RIGIDITY FOR LATTICES IN SEMISIMPLE · from 1955 to 1975. Most of this work has an algebraic flavour, but ergodic theory started creeping in around 1970. We present some results on

REND. SEM. MAT. UNIVERS. P0LITECN. TORINO

Vol. 47 ° , 1 1989

M. Cowling

RIGIDITY FOR LATTICES IN SEMISIMPLE LIE GROUPS: VON NEUMANN ALGEBRAS

AND ERGODIC ACTIONS

This paper is about semisimple Lie groups and lattice subgroups therein. The leitmotiv is how discrete structures approximate continuous ones, or vice versa.

If we look at an object such as a crystal with light whose wavelength is longer than the interatomic distances, we perceive it as a continuous rather than a discrete structure. While we can no longer see the local structure, we can still see the global structure. In applied mathematics, structures which are discrete at the atomic level are modelled by continuous structures (e.g., in the equations of fluid dynamics), and nowadays these equations are "solved" by numerical analysis using another discrete approximation (e.g., the finite element method). The first of these jumps between the continuous and the discrete is "justified" by the belief (born out empirically) that the macroscopic properties of the discrete fluid and those of the continuous model are the same, while the second approximation is (sometimes) justified by the functional analytic methods of computational mathematics.

We consider structures with symmetry, such as cosmological space-time for which an initial assumption is that the universe is homogeneous (this assumption is expressed by saying that the laws of physics are invariant under certain changes of coordinate system). In such structures, the symmetries form a continuous group, typically a Lie group. The space itself is usually a quotient space (homogeneous space) of the Lie group. Our basic question is

AMS (MOS) 1980 Classification: 43A70, 46L10 \

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then how a homogenous space can be "naturally" approximated by a discrete structure. In particular, we ask how Lie groups can be approximated by discrete structures, and in this case the natural discrete structures are the discrete subgropups. In some sense, we are concerned with the problem of "seeing" the global properties of graphs which embed "nicely" in manifolds, because the lattices of the title are discrete groups, and to a discrete group we can associate its Cayley graph. Alternatively, we are looking for ways of comparing the global geometry of a Lie group with that of a discrete subgroup.

A lattice P in a Lie group G (always assumed connected, and usually algebraic) is a discrete subgroup of G such that G/T has finite G-invariant measure. (For this to be possible, G must be unimodular). We say r is uniform or cocompact if G/T is compact. Two lattices IY and 1^ in G are called commensurable if T\ h l Y Is of finite index in both. One would expect commensurable lattices to have similar global structure. We return briefly to one of our physical paradigms, and recall that, according to the chemists/ a water crystal has the structure in Diagram 1 (below), where the large dots represent oxygen atoms and the small dots represent hydrogen atoms. If we could only see the larger oxygen atoms, then the structure would be that obtained by ignoring the small dots, which is that of a diamond crystal. Ice and diamond are quite similar materials. The relation between water crystals and diamond is analogous to the relation between a discrete group and a 'subgroup of finite index. "~"

•rk Diagram 1

We describe some of the development of the theory of lattices in semi-

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simple groups in this paper, which, like Gaul, can be conveniently divided into three parts. In the first of these, we outline some of the basic algebraic results about lattices in Lie groups obtained (roughly) between 1955 and 1975, and sketch some of the more recent analytic developments of the theory. In the second part, we describe in more detail the work of U. Haagerup, of Cowling and Haagerup, and of Cowling and R.J. Zimmer on von Neumann algebraic and ergodic theoretic rigidity for lattices. This work is based on the notion of a completely bounded operator on the von Neumann algebra of a group or of an ergodic action. Part HI contains some new work on completely bounded multipliers.

I. An Outline of Rigidity

In this part, we sketch some of the significant work on lattices from the period from 1955 to 1975. Most of this work has an algebraic flavour, but ergodic theory started creeping in around 1970. We present some results on nilpotent groups as well as those on semisimple groups, our main interest. Everything we describe here may be found in the books of M.S. Raghunathan [41] and of R.J. Zimmer [45].

1.1 Examples of lattices

It seems worth giving a few examples of lattices. 1. The prototype of a lattice is the subgroup Zn of R". More generally,

take a basis {ei, . . . ,en} of R", and form the lattice L(e\}... ,en), given by the rule

L(ei, , . . , en) = {??iiei -f mnen : mi , . . . , mn G Z} .

2. Let U be a subring of C, and let Ha(H) be the Heisenberg group over

.J/1- a c\ )

l\o o \) J Then /73(Z) is a lattice in //3(R), and #3(Z + tZ) is a lattice in //3(C).

3. SL(2,Z) is a lattice in SL(2,R). Further, there are lots of cocompact lattices in 5L(2,R), associated with the regular tilings of the-hyperbolic disc (see the book of A.F. Beardon [2], or the paper of J.W. Cannon [8]).

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We may assume that most mathematicians have some understanding of the relation of Zn and R". However, the structure of lattices in nilpotent Lie groups or in semisimple Lie groups is rather less well known. We now summarise what is known about the nilpotent case; most of these results are due to A.I. Mal'cev [31]. Later we shall discuss the semisimple case.

1.2 Nilpotent Lie groups

Suppose N is a nilpotent Lie group. Then N may or may not contain a lattice. For example, in h^iC), the Lie algebra of //3(C), we may take a basis over R as follows:

/ 0 1 0 \ / 0 0 0 \ / 0 0 1 \ a 1 = [ 0 0 0 I" b i = ( 0 0. 1 ci = I .0 0 O l

\ o 0 0 / \o 0 0 / \ o 0 0 /

a2 •= iai b 2 = ibi c2 = ici

Then [ a ^ b j ^ c i [a1,b2] = c2

[a2,bi] = c2 [a2,b2] = ~ci ,

arid all unspecified Lie brackets are 0. (This means that [bi,ai] = —[a^bj = —ci, and so forth, but [ai,ci], which cannot be determined from the above list, is 0). We have already remarked that //3(C) admits #3(Z + iZ) as a lat­tice; this lattice is commensurable with the lattice exp(L(ai,a2,bi,b2,ci,c2)), where L(ai ,a2,b1 ,b2 ,c1 ,c2) is the lattice in the vector space ^(C) defined in Example 1 above.

We define a Lie algebra by a set of modified relations; more precisely, we define n. to be the span of {ai ,a 2 ,bi ,b 2 ,c i ,c2} and we take the commutation relations to be

[ai,bi] = ci [a2,b2] = c2

[a2, bi] = c2 [a2, b2] = \/2ci;

then the corresponding Lie group N admits no lattice subgroup. Mal'cev generalised this example, and proved the following theorem.

THEOREM I . l . A connected, simply connected nilpotent Lie group N contains a lattice subgroup if and only if the Lie algebra has a basis for which the structure constants are all rational. Such lattices are cocompact.

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1.3 Semisimple Lie groups

Now let us discuss lattices in noncompact semisimple Lie groups; our basic example is SL(2,Z) inside SL(2,R). It has been shown that both cocompact and noncocompact lattices exist inside every noncompact semisimple algebraic Lie group. The group theoretic way of characterising the cocompact lattices is that they contain no unipotent elements (A in GL(n, R) is said to be unipotent if A — I is nilpotent). The fundamental papers here are by Dorel and Uarish-Chandra [5] and G.D. Mostow and T. Tamagawa [35].

The question, to what extent and in what ways does a lattice F in a noncompact semisimple Lie group resemble the ambient group G, has answers from arithmetic, group theory, algebraic geometry, probability,-operator alge­bras, and representation theory. We shall describe these. But first, we recall a negative result, due to D.A. Kazhdan and G.A. Margulis [28] (expounded in English by Raghunathan (op. cit.)), namely, if the Haar measure of G is fixed, then the volume of G/T is bounded below. More precisely, if G has no compact factors, then there is a neighbourhood U of the identity e in G such that every lattice in G is conjugate to one whose intersection with U is just e.

Many of the theorems in this subject admit a much simpler formulation for the case where the ambient Lie group is simple, rather than semisimple, and we give the simpler version. For instance, the arithmeticity theorem of Margulis is stated below for a lattice in a simple Lie group. Margulis' result was proved in the more general situation of an irreducible lattice in a semisimple Lie group of real rank at least two. By stating the theorem in the "simple" case, we may avoid giving the precise definition of irreducibility. It is worthwhile pointing out that this definition is formulated to exclude products of lattices in the factors of the semisimple group, and, more generally, that it is "only" technicalities of this nature that we are brushing under the carpet. The reader should consult Raghunathan and Zimmer (opera citato), or the original papers, for more complete versions.

1. Arithmetic.

A subgroup G of GL(n}C) is said to be defined over R (or Q) if there are polynomials pi,...,pk with real (respectively rational) coefficients in the n2 + 1 variables p,j, 1 < i,j < n, and det(#)~"1, such that

G={ge GL(n, C) : Pl(g) = . . . .= pk(g) = 0}.

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The real algebraic group G R is the set GnGL(n,R), where G is defined over R.

A subgroup T of a real algebraic group G, defined over Q, is called arithmetic if there exists A in GL(n,Q) such that T is commensurable with A-lGL{n,7>)AnG.

An alternative definition, for a lattice in a semisimple Lie group with no given Q-structure (i.e., no choice of p i , . . .,/?*), is that V is arithmetic if there exists a real algebraic subgroup / / of GL(n,R) and a homomorphism 7r: / / —• G with compact kernel, such that n(H C)GL(n,Z)) is commensurable with T.

When we say that T is arithmetic, we are essentially saying that T looks like a lattice in R" : we take the right basis (this is what A does in the first definition) and then take the integer points.

It is clear that lattices in Rn and in nilpotent Lie groups (when they exist) are arithmetic. • /

In the semisimple case, we have the following results, due to Borel and Harish-Chandra [5] (part (a)) and Margulis [32] (part (b)).

THEOREM 1.2. Let G be a real simple algebraic group. (a) Every arithmetic subgroup of G is a lattice. (b) If the rank of G is at least two, every lattice is arithmetic.

These results are described in the books of Raghunathan and of Zimmer (opera

ciiata). Margulis' result is a tour de force which is not yet completely under­stood. . . ••

When the real rank of G is one, G is 50( l ,n ) , SU(l,n), Sp(l,n) or ^4(-2o)- 1^ this case, the situation regarding arithmeticity of lattices is pre­sently uncertain, but it seems likely that all lattices in 5p(l, ?i) and in ^4(-^p) are arithmetic (K. Corlette, from the University of Chicago, has apparently just made some important progress here), while nonarithmetic lattices pro­bably exist inside all the groups 50(1,n) and SU(l,n). Indeed, examples of nonarithmetic lattices are known inside 50(1,rc) and SU(l,n) for very small n.

2. Group theory.

Now we look at lattices from a different point of view, that of group theory. The rigidity theorem of Mostow, and others, is the following.

THEOREM 1.3. Suppose that T,- is a lattice in G,-, where G{ is a connected non-compact simple Lie group with trivial centre (i = 1,2), and that G\ ^ P5X(2,R).

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Suppose also that h : Fi —• V2 is an isomorphism. Then h extends to an iso­morphism from G\ to G2.

Mostow's original theorem [33], [34] was for cocompact lattices in groups of higher rank. The extension to noncocompact lattices in the higher rank case is due to Margulis [32], and to the rank one case to G. Prasad [39], [40]. Margulis' extension is a very powerful result: under the assumption that the real rank of G\ is at least two, the word isomorphism may be replaced by homomorphism, whether the r, 's are cocompact or not. This "superrigidity theorem" is a key step in the arithmeticity result already mentioned.

Similar rigidity theorems hold for lattices in nilpotent Lie groups, and of course for lattices in Rn.

These rigidity theorems have interesting geometric applications. A com­pact nilmanifold is defined to be a manifold of the form N/T, where T is a lattice in a connected simply connected Lie group TV. If N1/V1 and N2/V2

are homeomorphic compact nilmanifolds, then their fundamental groups Ti and T2 are isomorphic, and so Ni and N2 are also isomorphic.

A locally symmetric space is a connected Riemannian manifold M which admits a local isometric symmetry at each point, i.e., given p in M, there exists a neighbourhood U of p and an isoinetry <j>: U —• U which fixes p and whose differential d<j> satisfies d<j>\T (M) — —I- Locally symmetric spaces are of the form K\G/r, where K is a maximal compact subgroup of a se-misimple Lie group G with trivial centre, whence K\G is simply connected, and T is a discrete subgroup of G, isomorphic to the fundamental group of K\G/Y. Mostow's theorem says that homeomorphic locally symmetric spaces of finite volume are isometric.

3. Algebraic geometry.

The Zariski topology is frequently used in algebraic geometry. The Zariski closure of a subset S of an algebraic variety V is the smallest algebraic subvariety of V containing S (a subvariety of V is defined to be the set of zeroes of a finite number of regular functions (polynomials); V itself is defined to be the set of zeroes of some other regular functions). A subset is said to be Zariski dense if its Zariski closure is the whole variety.

Lattices are Zariski dense in connected simply connected nilpotent Lie groups, and in noncompact semisimple Lie groups.

A variant of this is the Borel density theorem, proved by Borel [4].

THEOREM 1.4. Let p be an irreducible finite-dimensional representation of the

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simple noncompact Lie group G, and let T be a lattice in G. Then the restriction p\r of p to V is irreducible.

4. Probability theory.

H. Furstenberg [18] (see also the papers cited there) established certain connections between the boundary of certain random walks on the lattice and that of certain Brownian motions on the ambient Lie group. N. Th. Varopou-los [42], [43] also proved some results which relate the behaviour of random walks on discrete groups to that of Brownian motion on ambient manifolds, and in particular, on ambient Lie groups.

5. Operator algebras.

Some years ago, A. Connes suggested that if i y is a lattice in the non-compact semisimple Lie group d (i = 1,2), and G\ is not locally isomorphic to G2, then the group von Neumann algebras VW(ri) and VN(T2) are not isomorphic. This result would be stronger than the Mostow rigidity theorem, as isomorphic groups have isomorphic von Neumann algebras, but the con­verse is false. It seems that Connes had an idea for a proof of the result, but this fell through. The results in this direction so far are due to Connes and V.F.R. Jones [9], to U. Ilaagerup [24], and to Cowling and Ilaagerup [12].

The idea behing the work of Connes and Jones is to reformulate Ka'zh-dah's. "property T" [27] in terms of "correspondences" on the von Neumann algebra of the group. It is possible to decide, if one knows the von Neumann algebra of a group, if the group has property T. The simple Lie groups without property T are 50(1,n) and SU(l,n), where n = 1,2,..., and the groups lo­cally isomorphic to these; all the other simple Lie groups have property T. Since a lattice has property T if and only if the ambient Lie group does, it follows that the von Neumann algebra of a lattice in a group with property T cannot be isomorphic to that of a lattice in a group without property T. This is not a very fine distinction between von Neumann algebras^ and cannot tell, for instance, if the von Neumann algebras of lattices in SX(m,R) and SL(n,K) are different when m, n > 3 and m^n.

Cowling and Ilaagerup [12] introduced a numerical invariant of a von Neumann algebra, which will be described in more detail below, that enables one to make somewhat finer distinctions. Unfortunately, while this invariant can differentiate between some von Neumann algebras with property T, it too fails to distinguish between the von Neumann algebras of SL(jn,R) and SL(n, R) when m, n > 3 and m ^ n. However, Cowling and Zimmer [14]

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showed that this invariant does admit generalisation to give a useful ergodic theoretic invariant.

Another area in which operator algebraic techniques are being applied is the so-called KK-theory of G.G. Kasparov [26], So far, however, no pro­gress on the benchmark question (distinguishing the von Neumann algebras

"of SL(m,R) and SX(n,R)) has been reported.

6. Representation theory.

Here is a recent representation theoretic result due to Cowling and T. Steger [13], somewhere between the Borel density theorem and the operator algebraic results just described.

THEOREM 1.5. Suppose Y is a lattice in a noncompact simple Lie group G. If 7r is an irreducible unitary representation of G, then

(a) if IT is a discrete series representation, then 7r|p is reducible (b) if 7T is not a discrete series representation, then 7r|r is irreducible.

Here is an outline of the proof. Let A denote the regular representation of G on L 2(G/r) . We show that if T £ 0 and T intertwines 7r|r and p\r, then there exists T which intertwines TT and /><8>A, such that T ^ 0. (This is standard — the Frobenius reciprocity theorem associates an intertwining operator between TT and ind^o- to every intertwining operator between ?r|r and c, and the Mackey subgroup theorem shows that indp(/>|p) and p <g> A are unitarily equivalent). Thus if 7r|r is irreducible, there are nontrivial intertwining operators between n and TT <g> A. This much of the proof works for any lattice r in any Lie group G.

The second part of the proof works for simple Lie groups. We write A as a direct integral:

A== l®G\{\}mTTdv(T)> whence n <g> A = n <g> 1 0 - mT(7r ® r) dp(r).

We show that, if w is not a discrete series representation, then TT ® r cannot contain a copy of TT, because the matrix coefficients of TT <g> r vanish faster than those of TT. Thus the only intertwining operators between w and 7r<g> A are the trivial operators intertwining ir with 7r<g> 1.

This theorem tells us that the unitary dual V of T contains a copy of G, or at least of a large part thereof. The topological structure of G is enough to determine G, so there is reason to hope that the unitary dual F determines G.

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Finally, we mention a recent result of C. Bishop and Steger [3]. In PSZ,(2,R), it is easy to find pairs of hyperbolic transformations on the Poin-care disc D which generate free lattice subgroups of PSL(2, R). The natural isomorphisms between these free groups do not, in general, extend to iso­morphisms of the ambient group PSL(2,TV) (this is the case specifically ex^ eluded by the rigidity theorem). There is a natural unitary representation of PSL(2,Tl) on L2(dD)f and restriction of this representation to one of the above-mentioned free subgroups of PSL(2,H) gives an irreducible unitary re­presentation of this free group.

By identifying an abstract free group with two different free lattices in PSL(2, R), we obtain two irreducible unitary representations of the free group on L2(dD). Bishop and Steger prove that if these two representations are unitarily equivalent, then the two free lattices in P5L(2,R) are conjugate in that group.

II. Completely bounded maps and approximate identities

In this part, we define and discuss the completely bounded maps of a von Neumann algebra, and in particular the completely bounded multipliers of the von Neumann algebra of a locally compact group. Next, we outline va­rious applications of the completely bounded maps to characterising groups, von Neumann algebras, and ergodic actions, following U. Ilaagerup [24], Cow­ling and Ilaagerup [12], and Cowling and R.J. Zimmer [14]. Part II should perhaps be considered as an introduction to these papers; in particular, as the second-named paper is frequently quoted here, the reader is encouraged to have a copy at hand. The unifying theme of these papers is the existence (or nonexistence) of approximate identities of completely bounded maps having certain additional support properties.

I I . 1 Completely bounded maps.

If 7ix and 7 2 are Hilbert spaces, then their Hilbert tensor product Hi (g>2 ft2 *s the completion of the algebraic tensor product Tii <g> 7i2 in the metric derived from the natural inner product thereon, which, being sesquili-near, is determined by the condition that

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Alternatively, we may think of Hi <g>2 ^2 as the Hilbert space with basis {e,<8>f,}, where {e»} and {f,} are bases for Hi and H2 respectively. If H is identified with a space L2(X) for some measure space, then W®2W may be identified with L2(X x X). It is not difficult to see that if T\ and T2 are bounded linear operators on Hi and Hi respectively, then Ti^T2 is bounded on 7ii®2ft2, and the norm of Ti<8>T2 is the product of the norms of Ti and 22 . Indeed, on the one hand, (Tx ®T2)((i <8>f2) = Tifi ©T2£>, whence ||Ti®T2|| > ||Ti|| ||T2||, and on the other, Ti®T2 is the composition of Ti®I and 7®T2 , whence it suffices to show that ||T\ <g> I\\ < ||7i||; this is a consequence of Fubini's theorem.

It follows from this observation that if M and A/* are von Neumann algebras of operators on Hilbert spaces HM and HM respectively, the alge­braic tensor product M ® N may be considered to be a space of bounded operators on HM ®2 Af- The spatial tensor product of M and Af is the von Neumann algebra M <8>sp X of operators on HM ®2 Wtf generated by the algebraic tensor product (i.e., its weak operator topology closure).

We close this first discussion of topological tensor products by descri­bing the projective tensor product Hi §>H2. This space is the subspace of

' Hi ®2 H2 of vectors t which may be represented in the form EneN&i ® rjn, where JZneN IK»II ll7/"!! < °°- The n o r m °f * ls the infimum of the expressions STneN ll£"ll \\rln\\ over all such representations. If Hi and H2 are identified with spaces L2(Xi) and L2{X2) for measure spaces X\ and A'2, then Hi§>H2

may be considered to be the subspace of L2(Xi x A'2) of functions of the form

£ n € N * " ® * n = (*,V) ^ E „ 6 N M * ) * n O / ) > w h e r e E n € N l l / l " H H M < ° ° ' T l l e

norm of such a function is similarly expressible as an infimum. Analogous realisations hold if Hi and H2 are identified with spaces ^2(A'i) and P(X2). The dual of Hi ®H2 can be identified with B(Hi,H2)\ to do this we choose a conjugate-linear isometric involution J of H2, which gives rise to a bilinear pairing ( , ) on H2, defined by the rule (£,77) = (£, JTJ) for all vectors £,77 in H2. The duality between T in B(Hi,H2) and t in H§>H is then given by the rule

where / = E«eN hn-®kn is a representation of t. It should be noted that this duality depends on the choice of the bilinear pairing on H2. When dealing with a Hilbert space of the form L2(X)\ it is natural to use complex conjugation of functions for J.

Here, and elsewhere in this paper, we use parentheses for bilinear pai­rings, as between a space and its dual, or between LP(X) and Lq(X), and

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angled parentheses (( , )) for sesquilinear pairings.

If M is a von Neumann algebra, and T : M -+ M is a continuous linear operator, and if N is another von Neumann algebra, then the map T <g> 7, well defined on the algebraic tensor product M<S>J^} may or may not extend to a bounded linear operator on the spatial tensor product M<S>gpJ\f. If T®I extends to a bounded operator on the spatial tensor product when M is the set B{H) of all bounded linear operators on a separable Hilbert space H, then extension is possible for any von Neumann algebra Af, and T is said to be completely bounded. The completely bounded norm of T, written ||T||ct, is then defined to be the operator norm of T <g> I on M ®sp B(H)\ this norm dominates the norm of T<S>I on M<2>apAf for any von Neumann algebra N.

An alternative formulation of complete boundedness is a consequence of a theorem proved independently by Ilaagerup [23], V.I. Paulsen [37], and G. Wittstock [44] (expounded nicely by Paulsen [38]); if M is a von Neumann algebra of operators on HM, a i l ( l T Is a bounded linear operator on M, then T is completely bounded if and only if there exists an extension of T to a bounded operator on B(%M)- The least norm of such an extension is exactly ||T||C&. For a weak-star continuous operator on M, this theorem may be rephrased. The predual of B(7i) can be identified with the projective tensor product H § %, and then the predual of a von Neumann subalgebra M is a quotient Q of H®%. A weak-star continuous linear map T : M —• M is the transpose of a continous linear map Tt : Q —• Q\ this map extends to a bounded linear map on Ti^H exactly when T is completely bounded.

We shall be concerned with approximate identities of completely boun­ded operators on von Neumann algebras M. By this we shall mean a net {Tiiiel} of completely bounded operators On M with the properties

(La) There exists a constant C such that ||7i||c& < C, for all i in / .

(l.b) For all i in 7, T} : M —• M is continous when M is endowed with the weak-star topology.

(2) For all x in M, lim,-€/ TiX = x in the weak-star topology. We impose the extra restriction on the operators 7} that their images, i.e., the spaces T{(M), be finite-dimensional, and then define M(A-f) to be the infimum of all C s for which such a net exists. If no such net exists, M(A^) is defined to be -foo. An important equivalent formulation of the definition of M(M) involves approximate identities on the predual M+ of M. We seek nets of bounded operators {Ti : i G /} on M+ with the properties

(1) The transposes T- of the operators Ti are all completely bounded operators on M, and tliere exists a constant C such that ||T?||C4.<

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Cj for all i in J. (2) For all it in M+, lim^/ 7 > = u.

The support restrictions can be similarly reformulated. Then M(A4) is the infimum of all C's for which such nets exist.

I I .2 Locally compact groups

Here we discuss the case where M is the von Neumann algebra of a locally compact group G. We begin by sketching the theory of the Fourier algebra of G, following P. Eymard [16]. The reader who finds our outline requires fleshing out should consult this paper.

Let G be a locally compact group, equipped with a left invariant Haar measure, written dx. We denote the usual Lebesgue spaces relative to this measure on G by LP(G). We define Rep(G) to be the set of all unitary representations of G (in this paper, all representations are implicitly assumed to be continuous); the reader who prefers may equally well work with this set modulo the relation of unitary equivalence. We write B(G) for the space of matrix coefficients of unitary representations of G\ a function M on G belongs to B(G) if and only if there exists a unitary representation IT of G on a Hilbert space 7in, and vectors £ and 77 in Hw such that

u(x) ••= (n(x)i,rf} VxeG.

We write u = (7r()£, 77) to abbreviate this equality. The sum and tensor product of two unitary representations is again a unitary representation, aiid consequently B(G) is closed under pointwise sums and products. It is easy to see that, equipped with the norm || ||#, defined by the rule

\\u\\B = inf{|K||- \\n\\.i u = (^(.)^^>, TT e Rep(G), f,r; G U« },

B(G) is a Banach algebra. It can be shown that the infimum is attained in this formula for \\U\\B.

An alternative definition of B(G) is sometimes useful. The (big) C*-algebra of the group G is defined to be the completion of the Banach convo­lution algebra Ll(G) in the C*-norm:

| | / | | c . = Bup{||ir(/)|| : ir €Rcp(G)}:

The dual of this Banach algebra is naturally identified with B{G).

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The closed ideal of B(G) generated by the compactly supported B(G)-functions, denoted ^(G), can be identified with the space of matrix coefficients of the left regular representation A of G by translations on L2(G). Thus it is in A(G) if and only if there exist functions h and k in L2(G) such that

u(x) = (\(x)h,k) VxeG;

moreover \\U\\D = rnin{|H| ||^|| •' u = (H')h> fy }•

The identification of the Banach algebra A(G) with a set of matrix coefficients which is a priori not even a vector space is one of the more subtle parts of the theory of the Fourier algebra A(G), and relies on the polar decomposition in the preduals of von Neumann algebras. It is possibly more natural to use the equivalent definition that A(G) is the image of the projective tensor product L2(G)®L2(G) under the linear map V which is defined by the condition that V(h <g> k) = (A(-)/i,&), for h and it in L2(G). One often writes HMH^ rather than ||U||JB to emphasize that u is an element of A(G). The dual space of A(G) can be naturally identified with V7V(G), the von Neumann algebra of all bounded linear operators on L2(G) which commute with right translations. We recall that , for / in Ll(G)% the map g H- f*g is in VN(G). The "reduced" C*-algebra of G, C*(G), is the norm closure of Ll(G) in VN(G) : C*(G) is strong operator topology dense in VN(G). The theory of C* -algebras shows that there is a canonical projection of C*(G) onto C*(G), and the dual space Br(G) of C*(G) is a closed subspace of B{G). It is easy to show that Br(G) is the subset of B(G) of limits in the topology of duality with Ll(G) (or in the topology of uniform convergence on compacta) of bounded nets of A{G)-f unctions.

On any locally compact group G, A(G) is dense in C'o(G), the space of continuous functions on G which vanish at infinity, equipped with the uniform norm. Consequently, an element <I> of the dual of A(G) can be considered to be a distribution on G, and convolution with this distribution, i.e., the map h H-+ $ * hy is a typical operator on L2(G) which commute with right translations.

It is always true that A(G) C Br(G) G B(G), and when G is compact, A(G) = B(G), but otherwise A(G) C B(G), as constants belong to B(G) but not^to A(G).

The group G is said to be amenable if it satisfies one (and hence all) of various equivalent conditions:

(1) There exists an invariant mean on L°°(G); an invariant mean M is a positive linear functional on L°°(G) which maps the constant

1,5

function 1 to 1, and is translation-invariant, i.e., M(\(x)f) = M(/) , for all x in G and / in L°°(G).

(2.a) There exists a net of A{G)-functions {w» : i G /} such that

IKIU<c Vie/ and limllu.-i/ - v\\A = 0 Vt/ G .4(G).

(2.b) There exists a net of A(G)-functions {«,- : i e 1} such that

IMU < c Viei and lirnw,- = 1 locally uniformly

(3.a) The C;{G) and C*(G) norms agree on L\G). (3.b) The spaces Dr(G) and B(G) coincide.

It should be pointed out that in the approximate identity conditions (2.a) and (2.b), we may take C equal to 1, and we may also require that the functions m be compactly supported and positive definite (i.e., of the form (()\hi,hi) for suitable L2(G)-functions /?,). It should also be pointed out that a net which satisfies condition (2.a) automatically satisfies condition (2.b), but not vice versa. For more details about amenability, the reader is invited to consult Eymard [16] or the book of F. Greenleaf [21],

Another property of locally compact groups which is defined in terms of approximate identities is D.A. Kazhdan's property T [27]. We preface our definition with the observation that every unitary representation may be written as a direct sum of a trivial representation and of a representation with no G-fixed vectors. Corresponding to this decomposition of unitary representations of G, there is a decomposition of Z?(G)-functions. Every u in B(G) may be written in the form wi 4- «2, where u\ is a constant and u2 is a coefficient of a unitary representation without G-fixed vectors. This decomposition can also be described in terms of the Z?(G)-norm: uj is the constant function for which ||u - tzi||# is as small as possible. We write B(G) = C e B°(G) to indicate this decomposition of B{G).

A locally compact group G is said to have property T if the constant function 1 cannot be approximated uniformly on compacta by a net of positive definite B°(G)-functions. In terms of the topology on the dual G of G, which we shall not describe here, G has property T exactly when the trivial representation is isolated in G.

The last property of locally compact groups that we shall describe in terms of approximate identities is weak amenability. Cowling and Haage-rup [12] define a group G to be weakly amenable if there exists a net of

16

A(G) -functions {*/,•: i e 1} such that

\M\MoA <C

and either limu,-= 1 locally uniformly

or limlltiiv - v\\A = 0 Vw 6 A{G), » € /

where || \\M0A denotes the completely bounded multiplier norm, which we shall shortly explain.

It should be pointed out that , in contrast to the definition of amenability, the value of the constant C is crucial — indeed, the Haagerup number A(G) is defined to be the iniimum of all C for which such nets exist, and this number can certainly be any odd positive integer, or -foo. We may also require that the functions u, be compactly supported. It should also be pointed out that a net which is an approximate identity for multiplication (the first of the alternative conditions) converges to 1 uniformly on compact a, as, given any compact set A', we can find a compactly supported A{G)-function taking the value 1 on K, but the converse is not necessarily true.

The space MA(G) is defined to be the space of all functions on G which are bounded multipliers of A(G). In other words, a function u on G lies in MA(G) if uv is in A(G) whenever v is. The corresponding norm, written || ||AM, is just the operator norm, of the map v i-+ uv. The transpose of this map, which we consider to be the multiplication of the distributions in VN(G) by i/, is a bounded map on VN(G)\ if it is completely bounded, then we say that u is a completely bounded multiplier of A(G). The norm on MQA{G)

is taken to be the completely bounded norm of the transpose multiplication map.

The key result on M0A(G) is the following characterisation, which synthesizes results of various authors, in particular C.S. Herz [25], J.E. Gil­bert [20], G. Fendler [17], J. De Canniere and Haagerup [15], M. Bozejko and Fendler [7], and Haagerup [24]. The reader should note that the completely bounded multipliers of A(G) are called Ilerz-Schur multipliers in some of these references, and in several other papers on the subject. Before we state the theorem, we give one last definition: for a function u on G, Mu : G x G —*• C, is the function on G x G given by the rule Mu(x,y) = u(y}x~l) for all x,y in

THEOREM II. 1. The following conditions are equivalent: (i) ueM0A(G);

(ii) the function u <g> 1 is in MA(G x II), where II is the group SU(2);

17

(iii) the function u<g> 1 is in MA(G X H), where H is any locally compact group; (iv) there exist bounded continuous mappings H, II : G —*% (7i a Hilbert space)

such that the function Mu can be represented in the form

Mu(*} y) = (E(x), H(jf)) Vz, y <= G;

(v) the function A4u multiplies pointwise the projective tensor product L'2(G)<g> L2(G).

(vi) the function Mu is continuous and multiplies pointwise the projective tensor product i2(G)%f{G). The natural norms associated to these conditions coincide. Thus

\H\M0A(G) = \\u\\'MA(GxSU(2))

= sup{\\u\\AU(GxHy. II a locally compact group}

= inf{||E||eo | |H||00:>fu=:(S>H>}

= WMUWM(L>®L>) •

It should be observed that condition (v) is a strong form of the predual version of the Haagerup-Paulsen-Wittstock theorem mentioned above, and that condition (iv) can be linearised by replacing it by the condition that Mu is in the image of C(G\'H) §> C{G\H)) under the linear map II into C(GxG) defined by the condition that n(S.® II) = (S,H). Indeed, if Mu = !Cn€N-(-n»^n)» w n e r e 2„ and IIn are in Q(G\H) for some Hilbert space H a u d En€N!ISn||oo||H„||oo < oo (and ||H„||oo'||Hn||oo £.0 for any n in N), then a representation of Mu of the form of condition (iv) is Mu = (0,Z), where 0 = {onHn} and Z = {a^Kh} in the "bigger" Hilbert space C2(N;7i)', here «n=| |H n | | - 1 | |H n | | 0 0 .

It is known that B(G) C MQA(G) CMA(G), with equality holding if G is amenable. From the work of Bozejko [6], C. Nebbia [36], and V. Losert [3,0], it seems likely that these inequalities are all strict and the norms inequivalent when G is nonamenable.

In the papers of Ilaagerup [24] and Cowling and Haagerup [12], most of the following results are established.

THEOREM II.2. Let G be a locally compact group, and let A(G) 6e as defined above. Then

(i) if G\ and Gi are isomorphic, A(Gi) = A(G2);

(ii) if G\ is a closed subgroup of G2, A(Gi) < A(G2); if G2/G1 has finite G2-iiivariant measure, then equality holds, and in particular, equality holds if G\ is of finite index in G^ or if G\ is a lattice in Gi-

18

(iii) if G = Gi x G2, A(G) = A(Gi) x A(G2);

(iv) if G is discrete, and Z is central in G, A(G) < A(G/Z).

The only new result of this theorem is part (iv), which we shall prove in the next part . It seems very likely that equality holds in (iv), but we cannot prove that here.

THEOREM II 3. If G is an amenable group, A(G) = 1. If G is a connected noncompact simple Lie group, then A(G) is equal to 1 if G is locally isomorphic to SO(l,n) or SU(l,n), to 2n — 1 if G is locally isomorphic to Sp(l,n), to 21 if G is locally isomorphic to /*4(_20), a"d to -f.oo if G is of real rank at least two.

These two theorems permit the computation of A(G) for any connected semi-simple Lie group G with finite centre, and for some with infinite centre, and for any lattice in such a group. It should be pointed out that the calculation of A(SO(l,7i)) was carried out by J. De Canniere and Ilaagerup [15], and that of A(SU(l,n)) by Cowling [11]. It should also be pointed out that M. Lemvig Hansen [29] showed by means of direct calculation that A(G) = 1, for the case where G is the infinite covering group of SU(l,n). This result also fol­lows from the calculation of A(SU(l,n)) and parts (ii) and (iv) of Theorem II.2, and the fact that there are lattices in SU(l,n), but Lemvig Hansen has priority.

The calculation of A(G) for a semisimple Lie group G is a messy busi­ness. The bulk of the paper of Cowling and Ilaagerup is dedicated to obtaining estimates for A(G), for G of real rank one, with finite centre, while the paper of Ilaagerup deals with SL(3,R) and' Sp2(R), which, up to a finite central factor, are subgroups of any simple Lie group of higher rank. The key to these computations is Haagerup's observation [24] that, by averaging over the cosets of a maximal compact subgroup K, it can be shown easilythat the existence of an approximate identity for multiplication in M0A(G) implies the existence of a /v-biinvariant approximate identity, and that for a A'-biinvariant func­tion u on G, IMU/oA = ||«UAH|Z?, where ,47V is the solvable subgroup arising in the Iwasawa decomposition. The estimate of A(G) from above in Cowling and Ilaagerup [12] is obtained by producing an explicit approximate identity, using the so-called spherical functions <j>\ on G. These have a representation — the so-called noncompact picture — involving an integral over N, which is closely related to the unitary representations of the group AN, and which permits estimation of | |^AUW||B-' ^ a e estimation of A(G) from below involves proving an inequality of the form

liminf (I^ITVIU > A(C), »€/

19

for everv approximate identity for multiplication {wt- : x €.,/} of K -bi in variant «

A(G)-Lunctions. This inequality is proved by finding an explicit element $ of VN(N) of norm one such that

l iminf | (^ ,$) |>A(G);

this $ is a distribution which involves integrating over some directions in N and differentiating in others. These distributions have cropped up in harmonic analysis on Heisenberg groups in various ways; see, e.g., D. Geller and E.M. Stein [19].

II.3 Applications

THEOREM II.4. Let M be a von Neumann algebra, and let M(M) be defined as above. Then

(i) if Mi and M2 are isomorphic, M(M\) = M(jM2);

(ii) if Aii C A^2 and there is a conditional expectation from M2 to Aii,

M(Mi)<M(M2)\

(iii) if T is a discrete group, M(V7V(r)) = A(r).

The first part of this theorem is obvious, and the second is proved in Cowling and Haagerup [12], while the last is due to Haagerup [24]. This theorem allows us to compute M(M) for certain M — the von Neumann algebras of lattices in sernisimple Lie groups. It is worth noting explicitly that (iii) fails for connected groups. Indeed, if G is Sp(l, n), then VN(G) is a direct integral of 1^ factors, associated to the irreducible unitary representations of G, and it is easy to show U{VN\G)) = 1, although A(G) = 2n - 1. The point is that operators on VN(G) with finite-dimensional range spaces need not give rise to pointwise multiplier operators on VN(G) or on A(G), and it is rather surprising that (iii) holds at all.

It seems likely that M(A4i ®sp M2) = M(Mi) • M(M2)> but there is no proof known to the author at the time of writing. The proof of the analogous result for groups hinges on a little extra structure; it seems that this structure also underlies the proofs in Part III.

It will be clear how to combine these theorems to obtain the following corollary.

COROLLARY II.5. Let Ti and T2 be lattices in 5p(l,fti) and Sp(l,n2) re­spectively, where nj ^ n2. Tiien VAT(ri) and V7V(r2) are nonisomorphic.

20

As we observed in Part I in our discussion of Connes' conjecture in the theory of operator algebras, this is a strong form of rigidity theorem.

Here is a corollary due to Haagerup, which should have appeared in Cowling and Haagerup [12]. We give a sketch of the proof which the reader may wish to fill out — if so, the necessary details are in [12]. -

COROLLARY II.6. There is an increasing sequence of discrete groups {Tn : n E N}, whose von Neumann algebras {Mn '- n G N} form an increasing sequence of pairwise nonisomorphic factors of type Hi.

Proof. We identify Sp(l,n) with the group of all (n+1) x(n-f 1) matri­ces, with entries in the set of quaternions H, which preserve the quaternionic\ form Q —-

Q(v',v) = v'0v0-v'1vl- v'nvn Vv,t/ .€Hn + 1 ,

where v = (v0, v r , . . . , vn) in H" + 1 and (a + fci + cj-f dk)~ = a — M-c j -dk , for all a,b,c,d in R. We let Tn be the subgroup of Sp(l,n) of all quaternionic ma­trices M such that M — I has entries in 3Hjnt, where Hjnt = {a + 6i + cj + c/k : a,b,c,d e Z}. Then Ti C T2 C • • • C Tn C • • •, and Tn is a lattice in Sp(l,n), so that A(rn) = 2n— 1. Last, but not least, the centre of Tn is trivial, so that V7V(rn) is a Hi-factor. 0

An alternative approach to similarities and differences between groups is through ergodic theory. Discussion of the ergodic theoretic version of these ideas in requires some more notation and definitions.

A discrete group T is said to act on a measure space A' if there is a mapping (y,x) »—• 7 • x from V x X to X, such that x>-• i • x is measurable for each j in T, ex = x for all x in X (where e is the identity of F ) , a n d 7i • (72 • x) = (7172) • x for all x in X and 71,72 in P. The action is said to be ergodic if the only r-invariant subsets of X are null or conull (i.e., their complements are null), and free if 7-2? = x for some x only if 7 is the identity. We say that groups Ti and T2 have orbit equivalent finite ergodic theory if there exist finite measure spaces X\ and X2 on which Pi and T2 act freely and ergodically, together with a measure-preserving bijection <j> : X\ —+ X2

which maps Ti-orbits onto T2-orbits. It is known that all countably infinite discrete amenable groups have orbit equivalent finite ergodic theory, so that not having orbit equivalent finite ergodic theory is another strong expression of the difference of two groups. Zimmer, extending the ideas of Margulis, has proved ergodic theoretic rigidity theorems for lattices in semisimple Lie groups

21

of higher rank. (See Zimmer's book [45] for a useful discussion of semisimple groups and ergodic theory). In Cowling and Zimmer [14], we show (inter alia) that lattices in 5p(l,n), for different values of n, do not have orbit equivalent finite ergodic theory.

Suppose that T is a locally compact group, acting ergodically on the measure space X. Then we consider the Hilbert space L2(T x X), on which T acts by the formula

WT)/)(T ,^) = / ( T - V , 7 - 1 - ^ ) V*€X, vT,T 'er,

for any / in L2(T x X), and on which L°°(X) acts by the rule

(ir(a)f)(1)x) = a(x)f(1)x) Vx € X, VT € I\

for any a in L°°(X) and any / in L2(r x X). We denote by A the commu­tative algebra of operators on L2(V x A") generated by 7r(L°°(A')), and by M the larger algebra generated by A together with the operators 7r(7) with 7 in T. In ergodic theory, it is natural to deal with the pair (M,A). Unfortunately, we have difficulty in doing this, and we deal with the pair (r, Ar), where and X has finite invariant measure; however, we also need to use M and A.

The definition of N(r, X) depends on the existence of nets of completely bounded operators {7} : i € 1} on M} having the properties that

imiU<c Vie/

Tim —• ?TI weak-star V?n€*M,

and for each i, there exists a finite subset Si of V such that

Ti(M) C "£ *(7)A-

These three conditions correspond naturally to the conditions imposed in the definition of A(G) and M(M) for a group G and a von Neumann algebra M. We define N(Y%X) as follows:

N(r,Ar) = inf {C G [l,oo) : the above conditions hold for some net {X}}}.

It is quite easy to modify the techniques developed for von Neumann algebras to deal with N(r,A") and prove the following result.

22

THEOREM II.7. Let T denote a discrete group acting freely and ergodically on a finite measure space X. Then

(i) if(TuXi) and (T2lX2) are orbit equivalent, N(ruXi) = N(r2, A'2);(

(ii) A(r) = N(r,x).

This theorem, combined with the previous ones, proves the statements made earlier about the nonorbit equivalence of the finite ergodic theories of lattices in different groups Sp(l, n). This, and other rigidity results on actions of lattices in Sp(l,n), appear in [14].

It seems likely that these results could be developed; the case of a con­tinuous group action might be treated, the role of the so-called full group is , not yet understood, and there are indications that a theory in terms of M and A could be worked out which would generalise to treat the so-called cross products of groups and C* -algebras; however, we shall not attempt this here.

,(

III. Factoring out central subgroups

Our aim in this part is to prove that A(G) < A(G/Z) if Z is any central subgroup of the discrete group G.

III. 1. Introduction.

Let G be a discrete group, with the counting measure. Write Sx for the function on G which is 1 at x and 0 elsewhere, and A for the left regular representation of G on £2(G) —

[\(x)f](y) = f(x-1y) Vx,yeG.

The reduced C*-algebra of G, C*(G), is the completion of Cl(G) in the norm | | / | |c . (||/||c. = \\\(f)\\op; here A(/), for / in i*(G)x is the operator E,eG/(*)A(*).on C2(G));

Let Z be a central subgroup of G. We denote by x the coset xZ of x in G/Z, and write <r : G/Z —• G for a cross-section, i.e. a map such that <T(X) 6 x for all x in G/Z. Abelian harmonic analysis on Z enables us to partially decompose C*(G).

vTake a character \ °f Z ( w e write \ 6 Z). The partial Fourier tran­sform fx of an £l(G)-function / is the £l (G/Z)-function given by the rule

fx(i) = £ . /M±)*)x( i ) Vi € G/Z.

23

It is easy to check that / can be reconstructed from the functions fx, as x varies over Z. Indeed,

f(x) =z fx(i)x(^(i)~ix)dx

(the Ilaar measure dx on the compact group Z is normalised to have total mass one). Furthermore, from Fubini's and Plancherel's theorems,

( ? l l /x l l^-v) I / 2 = ( E z\k(i)?dx)"' ieo/z

= ( E Ei/w^)i2)1/2

ieG/z zez

= H/ll2; it follows that £2(G) is naturally isomorphic to L2(Z;£2(G/Z)).

We define the twisted convolution of f(G/Z) -functions / and g by the formula

f*x9(*)= Y, f(y)9ty~l*)xHy,y~x*)) v*€G/zt y€G/Z

where u(x,i)) = <r(y)<r(xy)~lcr(x) Vx,y (= G/Z.

The definition of twisted convolution is motivated by the fact that , for ^ (G) -functions / and g,

(f*g)x = fx *x gx.

Indeed, for x in G/Z,

zeZyeG

= £ E Eyw»)*')»(*'"M»)-?«(i)*)x(«-)-zeZyeG/z z'ez

= E E E/w»)*')x(*')»w»rMi)»^1)3f(**'-1)

zeZyeG/z z'ez

zeZyeG/z

*eZyGC?/Z

yGG/Z

24

as required. This allows us to decompose C*{G) into an integral of C* -algebras over

Z. Motivated by the fact that \(x)f = 6X* f for x in G and / in i2(G), we define the twisted translation operators Ax(x), for x in G/Z, by the rule

[Ax(A)/].(y) = («**x/)(y).

= xM«.*" 1 y) ) / ( i " 1 ») Vx,y€G/Z.

Then A(x)/ can be recovered by integrating together the various Ax( i) /X ' s . We denote by CX{G/Z) the C*-algebra obtained by completing £1{G/Z) in the norm ||/ | |c», defined by the rule

ll/llc; = Px(/)||.

From the definitions, extended to allow (?{G)-functions, for / in £l(G), and 9 in £2(G),

W)9)x = *x(fx)9x-

It is standard that the map / i-+ fx extends to a map of C*{G) onto CX(G/Z), and that | | /x | |c* < | | / | |c ; - It is also known that for any g in CX(G/Z), there exists h in C*(G) of the same norm such that hx=g.

We shall need the following .well-known result later.

LEMMA III.l. Suppose f is in C*(G). Then fx(x) is well defined for every x in G/Z and \ in Z, and fx(x) depends continuously on x m % for every fixed x in G/Z.

III .2 . The spaces A(G) and AX(G/Z).

Recall that A(G) denotes the Fourier algebra of G; A(G) is the space of all functions u on G such that, for some hn,kn in £2(G) (n = 1,2,...),

oo

Ell*-IWIU»<'oo (i) n = l oo

and u(x) = Y^{X(x)hn, kn) Va;6G. (2) n = l

The norm of u in A(G) is then the infimum of all sums (1) over all represen­tations (2) of u. We define AX(G/Z) similarly; a function u on G/Z belongs

25

to AX(G/Z) if and only if, for some hnikn in t2(G]Z) (n = 1,2,...),

oo

£' | |A»IWMI» < 00 n = l

oo

and u(x) = ^(Ax(x)/in,ibn> Vi G G/Z. n = l

The norm on AX(G/Z) is defined analogously to that on A(G). Such spaces were first studied by G. Arsac [1]'.

Because A is a direct integral of Ax 's, every u in A(G) can be de­composed as an integral of ux 's. Indeed, if u is in A(G) (and has compact support, at least initially), then we may represent u thus:

oo

n = l

whence ux = ]T](Ax(-)/i»x, ^ x ) , r » = l

and u(x) = ~ ux(x)x((r(x)~1x)dx. (3)

oo

Note that z(Y^\\knXh\\h„x\U)dx n = l

oo

= £#»»XIWI*»xlMX • n = l

CO 1 / 9 1 / * )

< £ Gll*-xll! rfx) (~ll*»xll»<fo) n = l

oo

= EW2l|fc-ll2; n = l

consequently z l W U * dX < ||WIU-

Conversely, it can be shown that if (3) holds and £||%|UX d\ < oo, then u is in A(G) and

IMU <z\\ux\Wdx-

We now state and prove a technical result.

26

LEMMA III.2. Suppose that S is a finite set, of cardinality \S\. Given any func­tion d : S X S —* C, there exist functions rm,sm : S —• C (m = 1,2,...) such that

CO

r n = l CO

and £||rm|U|«m||»<|S|s|MU.-m = l

Proof. Label the points of S x S by the integers 1,2,..., |5 | 2 . If m > |5 | 2 , put rm = sm = 0. If 1 < m < \S\2, and m corresponds to (a:0,yo), then let rm — d(x0, y0) 6So and 5 m = ^ 0 . D

It should be noted that better results are possible here. These are related to early work of A. Grothendieck [22].

We now prove a series of lemmas which will show that, for a finitely supported function u on G/Z, \\U\\AX varies continuously with x, and will show the existence of special representations of u.

LEMMA III.3. Suppose K is a finite subset of G/Z, x JS in Z, and c is in R+ . Then there exists a finite subset S(K, x, €) °f G/Z such that, for any function u on G/Z with supp(w) C K, there exists a representation, u = ]CnLi(^x(')^n> n)> of u such that

»

supp(/in), supp(fcn) C 5(/C,x,c) CO

a«d EH*»H»ll*»ll2^(i + 0 N k -n = l

Proof. An argument of Cowling [10} shows that a single function has such a representation. An e-covering of the unit ball of the subspace of AX(G/Z) of functions supported in K is then used. (For a detailed example of the use of c-coverings, see Lemma III.6). Thanks are due to U. Haagerup for pointing out this extension of my argument. D

LEMMA III.4. Suppose that K is a finite subset of G/Z, x iS J*11 Z-t a n ( l e is in R+ . Then there exists a neighbourhood U of 1 in Z such that, for any x' in

U and any function u on G/Z with supp(u) C A', there exists a representation,

u= E^ilVxl')^'^). of u sucli that *

supp(/*/n), supp(^) C S(K, x, 0

oo

and £lWJWIO»<(i.+ 0aIMk. n = l

27

Proof. By the previous lemma, we may represent u, which certainly belongs to Ax{GfZ)y in the form

oo u0*0 = YlM^n^n)

n = l oo

J2 J2 x(w(*,« ly))hn(i ly)kn(y)

n = l y € G / Z

oo

n = l y € G / Z

where supp(^„), supp(fcn) C S(K, x\c) and CO

•DW*ll*..ll2<(i + <)IMk.. n = l

Consider the function D : S(A7x»f) x S(A',x,e) —• C defined by the rule

^(i,2/) = X/(^(i'*"1^))-As i and y vary, (^(yi'"1,*) varies over a finite set. Therefore there exi­sts a neighbourhood U of 1 such that, for any x' in ". £/", |D(ir,r/) - 1| < e/|5(AT,x,e)|2. We apply Lemma 1II.2 and renumber: there exist functions rm,sm : S(K,x>0 —• C (rn = 2,3,...) such that

CO

£>(£,?/) - 1 = 5 ^ rm(i)sm(y) m = 2

oo

and 5Zlkm||oo||vlloo'<^'' m = 2

Now let ri'= «i = 1. We may now represent u as follows:

CO OO

n = l m = l y€G/Z

CO CO

n = l rn=l

clearly supp(/inrm) G 5(A',X,e) and supp(fcnsm) C S(A',;v:,<f)> a n c l

CO CO

] C £ J | f c n r T O | f 2 | | f c n * m | | 2 n = l m = l

CO CO

<EEii*»iwi iwir™ii»ii»-»ii' n = l m = l

v2 <U + OaIMk- a

28

We now show that, for a fixed, finitely supported function u on G/Z, the function x H-+ llwIUx *s continuous. We use Lemma III.4 to do so, but later feed the continuity result back to strengthen Lemma III.4.

LEMMA III.5. Fix a finitely supported function u on G/Z. The function x h~+

|IWIUX is continuous.

Proof. It follows from Lemma III.4 that the map x *~* IMUX *s upper semicontinuous. Suppose it is not continuous. Then we can find a net x»> converging to x lu Z, with the property that

NU X >limsup||ti||i4xi..

By the Hahn-Banach and Kaplansky density theorems, there exists a finitely supported / on G/Z such that ||/||c* = 1 and

| ] £ /(iMi)|>lims»p|HUx.. (4) ieG/z Xi~*x

Now take g in C*(G) such that gx — /, and \\g\\c; — 1. On the one hand, gXi —• gx pointwise as xi —* X> s o ^ i a t

x]jl2}x Yl ^ ( i ) l i ( i ) = Yl /(^M*) (5)

xeG/z „_ x^G/z

(from Lemma III.l). On the other hand,

i £ fe.(*w*)i<n*xiiic»iii««iu„ x£G/Z

<IHk,. . (6)

Formulae (4), (5) and (6) are contradictory, so the assumption of the noncon-tinuity of \ »-* ||«||x, must be wrong. D

LEMMA III.G. Fix a compact subset K of G/Z, x in Z, and small e in R+. Then there exists a neighbourhood U' of 1 such that,

i - 3 c „ 1 + 6.. M J^M\AX<\\U\\AX,X<~\\U\\AX

for all u on G/Z supported in K and for all x' m U''.

Proof. It suffices to show that there exists U' such that

(1 - 3c)/(I - e) < ||«|Ux,x < (Z + 0/(1 - 0

29

for all it in the unit ball of Ax, with support in K, and all \* in U'. As the space of functions supported in K is finite dimensional, there

exist ui , . . . ,uy in the unit ball of Ax(G/Z), vanishingoff K, such that for any function u in AX(G/Z) which vanishes off K, there exists j such that ||it - UJ\\AX < c. Now since x •-+ llMilUx Is continuous, by Lemma III.5, there exists a neighbourhood U' of 1 such that

l + ^ > l l « i l U x v , ' > ( l - - ^ V x ' G ^ Vj6 {1- . . . , J } .

Take it in AX(G/Z) of norm 1, vanishing off A'. Choose ?t;i such that ||u - UJJIAX < f> set ai — 1, and then, inductively define c*2, • • • and « j a , . . . :

N

<xN+i= \\u-^2anujn\\ n = l

JV + 1

n = l OO

T h e n it = /JcywUf-w.

n = l

oo

It follows that IMUX,X > | |u;JUx,x - £ ^ , | h n | U x , , n = 2

oo

>(l-f)-£f-'(l + e)

and

_ 1 - 3e

" 1 - e '

CO °° 1 I

IMk<, < 2>dWk.x •'< (i + ^ E ^ - 1 = 137-n = l n = l

as required. D

THEOREM III.7. Suppose that K is a finite subset of G/Z, and c is in R+. Then there exists a finite subset F(K,c) of G/Z such that, for any u on G/Z which vanishes off K, and any \ in Z, there is a representation u = 52n=i(^x( ) »> n)> where supp(hn),supp(kn) C F(K,e) and

DwwiMi»<u+oiMk-

30

Proof. From Lemmas III.4 and III.6, for each xo m Z, we can find a neighbourhood U(xo) and finite set S(K} xo, U(xo)) such that every u with support in K admits a "good" representation in Ax{GjZ), for all % in U(xo). Because Z is compact, it can be covered by finitely many such U(xqh.-We take F(K} e) to be the union of the corresponding S(K}xo>U(xo))- D

COROLLARY III.8. Suppose that K is a finite subset of G/Z and e is in R+. Then there exists a neighbourhood U" of the identity in Z such that

\MAX,X < (1 + e)\\u\\Ax Vu £CK(G/Z) VX' € U".

Proof. By Theorem III.7, there exists a finite subset F(K}c) of G/Z such that every u in CK(G/Z) has a representation

CO

n = l oo

where ^ I I M W I ^ I h < (l + * ) I M k n = l

and supp(/?.n), supp(fcn) C £/".

As in Lemma III.4, we may write

u ( i ) = X ] 1Z X /(^(ir^1y))^n(«"1y)^a(j/)xX /(o;(x,i'17/)); n-ly^GjZ

continuing the argument of Lemma 111.4, with the additional uniformity in the support set yields the corollary. •

III.3. Completely bounded multipliers.

In this section, we shall prove our main new result.

THEOREM III.9. If Z is a central subgroup of a discrete group G, then

A(G)<A(G/Z).

Proof. We recall that A(G) is the infimum of all C such that there exists a net {«< : i € 1} of (compactly supported) A(G)-functions with the properties

INko* < c (7) Ui —• 1 pointwise. (8)

31

Take an appropriate identity of compactly supported A(G/Z)-functions, {u{ : i e 1} say, such that (7) and (8) are satisfied. We extend «,• by periodicity to a function on G. For x in Z, define vx : G —• T (T is the unit circle in the complex plane) by the rule vx(x) = x(<T(%)~ix) for all x in G.

We claim that w< t/x is in M0A(G) and that

IK^x||Af0>i <C(i,x),

where \im C(i,x) = \\ui\\M0A> '

Assuming this to be true for the moment, we choose a net {m,j : j e J} of A(Z)-functions (whence fhj € tl(Z)) whose supports shrink to {1}, with the properties that m.j > 0 and ^nij(x)dx = 1. Define the lexicographically ordered net {wij : i G I,j € «/} by the formula

w i,j =5uivXmj(x)dx-

Since K j M I < K M I l ^ x ( * ) ^ ; ( x ) ^ l

= MayJIImjHi)-1^)! ,

and u,\ vanishes off finitely many cosets of Z in G, while TB,- is summable, i^ij is in ^ (G) , a subspace of A(G). Further,

ll^i,iiU/o>i <^ ||W«^X||A/OJ4

<z C(i>x)™j(x)dx

and \\m ^C(i}x)^j(x)dx = ||WI||A/0>I-

Finally, limif;X|j(x) = limu,-(z) fhj (x cr(x))

= Ui(x),

by the choice of mj. It follows that, if dtj = llwijIlj^llujIlAfo/tj then {dj Wij i El, j G «/} is a net in A(G) satisfying the conditions

WCijWiAWoA <c

and linilimGijK/ij = 1 pointwise, i j

whence A(G) < C} and we are done. It suffices therefore to prove our claim.

32

To prove that

|kv*||AJo,4 <C( i ,x ) i '

where limC(i,x) = IKHM<M> w e firs* observe that it is enough to show that

||UIV*||AM has these properties (|| • ||MA denotes the multiplier norm, not the completely bounded multiplier norm). For by considering groups GxF, where F is a finite group, we can derive the result for the M0A-novm. Indeed, much as in the characterisation of M0A(G) in Theorem II. 1 (due to J. De Canniere and Haagerup [15]), we can show that for any function u on G, ||M||A/0/I(G) = suPneN \\u® M\M0A(Gxsn)i where Sn is the permutation group on n letters.

Now take w in A(G). We must estimate ||t/,f xt/||^ = ||^x(w««)||>4-Observe that supp(w,w.) C supp(i/,t), and ||i*«t*||>t < ||«i||A/^t|h'||.4- It is the­

refore sufficient to prove that, for any finite subset A' of G/Z (in particular, K can be supp(u;)), ||VVM||,4 < C(AT,x)IMU *° r a ^ u ul M&) w ^ n support in A', where lim C(K. y) = 1. /

We employ the partial Fourier transform:

ll(»x«)IU=2llK«)x'lk,<*x', and (vxu)x'(x) = J>(VXH)((T(X)Z)X'(Z)

zez

= 5^x(2r)w(ff("i)2:)x'(*) zez

= uxx,(x).

We apply Corollary III.8, and we deduce that

z l K ' I k x * ^ ' ^z l l«xx ' lk^x ' < C(K,X)z\\uAAx,dx!

where C(K, x) —• 1 a s X - + 1> a s required. D

T h e structure of MQA(G) and P(G).

In this section, we show that MQA{G) can be decomposed over the central subgroup Z in much the same way as B(G). We assume throughout this section that G is countable. Until further notice, we shall assume only that Z is an amenable subgroup of G.

Let Z be the maximal ideal space of £°°(Z)\ there is a natural correspon­dence between bounded functions on Z and (necessarily bounded) continuous

33

functions on Z. Left and right multiplication on Z extend to left and right actions of Z on Zy which gives rise to "left translation" and "right transla­tion" representations of Z on C(Z). Take an invariant mean M on £°°(Z); this corresponds to a Z-biinvariant Borel probability measure ft on Z. We denote by L2{Z) the usual Lebesgue space on Z constructed relative to /i. The left translation representation of Z on C(Z) extends to a unitary repre­sentation on L2(Z). We may even consider vector-valued functions on Z and obtain left translation representations on spaces of these. All such representa­tions will be denoted A . Similarly defined right translation representations will be denoted p.

Recall from the previous section that if u is a MQA(G)-function, then there exist a Hilbert space (which we may and shall assume is separable) ft and bounded ft-valued functions S and 11 on G such that

Af«(v) =<=(•), H(.)). ()

Take a basis {e, : i € /} for ft, and define E< : G -+ £°°(Z) by the formula

Ei(g)(z) = (E(gz)}ei) V<7€G, Vz € Z.

Clearly, for each g in G, E,(<7) can be extended to a bounded C(Z)-function, denoted E-(p), and it is easy to show (by considering arbitrary finite sets of indices) that

we conclude that S extends to a bounded function E" on G whose values are bounded, weakly measurable .'ft-valued functions on Z, defined by the formula E11 == £ i € / E | e , ; it is easy to see that E" lies in e°°(G;L'2(Z;7i)), and ll-8!! < Halloo- We define II8 similarly. Now, by definition, for any x,y in G and z in Z>

u(yx-l)=Mu(x,y) = {E(x),ll(y)) = (E(xz)}U(yz))

and by applying the invariant mean M to the right hand expression, which corresponds to integrating on Z with respect to /i, we see that

Mu(xyy) = ((~t(x)J\*(y)))

where (( )) denotes the inner product in L2(Z;ft). The function E" involved in this new representation of Mu has the property that

E\xz){z') = E(xzz') = E[(x)(zz') = [A^-^x) ]^ ' ) V* € G, Vz, z' 6 Z,

whence S*(zz) = A(z"1)EB(x').

34

The function H" satisfies a similar condition. This condition is similar to the condition which vectors in the representation space of an induced repre­sentation satisfy, and hence makes u resemble a matrix coefficient of such a representation. As the representation induced from a reducible representation 7r can be decomposed into the integral of the induced representations induced from the irreducible components of ?r, so can the matrix coefficients of such induced representations be decomposed, and analogous analysis will allow us to decompose the MoA(G)-function u.

Hereafter we suppose that Z is a normal amenable subgroup of G, and that the invariant mean M on Z is invariant under conjugation by elements of G, i.e., if v and w are £°°(Z)-functions and v(-) = w(x~x • x) for some * in G, then Mv = Mw. For any k in £1(Z)i and any bounded ft-valued function E on G, we define p(k)E to be the bounded H-valued function x »-»• Ylz € ZE(gz)k(z) on G. We may repeat the construction of E" given above, but starting with p(k)E, and obtain a function p(k)E* on G with values in L2(Z;7i), or we may define p(k) on L2(Z,7i) in the natural way, and hence define />(Ar)S*; the result is the same whichever way we proceed. We define p(k)ll* similarly. Notice that, for any k\ and k2 in £l(Z),

((p(^i)El,(a;1y)Jp(^2)H1,(x2y))> = ]T^ fc(u>i) ]PJc(w2)M2(E(xlyzw1),li(x2yzw2)) wi$Z wzE'Z

= ^Kw^) ^^{w2)Mz{E(xiyzy'~lywiy''ly),l\{x2yzy~lyw2y~ly\ w\G.Z w?€.Z

= ^ k(wi) y£'k(w2)Mz(Z(x1yzy~1ywiy~1),ll(x2yzy-1yw2y~1)) wi£Z wjCZ

• . - • ' • . = ^ k(wi) ^Je(w2)Mz(E(xizyw1y-1)yll(x2zyw2y-1)) ,

= ]T] k(y~1w1y) ^2 Ky~1W2y)M2(E(xizw1))]l(x2Zw2)) w\£Z Wi£Z

where ky is the function z >-• k(y"1zy). This calculation is parallel to the computations in the Mackey theory of induced representations which show that any irreducible unitary representation of G, when restricted to the nor­mal subgroup N, decompose into representations supported in a G-orbit in N (or something like an orbit when N is not type I).

35

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Michael COWLING - School of Mathematics University of New South Wales, P.O. Box 1, Kensington N.S.W. 2033 - AUSTRALIA