already know (cf. Theorem 11.26) that MU = L(B/IU ) is avon Neumann subalgebra of B(HU ) and that we have M ′U = R(B/IU ) and R(B/IU )′ = MU . Wewill view (abusively) τU as a functional on MU by setting for any y ∈MU

τU (y) = limU τm(ym),

where (ym) ∈ B is any element of the equivalence class of L−1(y) ∈ B/IU .Let x = x(m) | m ∈ N be a bounded sequence of n-tuples with x(m) ∈ M(m)n as before.

Equivalently, x is a sequence in Bn. Let x = (x1, . . . , xn) be the associated n-tuple in MnU . Then,

for any “word” w in S , we clearly have

τU (w(x)) = limU τm(w(x(m))).

Hence the distribution of x(m) tends pointwise to that of x along U , so we can write limU µx(m) = µx.The next (again elementary and well known) lemma connects limits in distribution with ultrafilters.

Lemma 18.8. Let x(m) | m ∈ N be a sequence of n-tuples as in the preceding Lemma. Thefollowing are equivalent.

(i) The distributions of x(m) converge pointwise when m→∞.

(ii) For any nontrivial ultrafilter U on N, the associated n-tuple x = (x1, . . . , xn) in (MU , τU ) hasthe same distribution (i.e. its distribution does not depend on U).

(iii) There is a tracial probability space (M, τ) and y = (y1, . . . , yn) in Mn such that x(m)→ y indistribution.

Proof. (i)⇒ (ii) is essentially obvious. (ii)⇒ (iii) is proved by picking any fixed nontrivial ultrafilterU and taking (M, τ) = (MU , τU ) (see Remark 26.4 for clarification). Lastly (iii) ⇒ (i) is againobvious.

With the notation in (11.14), let [u1(m), . . . , un(m)],m ∈ N be a sequence of n-tuples ofunitary matrices as in (18.6) (recall u1(m), . . . , un(m) are of size Nm ×Nm).For any subset ω ⊂ N, let

(18.8) Bω =(⊕∑

m∈ωMNm

)∞.

Let N = ω(1) ∪ ω(2) be any disjoint partition of N into two infinite subsets, and let

(18.9) u1j =

⊕m∈ω(1)

uj(m) ∈ Bω(1) u2j =

⊕m′∈ω(2)

uj(m′) ∈ Bω(2).

228

Theorem 18.9. Suppose that [u1(m), . . . , un(m)] converges in distribution when m → ∞ andsatisfies (18.6). Let

t =∑n

j=1u1j ⊗ u

2j .

We have then‖t‖min ≤ C and ‖t‖max = n,

and hence ‖t‖max/‖t‖min ≥ n/C, where the min and max norms are relative to Bω(1) ⊗ Bω(2).

Proof. We have obviously

‖t‖min = supm∈ω(1),m′∈ω(2)

∥∥∥∑uj(m)⊗ uj(m′)∥∥∥

hence ‖t‖min ≤ C. We now turn to ‖t‖max. Let U be a nontrivial ultrafilter on ω(1) and let Vbe one on ω(2). We construct the ultraproducts MU and MV as previously. Since the quotientmappings Bω(1) →MU and Bω(2) →MV are ∗-homomorphisms, we have

‖t‖max ≥∥∥∥∑uj ⊗ vj

∥∥∥MU⊗maxMV

where uj (resp. vj) is the equivalence class modulo U (resp. V) of⊕

m∈ω(1)

uj(m) (resp.⊕

m∈ω(2)

uj(m)).

Now, since we assume that [u1(m), . . . , un(m)] converges in distribution when m→∞, the twon-tuples (u1, . . . , un) and (v1, . . . , vn) must have the same distribution relative respectively to τUand τV . But this implies (see Remark 11.20) that there is a ∗-isomorphism π from the von Neumannalgebra Nu ⊂MU generated by (u1, . . . , un) to the one Nv ⊂MV generated by (v1, . . . , vn), definedsimply by π(uj) = vj . Moreover, since we are dealing here with finite traces, there is a conditionalexpectation P from MU onto Nu (see Proposition 11.21). Therefore the composition T = πP is aunital completely positive map from MU to Nv ⊂MV such that T (uj) = vj . Thus we find by (4.30)

(18.10)∥∥∥∑uj ⊗ vj

∥∥∥MU⊗maxMV

≥∥∥∥∑T (uj)⊗ vj

∥∥∥MV⊗maxMV

=∥∥∥∑ vj ⊗ vj

∥∥∥MV⊗maxMV

.

But then by (18.3) we conclude that ‖t‖max ≥ n.

Remark 18.10. The same reasoning shows that

‖t‖min ≥∥∥∥∑ vj ⊗ vj

∥∥∥MV⊗minMV

.

Proof of Theorem 18.4. Since there exist max-injective inclusions Bω(1) ⊂ B and Bω(2) ⊂ B, The-

orem 18.9 gives us a tensor t′ ∈ B ⊗B of rank n with ‖t′‖max/‖t′‖min ≥ n/C. Using B ' B, wefind a similar tensor in B ⊗B.

We now exploit the mere fact that C(n) < n for some n, which we know by Theorem 18.6:

Corollary 18.11. Recall B =(⊕∑

n≥1Mn

)∞

. Then the pair (B,B) (or the pair (B,B)) is not

nuclear and B (although it has the WEP) fails the LLP.

Proof. Since C(n) < n for some n, Theorem 18.9 tells us that (Bω(1),Bω(2)) is not a nuclear pair.We have inclusions Bω(1) ⊂ B and Bω(2) ⊂ B each admitting a c.p. contractive projection. ByProposition 7.19 (i), these inclusions are max-injective, therefore we have an isometric embedding

Bω(1) ⊗max Bω(2) ⊂ B⊗max B

229

so the min and max norms cannot coincide on B⊗B or equivalently (recall B ∼= B) on B⊗B. Sincefor the inclusion B ⊂ B(`2) there is also a c.p. contractive projection from B(`2) onto B, the sameargument shows they do not coincide on B(`2)⊗ B, which means that B fails the LLP.

Similarly:

Corollary 18.12. For B = B(`2), we have

B ⊗min B 6= B ⊗max B

or equivalently B (although it has the WEP) fails the LLP.

Proof. Since we have max-injective inclusions B ⊂ B this follows from Corollary 18.11.

More generally, we have

Corollary 18.13. A von Neumann algebra M has the LLP if and only if M is nuclear.A pair (M,N) of von Neumann algebras is nuclear if and only if one of them is nuclear.

Proof. Let B =(⊕∑

n≥1Mn

)∞

. If M is not nuclear, by Theorem 12.29 there is an embedding

B ⊂M , and since B is injective, the embedding is max-injective. Since B fails the LLP (by Corollary18.11), so does M (see Remark 9.3 or Remark 9.13).If, in addition, N is not nuclear, we have B ∼= B ⊂ N . Thus again, since (B,B) is not nuclear, wefind that (M,N) is not nuclear if none of M,N is nuclear, and the converse is trivial.

18.2 Proof that C(n) = 2√n− 1 using random unitary matrices

We start by an easy result from [206] estimating C(n) from below. The alternate proof we givehere of (18.11) is due to Szarek.

Proposition 18.14. Let u1, . . . , un be arbitrary unitary operators in B(H) (H any Hilbert space),then

(18.11) 2√n− 1 ≤

∥∥∥∑n

j=1uj ⊗ uj

∥∥∥min

.

Proof. Let S+ = t ∈ S2(H) | t ≥ 0 ‖t‖2 = 1. We will use the self-adjoint case of (2.11). Note:

(18.12) ∀a ∈ B(H),∀t ∈ S+ tr(tata∗) = tr([t1/2at1/2][t1/2at1/2]∗) ≥ 0.

Let T =∑n

1 uj ⊗ uj and let S =∑n

j=1 λFn(gj). In accordance with the identification of T with theoperator t 7→

∑ujtu

∗j acting on S2(H), we denote 〈t, T t〉 = tr(

∑t∗ujtu

∗j ) for any t ∈ S2(H). The

idea of the proof is to show that for any integer m ≥ 1 and any t in S+ we have

(18.13) 〈t, (T ∗T )mt〉 ≥ 〈δe, (S∗S)mδe〉 = τFn((S∗S)m)

where δe denotes the basis vector in `2(Fn) indexed by the unit element of Fn. We can expand(T ∗T )m as a sum of the form

∑w∈I u

w⊗uw where the uw’s are unitaries of the form u∗i1uj1u∗i2uj2 . . ..

Now for certain w’s, we have uw = I (and hence tr(uwtuw∗t) = 1) by formal cancellation (no matterwhat the uj ’s are). Let us denote by I ′ ⊂ I the set of all such w’s. Then by (18.12) we have for allt in S+

〈t, (T ∗T )mt〉 =∑w∈I

tr(uwtuw∗t) ≥∑w∈I′

1 = card(I ′).

230

An elementary counting argument shows that card(I ′) = 〈(S∗S)mδe, δe〉 = τFn((S∗S)m). Thus weobtain (18.13). Therefore (recalling (11.10))

‖T ∗T‖ ≥ lim supm→∞

〈t, (T ∗T )mt〉1/m ≥ lim supm→∞

(τFn((S∗S)m))1/m = ‖S∗S‖,

so that we obtain ‖T‖ ≥ ‖S‖, whence (18.11) by (3.20).

Corollary 18.15.

(18.14) 2√n− 1 ≤ C(n).

Proof. Let (uj(m)) be as in (18.6). Let B = (⊕∑MNm)∞. Let U be a non trivial ultrafilter on N.

Let I ⊂ B be the (closed self-adjoint two sided) ideal formed of all x ∈ B such that limU ‖xi‖ = 0.Let q : B → B/I be the quotient morphism. Let uj = q((uj(m)). We claim that∥∥∥∑n

j=1uj ⊗ uj

∥∥∥min≤ supm 6=m′ ‖

∑n

1uj(m)⊗ uj(m′)‖.

Using the claim we deduce (18.14) from (18.11). To check the claim note that for any C∗-algebraA we have A⊗min B = (⊕

∑mA⊗min MNm)∞ and hence for any (aj) in An

‖∑n

1aj ⊗ uj‖A⊗min(B/I) ≤ limU ,m′ ‖

∑n

1aj ⊗ uj(m′)‖A⊗minMNm′

.

Indeed, since IdA ⊗ q : A ⊗min B → A ⊗min (B/I) is a contraction we observe that the left handside is ≤ supm′ ‖

∑n1 aj ⊗ uj(m′)‖min. Moreover, let γ ⊂ N be a subset that belongs to U . If we

multiply m′ 7→∑n

1 aj ⊗ uj(m′) by the indicator of γ and apply the same observation we find

‖∑n

1aj ⊗ uj‖A⊗min(B/I) ≤ supm′∈γ ‖

∑n

1aj ⊗ uj(m′)‖A⊗minMNm′

,

and since this holds for any γ the claim follows. (Note: the claim merely spells out the fact that thenatural morphism (A⊗minB)/(A⊗min I)→ A⊗min (B/I) is always contractive, see the discussionin §10.1) Applying the claim twice gives us

‖∑n

1uj ⊗ uj‖min ≤ limU ,m ‖

∑n

1uj(m)⊗ uj‖min ≤ limU ,m limU ,m′ ‖

∑n

1uj(m)⊗ uj(m′)‖min,

and obviously the last term is ≤ supm 6=m′ ‖∑n

1 uj(m)⊗ uj(m′)‖, which proves the claim.Alternate proof: Passing to a subsequence we may assume that (uj(m)) converges in distributionwhen m→∞. Then with the notation from Remark 18.10 we have

supm 6=m′

‖∑n

1uj(m)⊗ uj(m′)‖ ≥ ‖t‖min ≥

∥∥∥∑ vj ⊗ vj∥∥∥MV⊗minMV

and we again deduce (18.14) from (18.11).

The proof that C(n) ≤ 2√n− 1 is much more delicate. The first proof by Haagerup and

Thorbjørnsen in [119] was based on a fundamental limit theorem for Gaussian random matrices,which was a considerable strengthening of Theorem 12.24. We will use the following refinementdue to Collins and Male [56], valid for unitary matrices, which gives a more direct approach.

231

Theorem 18.16 ([56]). Let UN denote the group of all N × N unitary matrices (N ≥ 1). Let

U(N)1 , . . . , U

(N)n be a sequence of independent matrix valued random variables, each having as its

distribution the normalized Haar measure on UN . Let g1, · · · , gn be the free generators of the free

group Fn. For convenience we set U(N)0 = I (unit in UN ) and we denote by g0 the unit in Fn.

Then, for all k and for all a0, . . . , an in Mk, we have, for almost all ω

(18.15) limN→∞

∥∥∥∑n

0aj ⊗ U (N)

j (ω)∥∥∥

min=∥∥∥∑n

0aj ⊗ λFn(gj)

∥∥∥min

.

In particular, if a1, . . . , an are all unitary, for almost all ω

(18.16) limN→∞

∥∥∥∑n

1aj ⊗ U (N)

j (ω)∥∥∥

min= 2√n− 1.

The implication (18.15) ⇒ (18.16) follows from Remark 3.17 and (3.20).

Proof of Theorem 18.6. By (18.14) it suffices to show that C(n) ≤ 2√n− 1. Fix ε > 0. Obviously

it suffices to construct a sequence of n-tuples (uj(m))1≤j≤n | m ≥ 1 of unitary matrices (weemphasize that (uj(m))1≤j≤n is assumed to be an n-tuple of matrices of size Nm ×Nm) such that,for any integer p ≥ 1, we have

(18.17) sup1≤m6=m′≤p

∥∥∥∑uj(m)⊗ uj(m′)∥∥∥

min< 2√n− 1 + ε.

We will construct this sequence and the sizes Nm by induction on p. Assume that we already knowthe result up to p. That is, we already know a family (uj(m))1≤j≤n | 1 ≤ m ≤ p formed of pn-tuples satisfying (18.17). We need to produce an additional n-tuple (uj(p+ 1))1≤j≤n of unitarymatrices (possibly of some larger size Np+1 × Np+1) such that (18.17) still holds for the enlargedfamily (uj(m))1≤j≤m | 1 ≤ m ≤ p+1 formed of one more n-tuple. By (18.16), for any 1 ≤ m ≤ p,we have for almost all ω

limN→∞

∥∥∥∑n

1uj(m)⊗ U (N)

j (ω)∥∥∥

min= 2√n− 1.

Hence, if N is chosen large enough, we can ensure that, for all 1 ≤ m ≤ p simultaneously, we canfind ω such that ∥∥∥∑n

1uj(m)⊗ U (N)

j (ω)∥∥∥

min< 2√n− 1 + ε.

But then, if we set Np+1 = N and uj(p+ 1) = U(N)j (ω), the extended family (uj(m))1≤j≤n | 1 ≤

m ≤ p+ 1 clearly still satisfies (18.17). This proves C(n) ≤ 2√n− 1.

Fix ε > 0. Actually, by concentration of measure arguments (see e.g. [7, §4.4]), there is asequence of sizes N1 < N2 < · · · , for our random unitary matrices, such that

P

ω ∈ Ω | supm6=m′

∥∥∥∥∥∥n∑j=1

U(Nm)j (ω)⊗ U (Nm′ )

j (ω)

∥∥∥∥∥∥min

≤ 2√n− 1 + ε

> 1− ε.

Thus, provided the sizes grow sufficiently fast, with close to full probability a random choice yields(18.6) with almost the best C. We refer to [211, 212] for more details and for related estimates.

232

18.3 Exactness is not preserved by extensions

In this section we indicate a quick way to produce an example of a separable non-exact C∗-algebraA and a closed ideal I ⊂ A such that both I and the quotient C = A/I are exact. In this situationone usually says that A is an extension of C by I. Thus exactness is not stable by extension. WhenI and C are both nuclear, then A is nuclear (see Corollary 8.18). Thus in sharp contrast, nuclearityis preserved by extensions.

Recall B = (⊕∑

N≥1MN )∞. For any V ∈ B, for notational convenience, we will denote in this

section by V (N) ∈MN the N ’th coordinate of V .

Remark 18.17. Let I0 ⊂ B be the subset of all b = (b(N)) ∈ B such that limN ‖b(N)‖ = 0. Let A ⊂ Bbe a unital C∗-subalgebra. Let q : A→ A/A∩I0 be the quotient map. Then for any b0, · · · , bn ∈ Aand a0, · · · , an in Mk (n, k ≥ 1) we have

(18.18) lim supN→∞

∥∥∥∑n

0aj ⊗ b(N)

j

∥∥∥Mk(MN )

= ‖∑n

0aj ⊗ q(bj)‖Mk(A/A∩I0).

To justify this, let Q : B→ B/I0 be the quotient morphism. Since

(18.19) A/A ∩ I0 = A/ ker(Q|A) ' Q(A) ⊂ Q(B) ' B/I0,

it suffices to check (18.18) in the case A = B, for which the easy verification is left to the reader.

Kirchberg gave in [155] the first example of a non-exact extension of an exact C∗-algebra by thealgebra of compact operators on `2. Our example, based on the random unitary matrix model, willbe deduced from Theorem 18.16, but first we analyze the underlying deterministic matrix model.

Remark 18.18. Let V0 = 1B and let g0 be as before the unit in Fn. Let V1, . . . , Vn be unitaries inB that form a matrix model for MFn in the sense of §12.5. This implies that for any nontrivialultrafilter U on N, denoting as before by qU : B → MU the quotient map, the correspondenceλFn(gj) 7→ qU (Vj) (0 ≤ j ≤ n) extends to an isometric normal ∗-homomorphism embedding MFninto MU . In particular this implies that the following holds:

(18.20) ∀k ≥ 1 ∀aj ∈Mk ‖∑n

0aj ⊗ λFn(gj)‖Mk(MFn ) =

∥∥∥∑n

0aj ⊗ qU (Vj)

∥∥∥Mk(MU )

.

By Proposition 9.7 the latter property (18.20) implies conversely that the previous correspondenceextends to an isometric ∗-homomorphism from C∗λ(Fn) into MU .

Theorem 18.19. Let V0 = 1 and let V1, . . . , Vn be unitaries in B. Assume that they satisfy

(18.21) ∀k ≥ 1 ∀aj ∈Mk ‖∑n

0aj ⊗ λFn(gj)‖Mk(C∗λ(Fn)) = lim sup

N→∞

∥∥∥∑n

0aj ⊗ Vj(N)

∥∥∥Mk(MN )

.

Let A ⊂ B be the (unital) C∗-algebra generated by V0, V1, . . . , Vn, and let I = A ∩ I0. ThenA/I ' C∗λ(Fn), so that A/I and I are exact but A is not exact.

Proof. First observe that since I ⊂ I0 and I0 is obviously nuclear (e.g. because it has the CPAP, seeCorollary 7.12) it is immediate that I is exact (see Remarks 10.4 and 10.5). Let q : A→ A/I be thequotient morphism. Let E = span[λFn(gj) | 0 ≤ j ≤ n] ⊂ C∗λ(Fn). Similarly let Eq = span[q(Vj) |0 ≤ j ≤ n] ⊂ A/I, and let u : E → Eq be the linear map defined by u(λFn(gj)) = q(Vj) (0 ≤ j ≤ n).By (18.21) and (18.18) u is a unital completely isometric isomorphism. By Proposition 9.7 it extendsto a ∗-isomorphism π : C∗λ(Fn)→ A/I. Thus A/I is exact by Remark 10.21. As observed in (18.19)A/I ⊂ B/I0 and since I0 ⊂ IU we have obviously a canonical map rU : B/I0 → B/IU = MU , such

233

that qU = rUQ. Let σ : C∗(Fn) → A be the ∗-homomorphism defined by σ(UFn(gj)) = Vj(0 ≤ j ≤ n). Let J : A → B denote the canonical inclusion. Let ψ = qUJ : A → MU . Note thefactorization through B:

ψ : AJ−→B qU−→MU .

Assume for contradiction that A was exact. Then J would be (min→ max)-tensorizing by Corollary10.8, and hence so would be ψ = qUJ : A → MU . It would follow (see the diagram at the end ofthe proof) that the ∗-homomorphism Ψ : C∗(Fn)→MU defined by Ψ(UFn(gj)) = qU (Vj) (that can

be factorized as C∗(Fn)σ−→A ψ−→MU ) would also be (min → max)-tensorizing. But we claim that

this is not true when n ≥ 2, a contradiction that shows that A is not exact. To check the claim,observe that

‖∑n

0qU (Vj)⊗Ψ(UFn(gj))‖MU⊗maxMU

= ‖∑n

0qU (Vj)⊗ qU (Vj)‖MU⊗maxMU

= n+ 1

where the last equality follows from (18.3), and also since qU (Vj) = rU |A/I(q(Vj))

‖∑n

0qU (Vj)⊗ UFn(gj)‖MU⊗minC∗(Fn) ≤ ‖

∑n

0q(Vj)⊗ UFn(gj)‖A/I⊗minC∗(Fn)

= ‖∑n

0u(λFn(gj))⊗ UFn(gj)‖A/I⊗minC∗(Fn)

= ‖∑n

0λFn(gj)⊗ UFn(gj)‖C∗λ(Fn)⊗minC∗(Fn)

=√n,

where the last inequality follows from (3.21) and (3.13). Since 2√n < n+1 this proves the claim.

The preceding proof is summarized by the following diagram:

C∗(Fn)σ // A

J //

q

B qU //

Q

MU

C∗λ(Fn) ' A/I // B/I0

rU<<

Corollary 18.20. Let 1 = U(N)0 , U

(N)1 , . . . , U

(N)n be the random unitaries in Theorem 18.16, as-

sumed defined on a probability space (Ω,P). Let B = (⊕N≥1MN )∞. Let

Uj(ω) = (U(N)j (ω))N≥1 ∈ B.

Let Aω ⊂ B be the (unital) C∗-algebra generated by U0(ω), . . . , Un(ω), and let Iω = Aω ∩ I0. Thenfor almost all ω we have

(18.22) Aω/Iω ' C∗λ(Fn),

and Aω/Iω and Iω are exact but Aω is not.

Proof. By the preceding statement, it suffices to show that (U(N)j (ω)) satisfies (18.21) for almost

all ω. To check the latter assertion just observe that there is Ω′ ⊂ Ω with P(Ω′) = 1 such that forany ω ∈ Ω′ we have (18.15) for all k and all a0, · · · , an ∈ Mk. Indeed, we may restrict by densityto a0, · · · , an with rational entries, and then Ω′ appears as an intersection of a countable family ofevents each having full probability by (18.15).

234

18.4 A continuum of C∗-norms on B⊗ B

The preceding results from [141] only showed that there is more than one C∗-norm on B ⊗B orB⊗B. In [193] N. Ozawa and the author proved that there is actually a continuum of distinct such

norms. The proof for B ⊗ B is very simple and it yields a family of (maximal) cardinality 22ℵ0 ofdistinct C∗-norms. We include it in this section. Curiously, there does not seem to be a simpleargument to transplant the result to B ⊗B (as we did for min 6= max in Corollary 18.12). Thelatter case is more delicate, and we refer the reader to [193] for full details.

Let (Nm) be any sequence of positive integers tending to ∞ and let

B = (⊕∑

mMNm)∞.

Theorem 18.21. There is a family of cardinality 22ℵ0 of mutually distinct (and hence inequivalent)C∗-norms on B ⊗M for any von Neumann algebra M that is not nuclear.

Remark 18.22. Assuming M ⊂ B(`2) non-nuclear, we note that the cardinality of B(`2) and henceof B(`2) ⊗M is c = 2ℵ0 , so the set of all real valued functions of M ⊗ B(`2) into R has the same

cardinal 22ℵ0 as the set of C∗-norms.

Fix n > 2, and let [u1(m), . . . , un(m)] be a sequence of n-tuples of unitary Nm ×Nm matricessatisfying (18.6).By compactness (see Lemmas 18.7 and 18.8) we may and do assume (after passing to a subsequence)that the n-tuples [u1(m), . . . , un(m)] converge in distribution (i.e. in moments) to an n-tuple[u1, . . . , un] of unitaries in a von Neumann algebra M equipped with a faithful normal trace τ .Then, if U is any nontrivial ultrafilter, we can take for (M, τ) the ultraproduct (MU , τU ), and theresulting limit distribution along U does not depend on U .

For any subset s ⊂ N and any u ∈ B we denote by u[s] = ⊕mu[s](m) ∈ B the element of Bdefined by u[s](m) = u(m) if m ∈ s and u[s](m) = 0 otherwise.

We denote byπU : B →MU (or πU : B →MU )

the natural quotient map.Recall that if U ,V are ultrafilters on N, then U 6= V if and only if there are disjoint subsets

s ⊂ N and s′ ⊂ N with s ∈ U and s′ ∈ V (see Remark 26.5). In that case we have

(18.23) ∀u ∈ B πU (u[s′]) = πV(u[s]) = 0

Lemma 18.23. Let U 6= V be ultrafilters on N. Consider disjoint subsets s ⊂ N and s′ ⊂ N withs ∈ U and s′ ∈ V, and let

t(s, s′) =∑n

k=1uk[s]⊗ uk[s′] ∈ B ⊗B.

Then‖t(s, s′)‖B⊗minB

≤ C and ‖[πU ⊗ πV ](t(s, s′))‖MU⊗maxMV= n.

Proof. We have obviously

‖t(s, s′)‖min = sup(m,m′)∈s×s′

∥∥∥∑uk(m)⊗ uk(m′)∥∥∥

235

hence ‖t(s, s′)‖min ≤ C. We now turn to the max tensor product.Let uk = πU (uk[s]) and vk = πV(uk[s

′]) so that we have

‖[πU ⊗ πV ](t(s, s′))‖MU⊗maxMV=∥∥∥∑uk ⊗ vk

∥∥∥MU⊗maxMV

.

Since we assume that [u1(m), . . . , un(m)] converges in distribution, (u1, . . . , un) and (v1, . . . , vn)must have the same distribution relative respectively to τU and τV . Arguing as for (18.10) weobtain ‖[πU ⊗ πV ](t(s, s′))‖max = n.

For any nontrivial ultrafilter U on N we denote by αU the norm defined on B ⊗B by

∀t ∈ B ⊗B αU (t) = max‖t‖B⊗minB, ‖[πU ⊗ Id](t)‖MU⊗maxB

.

Theorem 18.24. There is a family of cardinality 22ℵ0 of mutually distinct (and hence inequivalent)C∗-norms on B⊗B. More precisely, the family αU indexed by nontrivial ultrafilters on N is sucha family on B ⊗B.

Proof. Let (U ,V) be two distinct nontrivial ultrafilters on N. Let s ⊂ N and s′ ⊂ N be disjointsubsets such that s ∈ U and s′ ∈ V. By Lemma 18.23 we have

αU (t(s, s′)) ≥ ‖[πU ⊗ πV ](t(s, s′))‖MU⊗maxMV= n

but since (πV ⊗ Id)(t(s, s′)) = 0 by (18.23) we have αV(t(s, s′)) ≤ C < n. This shows αU and αVare different, and hence (automatically for C∗-norms) inequivalent. Lastly, it is well known (see

e.g. [58, p. 146]) that the cardinality of the set of nontrivial ultrafilters on N is 22ℵ0 .

Proof of Theorem 18.21. If M is not nuclear, by Theorem 12.29 there is an embedding B ⊂ M .Moreover, since B is injective, there is a conditional expectation from M to B, which guaranteesthat, for any A, the max norm on A⊗B coincides with the restriction of the max norm on A⊗M(see Corollary 4.18 or Proposition 7.19). Thus we can extend αU to a C∗-norm αU on B ⊗M bysetting

∀t ∈ B ⊗M αU (t) = max‖t‖B⊗minM, ‖[πU ⊗ Id](t)‖MU⊗maxM

.

Since αU = αU on B ⊗B, this gives us a family of distinct C∗-norms on B ⊗M . Since B ' B, wecan replace B by B if we wish.

Remark 18.25. It is easy to see that Theorem 18.21 remains valid for any choice of the sequence(Nm) and in particular it holds if Nm = m for all m, i.e. for B = B.

18.5 Notes and Remarks

The main sources for §18.1 are [141] with the simplifications brought by [209]. The first proof in[141] that (B,B) is not nuclear was more indirect. It went through first proving that if B had theLLP then the set OSn of n-dimensional (n ≥ 3) operator spaces would be separable for the metricdcb (see §20), and (after some more topological considerations) that would contradict a certainoperator space version of Grothendieck’s theorem. A second proof was proposed in a revision ofthe same paper [141]. The latter proof used property (T) groups to show that C(n) < n for n ≥ 3,and deduced from that the non-separability of (OSn, dcb) (see §20 for more on this). Later on, A.Valette observed that the Lubotzky-Philips-Sarnak results were exactly what was needed to provethat C(n) ≤ 2

√n− 1 for any n = p+1 with p prime. Finally, the paper [120] proved using random

matrices that this bound remains valid for any n ≥ 2, and hence (by the easy lower bound in [206])that C(n) = 2

√n− 1 for all n ≥ 2. §18.4 comes from [193].

236

19 Other proofs that C(n) < n. Quantum expanders

In what precedes, we used random matrices to show that C(n) < n. In this section, we willdescribe a different way to prove the latter fact using more explicit examples based on the theoryof “expanders” or “expanding graphs”. Actually, we only use graphs that are Cayley graphs offinite groups. We will see that the more recent notion of “quantum expander” is particularly welladapted for our purposes. We should warn the reader that there is also a notion of quantum graphthat seems to have little to do with quantum expanders.

19.1 Quantum coding sequences. Expanders. Spectral gap

To prove that C(n) < n we must produce a sequence of n-tuples (u(m))m≥1 of unitary matrices ofthe same size Nm such that (18.6) holds for some C < n. Following (and modifying) the terminologyfrom [254] we call such a sequence a “quantum coding sequence” of degree n. See §19.4 for anexplanation of our terminology.

To introduce expanders we first recall some notation.Let π : G→ B(Hπ) be a unitary representation. Let

H invπ = ξ ∈ Hπ | π(t)ξ = ξ ∀t ∈ G

be the set of π-invariant vectors. Let S ⊂ G be a finite subset generating G (i.e. G is the smallestsubgroup containing S). We denote

(19.1) ε(π, S) = 1− |S|−1‖∑

s∈Sπ(s)P

Hinvπ⊥‖.

Definition 19.1. Let (Gm, Sm) be a sequence of finite groups with generating sets Sm such that|Sm| = n for all m ≥ 1. The sequence of associated Cayley graphs is called “an expander” or an“expanding family” (the terminology is not so well established) if |Gm| → ∞ and

infm≥1

ε(Gm, Sm) > 0.

The notion of “expanding graph” has had a major impact far beyond graph theory. For ourpurposes, we will only discuss Cayley graphs of groups. Given a group G with a finite symmetricset of generators S ⊂ G, the associated Cayley graph is defined as having G as its vertex set andhaving as edges the pairs (x, y) in G2 such that y−1x ∈ S.

For a general group G the spectral gap ε(G,S) can be defined by setting

(19.2) ε(G,S) = inf ε(π, S)

with the infimum running over all unitary representations π of G. Equivalently, this is the spectralgap of the universal representation. As anticipated in Proposition 17.3, ε(G,S) > 0 characterizesproperty (T) (see §19.3), and we will see in §19.3 that certain property (T) groups lead to expandingfamilies of graphs.

For the moment, let us assume that G is finite and that |S| = n. In that case we will show thatwe may restrict consideration to π = λG and we have

(19.3) 1− ε(G,S) = ‖n−1∑

s∈SλG(s)|1⊥‖

where 1 denotes the constant function on G (i.e. the element ξ ∈ `2(G) such that ξ(t) = 1 forall t ∈ G). The latter is an eigenvector for the eigenvalue 1 for the so-called Markov operator

237

n−1∑

s∈S λG(s). The number ε(G,S) measures the gap between that extreme eigenvalue and therest of the spectrum.Since the only vectors invariant under λG are those in C1, we have (using the notation (19.1))

ε(G,S) = ε(λG, S).

Indeed, as is well known, when |G| <∞, λG decomposes as a direct sum of a family formed of allthe irreducible representations (each with the same multiplicity as its dimension). Let G denotethe set of irreducible representations on G and let T be the trivial representation on G. As usualwe identify two representations if they are unitarily equivalent. With this notation we may write

λG '⊕π∈G

π

and also

(19.4) λG|1⊥ '⊕

π∈G\T

π.

Therefore

(19.5) 1− ε(G,S) = supπ∈G\T

|S|−1‖∑

s∈Sπ(s)‖.

Remark 19.2. For any finite group G we have

(19.6) ε(G,S) = infε(π, S),

where the infimum runs over all unitary representations π of G without invariant vectors.Indeed, by decomposing π into irreducibles the infimum remains unchanged if we restrict it toπ ∈ G \ T. Thus (19.6) follows from (19.5).

Remark 19.3. By the Schur Lemma 26.69 and (19.6), for any pair π 6' σ ∈ G we have

(19.7) |S|−1‖∑

s∈Sπ(s)⊗ σ(s)‖ ≤ 1− ε(G,S).

Proposition 19.4. Let π | π ∈ T be a finite set of distinct irreducible representations of G on acommon N -dimensional Hilbert space. Let ε = ε(G,S) and n = |S|. Then |T | ≤ (1 + 2/

√ε)2nN2

.

Proof. By (19.7), for any π 6= σ ∈ T we have |∑

s∈S tr(π(s)∗σ(s))| ≤ n(1− ε), and hence

∀π 6= σ ∈ T (n−1∑

s∈Str|π(s)− σ(s)|2)1/2 ≥

√2ε.

Thus we have a set of |T | unit vectors in a Hilbert space of dimension nN2 (and hence Euclideandimension 2nN2) that are mutually at distance ≥

√2ε. The proposition follows by a well known

volume argument: the open balls centered at these points with radius r =√

2ε/2 being disjoint,the volume of their union is = |T |r2nN2

vol(B) where B is the Euclidean ball of dimension 2nN2,and the latter union being included in a ball of radius 1 + r has volume at most (1 + r)2nN2

vol(B).This implies |T | ≤ (1 + 1/r)2nN2

.

Corollary 19.5. Let (Gm, Sm)m≥1 be an expanding family with |Sm| = n for all m ≥ 1. Let Nm

be the largest dimension dπ of a representation π ∈ Gm. Then Nm →∞.

Proof. Fix a number N . By the preceding proposition,∑

π∈Gm,dπ≤N d2π remains bounded. Since∑

π∈Gm d2π = |Gm| → ∞, when m is large enough we must have dπ > N for some π ∈ Gm. In other

words Nm > N when m is large enough.

238

19.2 Quantum expanders

Let H,K be Hilbert spaces. Let S2(K,H) denote the Hilbert space of Hilbert-Schmidt operatorsx : K → H equipped with the norm ‖x‖S2 = (tr(x∗x))1/2 = (tr(xx∗))1/2. Let u = (uj)1≤j≤n ∈B(H)n and v = (vj)1≤j≤n ∈ B(K)n. We denote by Tu,v : S2(K,H) → S2(K,H) the mappingdefined by

Tu,v(x) = n−1∑n

1u∗jxvj .

Since K ' K∗ (canonically), we may identify K⊗2H with S2(K,H). With this identification, Tu,vcorresponds to n−1

∑n1 v∗j ⊗ u∗j which has the same norm as n−1

∑n1 uj ⊗ vj (see Proposition 2.11)

so that‖Tu,v‖ = n−1‖

∑n

1uj ⊗ vj‖

where the last norm is in B(K)⊗min B(H) or equivalently in B(K ⊗2 H).Let us denote by L2(τN ) the space MN equipped with the norm

‖x‖L2(τN ) = (N−1tr(x∗x))1/2

associated to the normalized trace τN on MN (namely τN (x) = N−1tr(x)). Except for the nor-malization of the trace this is the same as S2(K,H) when K = H = `N2 . Therefore for anyv = (vj)1≤j≤n ∈Mn

N we have

‖Tv,v‖ = n−1‖∑

vj ⊗ vj‖MN⊗minMN= sup ‖n−1

∑v∗j ξvj‖L2(τN ) | ‖ξ‖L2(τN ) ≤ 1.

Note that for any ξ, η ∈ BL2(τN ) we have

(19.8) |n−1∑

τN (v∗j ξ∗vjη)| = |〈n−1

∑v∗j ξvj , η〉L2(τN )| ≤ ‖Tv,v‖.

Let T : L2(τN )→ L2(τN ) be a linear map with ‖T‖ = 1 such that T (I) = I and T ∗(I) = I whereI ∈MN denotes the identity matrix, so that 1 is an eigenvalue (with eigenvector I) of T and I⊥ isinvariant under T . The “spectral gap” of T is defined as

(19.9) e(T ) = 1− ‖T|I⊥‖.

Let u be an n-tuple in UN . We then set

(19.10) ε(u) = e(Tu,u).

Equivalently ε(u) is the largest number ε ≥ 0 such that for any ξ, η in BL2(τN ) with τN (ξ) =τN (η) = 0 we have

(19.11) |n−1τN

(∑n

1u∗jξ∗ujη

)| ≤ 1− ε.

Lemma 19.6. Let u = (uj)1≤j≤n ∈ UnN . Then for any k ≤ N and any v = (vj)1≤j≤n ∈ Mnk we

have

(19.12) ‖Tu,v‖ ≤ (k/N + 1− ε(u))1/2‖Tv,v‖1/2.

Proof. For simplicity we replace vj by vj ⊕ 0 ∈ MN . Thus we assume that the vj ’s are N × Nmatrices for which there is an orthogonal projection P ∈MN of rank k such that

vjP = Pvj = vj for all j.

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To prove (19.12) it suffices to show that for any ξ, η in BL2(τN ) we have∣∣∣n−1∑n

1τN (u∗jξ

∗vjη)∣∣∣ ≤ (k/N + 1− ε(u))1/2‖Tv,v‖1/2.

Since we may replace ξ, η by Pξ, Pη, we may assume ξ, η of rank ≤ k. Let ξ = U |ξ| and η = V |η|be the polar decomposition. Using the identity

τN (u∗jξ∗vjη) = τN

((|η|1/2u∗j |ξ|1/2) (|ξ|1/2U∗vjV |η|1/2)

)and Cauchy-Schwarz we find by (19.8)

(19.13)∣∣∣n−1

∑n

1τN (u∗jξ

∗vjη)∣∣∣ ≤ ∣∣∣n−1

∑n

1τN (|η|u∗j |ξ|uj)

∣∣∣1/2 ‖Tv,v‖1/2.Now let ξ′ = |ξ|−τN (|ξ|) and η′ = |η|−τN (|η|) so that ξ′, η′ ∈ BL2(τN ) but now τN (ξ′) = τN (η′) = 0.We have

(19.14) n−1∑n

1τN (|η|u∗j |ξ|uj) ≤ I + II

where I = τN (|ξ|)τN (|η|) and II = n−1∑n

1 τN (η′u∗jξ′uj). Since |ξ| and |η| have rank ≤ k (and

τ(P ) = k/N)) we have τN (|ξ|) ≤ τN (|ξ|2)1/2(k/N)1/2 and similarly for |η|. It follows that |I| ≤ k/N .Now using the definition of the spectral gap ε(u) in (19.11) we find |II| ≤ 1− ε(u). Putting thesebounds together with (19.13) and (19.14) we obtain (19.12).

In analogy with the theory of expanding graphs (or expanders) the following definition wasrecently introduced:

Definition 19.7. Let (Nm)m≥1 be a non-decreasing sequence of integers. For each m, consider

u(m) = (u1(m), ..., un(m)) ∈ UnNm .

The sequence (u(m))m≥1 is called a quantum expander if Nm → ∞ and infm≥1 ε(u(m)) > 0. Wecall n the degree of (u(m))m≥1.

The link between quantum expanders and the constant C(n) goes through the following.

Proposition 19.8. Let (u(m))m≥1 be a quantum expander. There is a subsequence of (u(m))m≥1

that is a quantum coding sequence.

Proof. Assume ε(u(m)) ≥ ε > 0 for all m. Just choose the subsequence m1 < m2 < ... so that

[Nmk/Nmk+1] + 1− ε ≤ 1− ε/2

for all k ≥ 1. Then by Lemma 19.6 we obtain (18.6) with C = (1− ε/2)n, which means (u(m))m≥1

is a quantum coding sequence.

Corollary 19.9. To show that C(n) < n, it suffices to know that there is a quantum expander ofdegree n.

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Remark 19.10 (From expanders to quantum expanders). Let (Gm, Sm)m≥1 be an expander in thesense of Definition 19.1. Recall |Sm| = n for any m ≥ 1. Let πm : Gm → B(Hm) be irreduciblerepresentations such that dim(Hm)→∞ (see Corollary 19.5). Then, by Schur’s Lemma 26.69, therepresentation [πm ⊗ πm]|I⊥ has no nonzero invariant vector. This implies that its decompositioninto irreducible components has only non trivial irreducible representations. By (19.4), the latterare all contained in the restriction of λGm to 1⊥, and hence

n−1‖∑

s∈Sm[πm(s)⊗ πm(s)]|I⊥‖ ≤ 1− ε(Gm, Sm).

In other wordsε((πm(s))s∈Sm) ≥ ε(Gm, Sm),

so that the sequence of n-tuples (πm(s))s∈Sm (m ≥ 1) forms a quantum expander.

19.3 Property (T)

We already introduced groups with property (T) in Definition 17.1. We will now show that theexistence of such groups leads to that of expanders, from which that of quantum expanders andquantum coding sequences follows. Compared with the random approach, the advantage of thismethod is that it produces explicit examples.

Proposition 19.11. A discrete group G generated by a finite subset S ⊂ G containing the unithas property (T) if and only if it has a non-zero spectral gap ε(G,S) (as defined in (19.2)) orequivalently if there is ε > 0 such that for any unitary representation π : G→ B(Hπ) we have

(19.15) ε(π, S) ≥ ε.

Proof. This is but a restatement of Proposition 17.3.

Lemma 19.12. Let G,S be as in Proposition 19.11, with property (T). Let ε > 0 be as in (19.15).

(i) For any irreducible unitary representation π : G→ B(H) with dim(H) <∞ we have

(19.16) ε((π(s))s∈S) ≥ ε.

(ii) Let σ be another finite dimensional irreducible unitary representation that is not unitarilyequivalent to π, then

(19.17) |S|−1‖∑

s∈Sπ(s)⊗ σ(s)‖ ≤ 1− ε.

Proof. (i) Let ρ(t) = π(t) ⊗ π(t). Observe that H invρ = CI by the irreducibility of π (see §26.21).

Therefore by (19.15)

(19.18) |S|−1‖∑

s∈Sπ(s)⊗ π(s)|I⊥‖ ≤ 1− ε.

Equivalently we have (i).(ii) By Schur’s Lemma 26.69, the representation π ⊗ σ does not have any invariant vector ξ 6= 0.Thus (19.17) follows from (19.15) applied with π ⊗ σ in place of π.

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Proposition 19.13. Let G be a property (T) group with S as in Proposition 19.11. Let n = |S|and let ε0 = ε(G,S). Assume that G admits a sequence (πm)m≥1 of finite dimensional distinctirreducible unitary representations with dimension tending to ∞. Then the sequence

(πm(s))s∈S | m = 1, 2, ...

is both a quantum expander and a quantum coding sequence and C(n) ≤ (1− ε0)n.

Proof. Let S = t0, ..., tn−1 with t0 = 1. Let Nm = dim(Hπm). Let

uj(m) = πm(tj).

For any m 6= m′ we have πm 6= π′m. By (19.17) this implies

n−1‖∑n−1

0uj(m)⊗ uj(m′)‖ ≤ 1− ε0,

and hence we have a quantum coding sequence. By (19.16) or (19.18), (πm(s))s∈S is a quantumexpander.

Remark 19.14 (From property (T) to expanders). In the preceding situation, assume that thegroup G admits a sequence of finite quotient groups Gm with quotient maps denoted by qm :G → Gm. Let Sm = qm(S). We assume that |Sm| = n. Let σm : Gm → B(Hm) be a unitaryrepresentation without invariant unit vector (e.g. a non trivial irreducible one) (m ≥ 1). Then πm =σmqm is a unitary representation without invariant unit vector on G. Moreover, ‖n−1

∑S πm‖ =

‖n−1∑

Smσm‖, and hence ε(σm, Sm) ≥ ε(G,S). Since this holds for any such σm on Gm this implies

ε(Gm, Sm) ≥ ε(G,S). Therefore if |Gm| → ∞ the sequence (Gm, Sm) is an expanding family (i.e.“an expander”). This shows that we can deduce the existence of expanding families from that ofa property (T) group with the required properties. We give an example in the next remark. Notethat for the present remark we could content ourselves with property (τ) for which we refer thereader to [173].

Remark 19.15 ((SL3(Zp)) is an expander). By Remarks 19.14 and 19.10, to prove that C(n) < n(or to produce quantum expanders) it suffices to produce a group G with property (T) admitting asequence of distinct finite dimensional irreducible unitary representations with unbounded dimen-sions. The classical example for this phenomenon is G = SL3(Z) (or SLd(Z) for d ≥ 3). For anyprime number p let Zp = Z/pZ. Recall this is a field with p elements. The group SL3(Zp) is afinite quotient of G. Indeed, we have a natural homomorphism qp : SL3(Z)→ SL3(Zp) that takesa matrix [aij ] to the matrix [aij ] where a ∈ Z/pZ denotes the congruence equivalence class of a ∈ Zmodulo p. It is known that this maps SL3(Z) onto SL3(Zp). This follows for instance from thewell known fact that, for any field k, SL3(k) is generated by the set Sk formed of the unit and thematrices with 1 on the diagonal and only one nonzero entry elsewhere equal to 1. When k = Z/pZ,it is obvious that such matrices are in the range of the preceding homomorphism. Therefore thelatter is onto SL3(Zp). We will use S = SZ so that n = |S| = 6. Clearly |qp(S)| = |S| = 6 if p > 1.Thus Remark 19.14 shows that the sequence (SL3(Zp), qp(S)) indexed by prime numbers p > 1 isan expanding family (i.e. “an expander”).

Actually, by Remark 19.14 for any finite generating set S in SL3(Z) the sequence (SL3(Zp), qp(S))indexed by large enough prime numbers p > 1 is an expanding family (i.e. “an expander”). Indeed,by taking p large enough we can clearly ensure that qp is injective on S and hence |qp(S)| = |S|.

The irreducible unitary representations on SL3(Zp) have unbounded dimensions when p →∞. This can be deduced from the property (T) of SL3(Z) using the same idea as for Corollary

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19.5. More explicitly, consider for instance the action of SL3(Zp) on the set of “lines” L(p) inZ3p, or equivalently the set of 1-dimensional subspaces in the vector space Z3

p (over the field Zp).By standard linear algebra, SL3(Zp) acts transitively on the latter set and this action defines aunitary representation πp of SL3(Zp) on `2(L(p)) that permutes the canonical basis vectors. Sincethe action on L(p) is transitive, the constant functions on L(p) are the only invariant vectors.Furthermore, again by linear algebra, the action is actually bitransitive, and hence (see Lemma26.68) the restriction π0

p = πp|1⊥ to the orthogonal of constant functions is irreducible, and ofcourse its dimension, equal to |L(p)| − 1 = p2 + p (see Remark 24.29), tends to ∞ when p → ∞.Thus we conclude by Remark 19.10 that for any finite unital generating set S in SL3(Z) with|S| = n the sequence of n-tuples (π0

p(s))s∈S indexed by large enough primes p > 1 is a quantumexpander.

Remark 19.16. Since it is known (see [251]) that SL3(Z) is generated by a pair of elements (togetherwith the unit), we may take |S| = 3. The preceding remark then gives us C(3) < 3. Since it canbe shown (exercise) that C(2) = 2 this is optimal.

19.4 Quantum spherical codes

We would like to motivate the terminology that we adopted to emphasize the analogy with certainquestions in coding theory. In [62, Ch. 9], Conway and Sloane define a spherical code SC ofdimension n, size k and minimal angle 0 < θ < π/2 as a set of k points of the unit sphere in Rnwith the property that

(19.19) ∀x, y ∈ SC, x 6= y x.y ≤ cos θ.

They discuss the reasons why it is of interest to find the maximal size A(n, θ) of such a code, andgive estimates for it.A similar variant of that problem is the search for a maximal set of vectors ξ(m) in the sphereof radius n1/2 in `n2 with coordinates all unimodular (e.g. equal to ±1) that are such thatsupm 6=m′ |〈ξ(m), ξ(m′)〉| < C for some C < n.

Such (finite) sequences are useful in coding theory: the family ξ(m) itself can be thought ofas a code. If we know that the message (i.e. a length n sequence of ±1’s) consists of one of theξ(m)’s then even if there are erroneous digits (=coordinates) but fewer than (n−C)/8 of them, wecan recognize which ξ(m) was sent in the message. (We leave the easy verification as an exercise).It is thus quite useful to have a number as large as possible of vectors ξ(m) given n and C < n.This becomes all the more useful when C/n is small, typically when C = δn with 0 < δ < n. Forthis (classical) problem, of course there can only be finitely many such ξ(m)’s since the sphere iscompact, but in the quantum version, the effect of compactness diminishes when the matrix sizeN → ∞, and the significant problem becomes to produce an infinite sequence such as the oneswe call quantum coding sequences. One can also fix the matrix size N and try to estimate (as afunction of N,n,C) the maximal number of n-tuples of N × N -unitaries that are separated as in(18.6) for some fixed C < n. Now the main point is the asymptotic behavior of the latter numberwhen N →∞ as in the forthcoming Theorem 19.17.

To illustrate better the analogy that we wish to emphasize, we revise our notation as follows.For any x = (x1, · · · , xn) and y = (y1, · · · , yn) in Mn

N we set x = (x1, · · · , xn) ∈MNn

and denote

x.y =∑n

1xj ⊗ yj ∈MN ⊗MN x.y =

∑n

1xj ⊗ yj ∈MN ⊗MN .

We view the set of x’s such that ‖x.x‖min ≤ 1 as a quantum analogue of the Euclidean unit sphere.Note that if x = (xj) ∈ n−1/2UnN then ‖x.x‖min = 1.

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When ‖x.x‖min ≤ 1 and ‖y.y‖min ≤ 1 we say that x, y are δ-separated if ‖∑n

1 xj⊗yj‖min ≤ 1−δ.This is analogous to the preceding separation condition (19.19) for spherical codes with cos θ = 1−δ.

The next result is a matricial analogue of some of the known estimates for spherical codes.We interpret it as an upper estimate of the size of quantum spherical codes of dimension n, angleθ = arccos(1− δ) and (this is the novel parameter) matrix size N . In addition, it gives us a ratherlarge number of δ-separated quantum expanders.

Theorem 19.17 ([212]). There are absolute constants β > 0 and δ > 0 such that for each 0 < ε < 1and for all sufficiently large integers n and N , more precisely such that n ≥ n0 and N ≥ N0 withn0 depending on ε, and N0 depending on n and ε, there is a subset T ⊂Mn

N with cardinal

|T | ≥ expβnN2,

such that∀x ∈ T (n1/2xj) ∈ UnN and ε(x) ≥ 1− (2

√n− 1/n+ ε)

∀x 6= y ∈ T ‖x.y‖min ≤ (1− δ).

Note that 2√n− 1 < n for all n ≥ 3 and hence for 0 < ε < 1 small enough 1−(2

√n− 1/n+ε) > 0.

Remark 19.18. As for lower estimates, by the same volume argument as for Proposition 19.4, forany set T with the property in Theorem 19.17 we have |T | ≤ (1 + 2/

√δ)2nN2 ≤ exp (4nN2/

√δ).

We refer to [212] for the proof, which makes crucial use of the following result of Hastings [132]:

Theorem 19.19 (Hastings). If we equip UnN with its normalized Haar measure PN,n, then for eachn and ε > 0 we have

limN→∞

PN,n(u ∈ UnN | 1− ε(u) ≤ 2√n− 1/n+ ε) = 1.

This is best possible in the sense that Theorem 19.19 fails if 2√n− 1 is replaced by any smaller

number. The proof of Theorem 19.19 in [132] is rather delicate; however a simpler proof based ona comparison with Gaussian random matrices is given in [212] with 2

√n− 1 replaced by c

√n− 1

where c is a numerical constant.

19.5 Notes and Remarks

The first explicit expanding graphs were discovered by Margulis around 1973. See [228] for a veryconcise introduction to the subject. The main reference for expanders and Kazhdan’s property(T) is Lubotzky’s book [172]. See Lubotzky’s survey [174] for a more recent update. The maininterest for us is the notion of spectral gap. We return to that theme with some more references in§24.4. Proposition 19.4 and Corollary 19.5 originate in S. Wassermann’s [260]. The term “codingsequences” was coined by Voiculescu in [254] for the sequences that we choose to call quantumcoding sequences to emphasize their non-commutative nature. While they suffice for our main goalto prove that C(n) < n, the notion of quantum expanders turns out to be more convenient becauseof the analogy with the usual expanders. Quantum expanders were introduced independently byHastings [132]–a mathematical physicist–and by two computer scientists Ben-Aroya and Ta-Shma[18]. See [19] for more on the connection with computer science. Hastings [132] proved the crucialbound appearing in Theorem 19.19 to exhibit quantum expanders using random unitaries. See also[212]. Quantum expander theory was further developed by Aram Harrow (see [130, 131]) who inparticular observed the content of Remark 19.10.

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20 Local embeddability into C and non-separability of (OSn, dcb)

In this section, we tackle various issues concerning finite dimensional operator spaces. While allthe spaces of the same dimension are obviously completely isomorphic there is a natural “distance”that measures to what degree they are really close.

When dealing with just Banach spaces E,F one defines classically the Banach-Mazur “distance”d(E,F ) as equal to ∞ if E,F are not isomorphic and otherwise as

d(E,F ) = inf‖u‖‖u−1‖ | u : E → F isomorphism.

Given two operator spaces E ⊂ B(H) and F ⊂ B(K) the analogous “distance” (called the cb-distance) is defined as

dcb(E,F ) = inf‖u‖cb‖u−1‖cb | u : E → F complete isomorphism.

If E,F are not completely isomorphic we set dcb(E,F ) =∞.In contrast, we have clearly dcb(E,F ) <∞ if dim(E) = dim(F ) <∞.

This is a “multiplicative distance” meaning that the triangle inequality takes the following form:for any operator spaces E,F,G we have

dcb(E,G) ≤ dcb(E,F )dcb(F,G).

Thus if we wanted to insist to have a bona fide distance we could replace dcb by δcb = log dcband then we would have the usual triangle inequality for δcb. Moreover, since the axioms of adistance include Hausdorff separation, it is natural to identify the spaces E and F if δcb(E,F ) = 0or equivalently if dcb(E,F ) = 1. If E,F are finite dimensional dcb(E,F ) = 1 (or δcb(E,F ) = 0)if and only if E,F are completely isometric. The last assertion is an easy exercise based on thecompactness of the unit ball of CB(E,F ).

Let us denote by OSn the set of all operator spaces, with the convention to identify two spaceswhen they are completely isometric, and let us equip it with the distance δcb. Then we obtaina bona fide metric space. Again a simple exercise shows that it is complete. The Banach spaceanalogue (that we could denote by (Bn, δ)) is called the “Banach-Mazur compactum” and as thename indicates it is a compact metric space for each dimension n (for a proof see [208, p. 334]).In sharp contrast (OSn, δcb) is not compact and actually not even separable! The main goal of thischapter is to establish this by exhibiting for each n large enough a continuous family of elementsof OSn that are uniformly separated.

But in practice we will usually not bother to replace dcb by δcb = log dcb and we will state the“distance estimates” in terms of dcb alone. The reader should remember that dcb(E,F ) = 1 is the“shortest distance” so that dcb(E,F ) > 1 + ε with ε > 0 means E,F are separated.

In analogy with the Banach space case, it is known (see [208, p. 133]) that

(20.1) ∀E,F ∈ OSn dcb(E,F ) ≤ n.

20.1 Perturbations of operator spaces

We include here several simple facts from the Banach space folklore which have been easily trans-ferred to the operator space setting.

We start by a well known fact (the proof is the same as for ordinary norms of operators).

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Lemma 20.1. Let v : X → Y be a complete isomorphism between operator spaces. Then clearlyany map w : X → Y with ‖v − w‖cb < ‖v−1‖−1

cb is again a complete isomorphism and if we let∆ = ‖v − w‖cb‖v−1‖cb we have

‖w−1‖cb ≤ ‖v−1‖cb(1−∆)−1 and ‖w−1 − v−1‖cb ≤ ‖v−1‖2cb(1−∆)−1.

Lemma 20.2 (Perturbation Lemma). Fix 0 < ε < 1. Let X be an operator space. Consider abiorthogonal system (xj , x

∗j ) (j = 1, 2, . . . , n) with xj ∈ X, x∗j ∈ X∗ and let y1, . . . , yn ∈ X be such

that ∑‖x∗j‖ ‖xj − yj‖ < ε.

Then there is a complete isomorphism w : X → X such that w(xj) = yj,

‖w‖cb ≤ 1 + ε and ‖w−1‖cb ≤ (1− ε)−1.

In particular, if E1 = span(x1, . . . , xn) and E2 = span(y1, . . . , yn), we have

dcb(E1, E2) ≤ (1 + ε)(1− ε)−1.

Proof. Recall (1.3). Let ξ : X → X be the map defined by setting ξ(x) =∑x∗j (x)(yj−xj) for all x

inX. Then ‖ξ‖cb ≤∑‖x∗j‖ ‖yj−xj‖ < ε. Let w = I+ξ. Note that w(xj) = yj for all j = 1, 2, . . . , n,

‖w‖cb ≤ 1 + ‖ξ‖cb ≤ 1 + ε and by the preceding lemma we have ‖w−1‖cb ≤ (1− ε)−1.

Corollary 20.3. Let X be any separable operator space. Then, for any n, the set denoted byOSn(X) of all the n-dimensional subspaces of X is separable for the “distance” associated to dcb.

Proof. Let (x1(m), . . . , xn(m)) be a dense sequence in the set of all linearly independent n-tuplesof elements of X. Let Em = span(x1(m), . . . , xn(m)). Then, by the preceding lemma, for any ε > 0and any n-dimensional subspace E ⊂ X, there is an m such that dcb(E,Em) ≤ 1 + ε.

Lemma 20.4. Consider an operator space E and a family of subspaces Ei ⊂ E directed by inclusionand such that ∪Ei = E. Then for any ε > 0 and any finite dimensional subspace S ⊂ E, thereexists i and S ⊂ Ei such that dcb(S, S) < 1 + ε. Let u : F1 → F2 be a linear map between two

operator spaces. Assume that u admits the following factorization F1a−→E b−→F2 with c.b. maps a, b

such that a is of finite rank. Then for each ε > 0 there exists i and a factorization F1a−→Ei

b−→F2

of u with ‖a‖cb‖b‖cb < (1 + ε)‖a‖cb‖b‖cb, and a of finite rank.

Proof. For the first part let x1, . . . , xn be a linear basis of S and let x∗j be the dual basis extended(by Hahn-Banach) to elements of E∗. Fix ε′ > 0. Choose i large enough and y1, . . . , yn ∈ Ei suchthat

∑‖x∗j‖ ‖xj − yj‖ < ε′. Let S = span(y1, . . . , yn). Then, by the preceding lemma, there is a

complete isomorphism w : E → E with ‖w‖cb‖w−1‖cb < (1+ε′)(1−ε′)−1 such that w(S) = S ⊂ Ei.In particular, dcb(S, S) ≤ (1 + ε′)(1− ε′)−1 so it suffices to adjust ε′ to obtain the first assertion.

Now consider a factorization F1a−→E b−→F2 and let S = a(F1). Note that S is finite dimensional by

assumption. Applying the preceding to this S, we find i and a complete isomorphism w : E → Ewith ‖w‖cb‖w−1‖cb < 1+ε such that w(S) ⊂ Ei. Thus, if we take a = wa : F1 → Ei and b = bw−1

|Ei ,we obtain the announced factorization.

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20.2 Finite dimensional subspaces of C

The results of this section are derived from [141].For any operator space X (actually X will often be a C∗-algebra), and any finite dimensionaloperator space E, we introduce

(20.2) dSX (E) = infdcb(E,F ) | F ⊂ X.

Of course if X = B(H) with dim(H) =∞, dSX (E) = 1 for all E.We will concentrate on the special case when X = C = C∗(F∞) and to simplify the notation

we set

(20.3) df (E) = dSC∗(F∞)(E).

Theorem 20.5. Let c ≥ 0 be a constant and let X ⊂ B(H) be an operator space. The followingare equivalent.

(i) df (E) ≤ c for all finite dimensional subspaces E ⊂ X.

(ii) For any finite dimensional subspace E ⊂ X and any ε > 0 the inclusion E ⊂ B(H) admits a

factorization through C of the form EvE−−−→ C

wE−−−→B(H) with ‖vE‖cb‖wE‖cb ≤ c+ ε.

(iii) For any H and any operator space F ⊂ B(H), we have

∀t ∈ X ⊗ F ‖t‖B(H)⊗maxB(H) ≤ c‖t‖min.

(iv) Same as (iii) with H = `2 and F = B(`2).

(v) Any mapping u : X → A/I into a quotient C∗-algebra that factorizes through B(K) (forsome K) as X

v−→B(K)w−→A/I with ‖v‖cb ≤ 1 and ‖w‖dec ≤ 1 is locally c-liftable.

Proof. Assume (i). Let u : X → B(H) be the inclusion map. Let E ⊂ X be a finite dimensionalsubspace. Let ε > 0. Since df (E) ≤ c there is E ⊂ C such that dcb(E, E) < c+ε. By the extension

property of B(H) there is a factorization of u|E : E → B(H) of the form u|E : EvE−→C

wE−→B(H)

with ‖vE‖cb‖wE‖cb ≤ c + ε. Thus (ii) holds. (ii) ⇒ (iii) follows from Kirchberg’s Theorem 9.6,(6.13) and (6.9). (iii) ⇔ (iv) is essentially trivial.Assume (iv). Let u be as in (v) of the form u : X

v−→B(K)w−→A/I, i.e. u = wv with ‖v‖cb ≤ 1

and ‖w‖dec ≤ 1. Let v : B(H) → B(K) be such that v|X = v and ‖v‖cb = ‖v‖cb ≤ 1. LetT = wv : B(H)→ A/I. Note T|X = u. Moreover ‖v‖cb = ‖v‖dec (see (6.9)) and hence ‖T‖dec ≤ 1(see (6.7)). By (6.13) we have for any t ∈ X ⊗B

‖(u⊗ Id)(t)‖(A/I)⊗maxB = ‖(T ⊗ Id)(t)‖(A/I)⊗maxB ≤ ‖t‖B(H)⊗maxB,

and by (7.6)

‖(u⊗ Id)(t)‖(A⊗minB)/(I⊗minB) ≤ ‖(u⊗ Id)(t)‖(A⊗maxB)/(I⊗maxB) = ‖(u⊗ Id)(t)‖(A/I)⊗maxB,

therefore using (iv) we obtain

‖(u⊗ Id)(t)‖(A⊗minB)/(I⊗minB) ≤ c‖t‖min.

By (ii) ⇔ (iii) from Proposition 7.48 u is locally c-liftable. In other words (v) holds.Assume (v). Assume B(H) = C∗(F)/I. Then by (v) the inclusion X → C∗(F)/I is locally c-liftable. Clearly this implies there is E ⊂ C∗(F) such that dcb(E, E) ≤ c. We may as well assumeE ⊂ C∗(F∞) (see Lemma 3.8). Thus we obtain (i).

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Applying the preceding theorem with X = E, we obtain:

Corollary 20.6. Let E ⊂ B(H) be an n-dimensional operator space. Then

df (E) = sup

‖u‖B(H)⊗maxB(`2)

‖u‖min| u ∈ E ⊗B(`2)

.

By Theorem 10.7 this implies:

Corollary 20.7. For any exact (in particular any nuclear) C∗-algebra A we have

(20.4) df (E) ≤ dSA(E)

for any finite dimensional operator space E.

Remark 20.8 (Completely isometric embeddings in C ). If a separable operator space X is suchthat df (E) = 1 for every finite dimensional subspace E ⊂ X and if in addition X admits a net ofcompletely contractive finite rank maps tending pointwise to the identity (in particular if X itselfis finite dimensional), then X embeds completely isometrically into C . Indeed, assume X ⊂ B(H),and also B(H) = C∗(F)/I. Then by (v) in Theorem 20.5, the inclusion X → C∗(F)/I is locally1-liftable. By Corollary 9.49 it is 1-liftable, and hence X embeds completely isometrically intoC∗(F), or equivalently (since X is separable, see Remark 9.14) into C . In particular, the operatorHilbert space OH from [205] embeds completely isometrically into C . Unfortunately however, wecannot describe the embedding more explicitly.

Remark 20.9. Let A be a nuclear C∗-algebra. By the preceding remark the condition df (A) = 1implies that A is completely isometric to a subspace of C . Note however that A need not embed asa C∗-algebra into C . For instance, let A be the Cuntz algebra, being nuclear it is a fortiori exact,so that df (A) = dSK (A) = 1, however A does not embed into C , because C embeds into a directsum of matrix algebras (see Theorem 9.18), hence left invertible elements in it are right invertible,and the latter property obviously fails in the Cuntz algebra. Nevertheless, by the Choi-Effros liftingTheorem 9.53 there is a unital completely positive (and completely contractive) factorization of theidentity of the Cuntz algebra (or any separable nuclear C∗-algebra) through C .

Let us record here an obvious consequence of Corollary 20.6:

Corollary 20.10. Let H = `2. For any n ≥ 1 we have

(20.5) sup

‖u‖max

‖u‖min| u ∈ B(H)⊗B(H), rk(u) ≤ n

= supdf (E) | dim(E) ≤ n.

Theorem 20.5 naturally leads us to introduce a new tensor product E1⊗M E2, both for operatorspaces and C∗-algebras, as follows.

Definition 20.11. Let E1 ⊂ B(H1), E2 ⊂ B(H2) be arbitrary operator spaces. We will denoteby ‖ ‖M the norm induced on E1 ⊗ E2 by B(H1) ⊗max B(H2), and by E1 ⊗M E2 its completionwith respect to this norm. Clearly E1 ⊗M E2 can be viewed as an operator space embedded intoB(H1)⊗max B(H2).

It can be checked easily, using the extension property of c.b. maps into B(H) (see Theorem1.18), that ‖ ‖M and E1 ⊗M E2 do not depend on the particular choices of complete embeddingsE1 ⊂ B(H1), E2 ⊂ B(H2). Indeed, this is an immediate consequence of the following Lemma.

Lemma 20.12. Let E1 ⊂ B(H1), E2 ⊂ B(H2), F1 ⊂ B(K1), F2 ⊂ B(K2) be operator spaces.

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(i) Consider c.b. maps u1 : E1 → F1 and u2 : E2 → F2. Then u1 ⊗ u2 defines a c.b. map fromE1 ⊗M E2 to F1 ⊗M F2 with ‖u1 ⊗ u2‖CB(E1⊗ME2,F1⊗MF2) ≤ ‖u1‖cb‖u2‖cb.(ii) If u1 and u2 are complete isometries, then u1 ⊗ u2 : E1 ⊗M E2 → F1 ⊗M F2 also is a completeisometry.

Proof. By Theorem 1.18 we may assume that each uj admits an extension uj ∈ CB(B(Hj), B(Kj))with the same cb-norm. By Proposition 6.7 ‖uj‖cb = ‖uj‖dec, therefore (i) follows from (6.15) and(6.6). Applying (i) to the inverse mappings, we obtain (ii).

Remark. When E1, E2 are C∗-algebras, then E1 ⊗M E2 can be identified with a C∗-subalgebraof B(H1) ⊗max B(H2), so that this tensor product ⊗M makes sense in both categories, operatorspaces and C∗-algebras.

The next result analyzes more closely the significance of ‖u‖M = 1 for u ∈ E ⊗ F . It turns outto be closely connected to the factorizations of the associated linear operator U : F ∗ → E througha subspace of C∗(F∞).

Proposition 20.13. Let E ⊂ B(H) and F ⊂ B(K) be operator spaces, let t ∈ E ⊗ F and letT : F ∗ → E be the associated finite rank linear operator. Consider a finite dimensional subspaceS ⊂ C∗(F∞) and a factorization of T of the form T = ba with bounded linear maps a : F ∗ → Sand b : S → E, where a : F ∗ → S is weak* continuous. Then

(20.6) ‖t‖M = inf‖a‖cb‖b‖cb

where the infimum, which is actually attained, runs over all such factorizations of T .

Proof. It clearly suffices (recalling Lemma 2.16) to prove (20.6) in the case when E and F are bothfinite dimensional, so we do assume that. Then since both sides of (20.6) are finite we may assumeby homogeneity that ‖t‖M = 1. Assume first T factorized for some S as previously. We claimthat ‖a‖cb‖b‖cb ≥ 1. Indeed, by Kirchberg’s Theorem 9.6, the min and max norms are equal onC∗(F∞) ⊗ B(K). Hence, by (ii) in Lemma 20.12, we have isometrically S ⊗min F = S ⊗M F , sothat if a is the element of S ⊗min F associated to a, we have ‖a‖M = ‖a‖cb and t = (b ⊗ IdF )(a).Therefore, by (i) in Lemma 20.12 we have 1 = ‖t‖M ≤ ‖b‖cb‖a‖M ≤ ‖a‖cb‖b‖cb which proves theclaim.We will now show that equality holds. Let F be a large enough free group so that B(H) is a quotientof C∗(F) (see Proposition 3.39) and let q : C∗(F)→ B(H) be the quotient ∗-homomorphism withkernel I. By the exactness of the maximal tensor product (see Proposition 7.15), if we view t assitting in B(H)⊗B(K), we have

1 = ‖t‖B(H)⊗maxB(K) = ‖t‖(C∗(F)/I)⊗maxB(K) = ‖t‖(C∗(F)⊗maxB(K))/(I⊗maxB(K))

≥ ‖t‖(C∗(F)⊗minB(K))/(I⊗minB(K))

By Lemma 7.43, since t ∈ B(H)⊗ F , we find

1 ≥ ‖t‖(C∗(F)⊗minF )/(I⊗minF )

and by Lemma 7.44 the tensor t ∈ B(H)⊗ F admits a lifting t in C∗(F)⊗ F with ‖t‖min ≤ 1. LetS ⊂ C∗(F) be a finite dimensional subspace such that t ∈ S⊗F . Note that since S is separable, byRemark 3.6 there is a subgroup G1 ⊂ F isomorphic to F∞ such that S ⊂ C∗(G1) ' C∗(F∞). Leta : F ∗ → S be the linear map associated to t, and let b be the restriction of q to S. Since t lifts t,we have T = ba, ‖b‖cb ≤ ‖q‖cb ≤ 1 and ‖a‖cb = ‖t‖min ≤ 1. Thus we obtain ‖a‖cb‖b‖cb = 1, whichproves (20.6) and at the same time that the infimum is attained.

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Remark 20.14. The preceding proof can be shortened using Proposition 9.59 applied to the operatoru : F ∗ → B(H) that is the same as T viewed as acting into B(H). We just need to observe that

(20.7) ‖IdB ⊗ u : B ⊗min F∗ → B ⊗max B(H)‖ = ‖t‖M .

Indeed, for any t′ ∈ BF ∗⊗minB with associated linear operator T ′ ∈ BCB(F,B) we have

(IdB(H) ⊗ T ′)(t) = (u⊗ IdB)(t′)

and hence

supt′∈BF∗⊗minB

‖(u⊗ IdB)(t′)‖B(H)⊗maxB = supT ′∈BCB(F,B)

‖(IdB(H) ⊗ T ′)(t)‖B(H)⊗maxB = ‖t‖M ,

and since the left hand side is equal to the one in (20.7) up to a transposition we obtain (20.7).

Applying the last result to T = IdE , we obtain

Corollary 20.15. Let E be a finite dimensional operator space. Let IE ∈ E ⊗ E∗ be the tensorassociated to the identity on E. Then ‖IE‖M = df (E). In particular we have

(20.8) df (E) = df (E∗).

Corollary 20.16. For any finite dimensional operator space E, there is a subspace E ⊂ C =C∗(F∞) and an isomorphism u : E → E such that

‖u‖cb‖u−1‖cb = df (E).

In particular, E satisfies df (E) = 1 if and only if E is completely isometric to a subspace of C .

20.3 Non-separability of the metric space OSn of n-dimensional operator spaces

Using the number C(n) it was proved in [141] that (OSn, δcb) is non-separable for any n ≥ 3 (thecase n = 2 remains open). See [208, Ch. 21] for a detailed proof. More precisely, there is acontinuous family (Et)t∈[0,1] in OSn and a constant cn > 1 such that dcb(Es, Et) ≥ cn for anys 6= t ∈ [0, 1]. When n is large the method in [141] based on C(n) gives this with cn ≈

√n. The

variant Cu(n) introduced next will lead us to the order of growth cn ≈ n when n → ∞, which isthe optimal one by (20.1).

We denote by Cu(n) the infimum of the numbers C > 0 for which there is a sequence of sizes(Nm)m≥1 and a sequence u(m) = (uj(m))1≤j≤n of n-tuples in UnNm such that for any unitary matrixa ∈ Un we have

(20.9) supm6=m′

∥∥∥∑n

i,j=1aijui(m′)⊗ uj(m)

∥∥∥ ≤ C.Clearly C(n) ≤ Cu(n) and Cu(n) ≤ n by (2.3).

Theorem 20.17. For any n ≥ 5 there is a continuous family (Et)t∈[0,1] in OSn such that

(20.10) ∀ε > 0 ∀s 6= t dcb(Es, Et) ≥ (n/Cu(n))2 − ε ≥ n/4− ε.

Corollary 20.18. The metric space (OSn, δcb) is not separable for any n ≥ 5 .

To prove Theorem 20.17 we will need several lemmas.

250

Lemma 20.19. Let U(N)1 , . . . , U

(N)n be a sequence of independent random unitary matrices uni-

formly distributed over UN as in Theorem 18.16. Then for any k and any x1, · · · , xn ∈ Uk we havealmost surely

(20.11) limN→∞ supa∈Un

∥∥∥∑n

i,j=1aijxi ⊗ U (N)

j

∥∥∥min≤ 2√n.

Proof. Let us first observe that for any a, b ∈ Un we have for any k, k′ and any xi ∈ BMk, yj ∈ BMk′

(20.12)∥∥∥∑n

i,j=1(aij − bij)xi ⊗ yj

∥∥∥min≤∑n

i,j=1|aij − bij |.

By compactness there is a finite ε-net Nε ⊂ Un for the distance appearing on the right hand sideof (20.12). By the triangle inequality it follows that for any k, k′ and any xi ∈ BMk

, yj ∈ BMk′

(20.13) supa∈Un

∥∥∥∑n

i,j=1aijxi ⊗ yj

∥∥∥min≤ sup

a∈Nε

∥∥∥∑n

i,j=1aijxi ⊗ yj

∥∥∥min

+ ε.

By (18.15) and Corollary 3.38 for each fixed a ∈ Nε we have a.s.

limN→∞

∥∥∥∑n

i,j=1aijxi ⊗ U (N)

j

∥∥∥min≤ 2√n