What is an Operator Algebra?

What is an Operator Algebra?

Aristides Katavolos

6th Summer School, Athens, 3-7 July 2017


Short answer:

It is an algebra of bounded linear operators on a Hilbert space.


Better short answer:It is a normed algebra (A ,‖·‖)that can be isometrically represented as an algebraof bounded linear operators on a Hilbert space.

So A is:• a vector space,

• a ring,

• a normed space with ‖ab‖ ≤ ‖a‖‖b‖ [usually complete].

[• Sometimes closed under weaker topologies.]

Need to consider all the (completely) isometric representations ofA as operators on Hilbert spaces.

The algebra B(H )

Let H be a Hilbert space. The algebra of all bounded linearoperators T : H →H is denoted B(H ). It is complete underthe norm

‖T‖ := sup{‖Tx‖ : x ∈ b1(H )}

Additionally, it has an involution T → T ∗ defined via

〈T ∗x ,y〉= 〈x ,Ty〉 for all x ,y ∈H .

[Theorem: There exists T ∗ ∈B(H ) satisfying this equality.]This satisfies

‖T ∗T‖= ‖T‖2 the C ∗-property.

The algebras C (K ),C0(X )

Let K be a compact Hausdorff [or metric] space.

C (K ) := {f : K → C : continuous}

• a vector space for pointwise operations,

• a ring for to pointwise multiplication,

• a Banach space for the supremum norm ‖f ‖∞

:= sup |f (t)|.

• has involution f → fwhich determines real functions (f = f ), positive functions f f .

Let X be a locally compact Hausdorff [or metric] space.

C0(X ) := {f ∈ C (X ) : ∀ε > 0 ∃K ⊆ X compact s.t. |f |K c |< ε}

Exercise

The algebra C0(X ) can always be faithfully represented as anoperator algebra (on some Hilbert space H ):

There exists an isometric *-morphism (a faithful *-representation)π : C0(X )→B(H ):

‖π(f )‖= ‖f ‖∞

π(f ) = (π(f ))∗

π(f + λg) = π(f ) + λπ(g)

π(fg) = π(f )◦π(g) f ,g ∈ C0(X ), λ ∈ C.

Abstraction: C*-algebras

Definition

• A Banach algebra A is a complex algebra equiped with acomplete submultiplicative norm:

‖ab‖ ≤ ‖a‖‖b‖ .• A C*-algebra A is a Banach algebra equiped with an involution1

a→ a∗ and a complete submultiplicative norm satisfyingthe C*-condition

‖a∗a‖= ‖a‖2 for all a ∈A .

1that is, a map on A such that (a+ λb)∗ = a∗+ λb∗, (ab)∗ = b∗a∗, a∗∗ = afor all a,b ∈A and λ ∈ C

The morphisms

A *-morphism φ : A →B between C*-algebras is a linear mapthat preserves products and the involution.

It can be shown that morphisms are automatically contractive, and1-1 morphisms are isometric (algebra forces topology).

Basic Examples of C*-algebras

CC (K ) : K compact Hausdorff: abelian, unital.

C0(X ) : X locally compact Hausdorff: abelian, nonunital (iffX non-compact).Commutative Gelfand-Naimark: All abelian C*-algebras canbe represented as C0(X ) for a unique X .

Mn(C) : A∗ = conjugate transpose,‖A‖= sup{‖Ax‖2 : x ∈ `2(n),‖x‖2 = 1}: non-abelian, unital.

B(H ): non-abelian, unital.Gelfand-Naimark: All C*-algebras can be represented asclosed selfadjoint subalgebras of B(H ) for ‘suitable’ H .

Other Operator Algebras: The disc algebra

A(D) := {f ∈ C (T) : f (k) = 0 for all k < 0}.

A closed subalgebra of the C*-algebra C (T) but not a*-subalgebra: A(D)∩A(D)∗ = C1: antisymmetric algebra.

Representations on Hilbert space• Restrict any *-representation of C (T); for instance,multiplication operators on L2(T). The C*-algebra generated bythis representation is abelian.

• But also, represent as operators on `2(Z+):f → [aij ] where aij = f (i − j).The C*-algebra generated by thisrepresentation is not abelian: It contains the unilateral shift S andhence also its adjoint S∗, which do not commute.

Other Operator Algebras:

Tn = {(aij) ∈Mn(C) : aij = 0 for i > j} (upper triangularmatrices).A closed subalgebra of the C*-algebra Mn(C) but not a*-subalgebra. Here Tn∩T ∗n = Dn, the diagonal matrices: amaximal abelian selfadjoint algebra (masa) in Mn.

Moo(C): infinite matrices with finite support.To define norm (and operations), consider its elements asoperators acting on `2(N) with its usual basis. This is aselfadjoint algebra, but not complete.Its completion is K , the set of compact operators on `2: anon-unital, non-abelian C*-algebra.

Other Operator Algebras: Group algebras

Let G be a (countable) group (think of Z or F2). The Hilbert space`2(G ) has o.n. basis {δs : s ∈ G}. The group G acts on `2(G ) via

t→ λt ∈B(`2(G )) where λt(δs) = δts , s ∈ G

(or λt(f )(s) = f (t−1s), f ∈ `2(G )).

• The reduced C*-algebra C ∗r (G ) := span{λs : s ∈ G}op - closedin the norm of B(`2(G )).

Each λs commutes with the right repr. ρ where ρt(δs) = δst .Hence C ∗r (G ) commutes with every ρt . Can consider

• The von Neumann algebra of GL (G ) := {X ∈B(`2(G )) : Xρt = ρtX ∀t ∈ G}.

This is larger than C ∗r (G ), when |G |= ∞ (why?)

What about a semigroup S ⊆ G??

Gelfand Theory

Theorem (Gelfand-Naimark 1)

Every commutative C*-algebra A is isometrically *-isomorphic toC0(σ(A )) where σ(A ) is the set of nonzero morphismsφ : A → C which, equipped with the topology of pointwiseconvergence, is a locally compact Hausdorff space. The map is theGelfand transform:

A → C0(σ(A )) : a→ a where a(φ) = φ(a), (φ ∈ σ(A )).

The algebra A is unital iff σ(A ) is compact.

Gelfand Theory

In more detail:σ(A ) is the set of all nonzero multiplicative linear forms(characters) φ : A → C, (necessarily ‖φ‖ ≤ 1 and, when A isunital, ‖φ‖= φ(1) = 1) equipped with the w*-topology: φi → φ iffφi (a)→ φ(a) for all a ∈A .

When A is non-abelian there may be no characters (considerM2(C) or B(H ), for example).

When A is abelian there are ‘many’ characters: for each a ∈Athere exists φ ∈ σ(A ) such that ‖a‖= |φ(a)|.When A is unital σ(A ) is compact and A is isometrically*-isomorphic to C (σ(A )).

Spectrum and Positivity

Let a be an element of a unital C*-algebra A . Its spectrum is

σ(a) := {λ ∈ C : λ1−a not invertible in A }.

This is a compact, nonempty subset of C.

Definition

An element a of a C*-algebra A is positive (a≥ 0) ifa is selfadjoint (a = a∗) and σ(a)⊆ R+.

Write A+ = {a ∈A : a≥ 0}.If a,b are selfadjoint, we define a≤ b by b−a ∈A+.

Examples

In C (X ): f ≥ 0 iff f (t) ∈ R+ for all t ∈ X because σ(f ) = f (X ).

In B(H ): T ≥ 0 iff 〈Tξ ,ξ 〉 ≥ 0 for all ξ ∈ H.

Positivity

Proposition

In a C*-algebra, every positive element has a unique positivesquare root.

Theorem

In any C*-algebra, any element of the form a∗a is positive.

(Obvious in B(H ), key result for Gelfand - Naimark Theorem)

The GNS construction

Definition

A state on a C*-algebra A is a positive linear map of norm 1, i.e.φ : A → C linear such that φ(a∗a)≥ 0 for all a ∈A and ‖φ‖= 1.A state is called faithful if φ(a∗a) > 0 whenever a 6= 0.

NB. When A is unital and φ is positive, ‖φ‖= φ(1).

Examples

• On B(H ), φ(T ) = 〈Tξ ,ξ 〉 for a unit vector ξ ∈H ,or φ(T ) = ∑i 〈Tξi ,ξi 〉 where the ξi are ⊥ and ∑‖ξi‖2 = 1(diagonal ‘density matrix’).

• On C (K ), φ(f ) =∫fdµ for a probability measure µ (in

particular φ(f ) = f (t0) for t0 ∈ K - Dirac measure at t0).

• For a C*-algebra A , if π : A →B(H ) is a *-representation andξ ∈H a unit vector, φ(a) = 〈π(a)ξ ,ξ 〉.

Conversely,


Conversely,

Theorem (Gelfand, Naimark, Segal)

For every state φ on a C*-algebra A there is a triple (πφ ,Hφ ,ξφ )where πφ is a *-representation of A on Hφ and ξφ ∈Hφ a cyclic 2

unit vector such that

φ(a) =⟨πφ (a)ξφ ,ξφ

⟩for all a ∈A .

The GNS triple (πφ ,Hφ ,ξφ ) is uniquely determined by this relationup to unitary equivalence.

2i.e. πφ (A )ξφ is dense in Hφ .

Consequence: The universal representation

Theorem (Gelfand, Naimark)

For every C*-algebra A there exists a *-representation (π,H )which is faithful (one to one).

Idea of proof Enough to assume A unital. Let S (A ) be the setof all states. For each φ ∈S (A ) consider (πφ ,Hφ ) and ‘addthem up’ to obtain (π,H ). Why is this faithful? Because

Lemma

For each nonzero a ∈A there exists φ ∈S (A ) such thatφ(a∗a) > 0.

... and then∥∥π(a)ξφ

∥∥2 =⟨π(a∗a)ξφ ,ξφ

⟩=⟨πφ (a∗a)ξφ ,ξφ

⟩= φ(a∗a) > 0

so π(a) 6= 0.

Complete positivity

For n ∈ N, each A ∈B(H n) gives n×n matrix [aij ] withaij ∈B(H ) given by

A

ξ1...

ξn

=

a11 . . . a1n...

......

an1 . . . a2n

ξ1

...ξn

(ξi ∈H )

The map A→ [aij ] : B(Hn)→Mn(B(H )) is a *-isomorphism.So Mn((B(H )) is a C*-algebra with the norm of B(Hn).

Hence if A ⊆B(H ) is any operator algebra, Mn(A ) becomes anoperator algebra.

If φ : A →B is a linear map between operator algebras, define

φn : Mn(A )→Mn(B) by φn([aij ]) = [φ(aij)].

If A ,B are C*-algebras and φ is *-linear, so is φn.If φ is a *-morphism, so is φn.

Complete positivity

φn : Mn(A )→Mn(B) by φn([aij ]) = [φ(aij)].

Definition

A map φ : A →B between C*-algebras is positive ifa≥ 0 ⇒ φ(a)≥ 0.

It does NOT follow that φn is positive. Example: take φ(a) = a†

(transpose) on A = M2; clearly positive. Then

A=

1 0 0 10 0 0 00 0 0 01 0 0 1

is +ive, but φ2(A) =

1 0 0 00 0 1 00 1 0 00 0 0 1

is not +ive.

Definition

A map φ : A →B between C*-algebras is completely positive if φn

is positive for all n ∈ N.

Stinespring: Operator-valued GNS

Examples of completely positive (cp) maps:A *-morphism π is positive (because π(a∗a) = π(a)∗π(a)≥ 0 ∀a).Hence a *-morphism is a cp map (because πn is a *-morphism).A map a→ V ∗aV is a cp map.(here A ⊆B(H ) and V ∈B(H ,K )).Hence a→ V ∗π(a)V is a cp map. There are no others:

Theorem (Stinespring)

For every unital cp map φ from a C*-algebra A to B(H ) there isa triple (π,K ,V ) where π is a *-representation of A on K andV : H →K is an isometry such that

φ(a) = V ∗π(a)V for all a ∈A .

We say the *-rep. π is a dilation of the cp map φ via theembedding V : H →K .[There is also a uniqueness condition.]

GNS and Stinespring

In the remainder of these notes, we will sketch the proofs of theGNS construction and of Stinespring’s theorem, to emphasize theidea that Stinespring’s theorem is in essence an ‘operator-valuedGNS construction’.


Theorem (Gelfand, Naimark, Segal)

For every state φ on a C*-algebra A there is a triple (πφ ,Hφ ,ξφ )where πφ is a *-representation of A on Hφ and ξφ ∈Hφ a cyclic 3

unit vector such that

φ(a) =⟨πφ (a)ξφ ,ξφ

⟩for all a ∈A .

The GNS triple (πφ ,Hφ ,ξφ ) is uniquely determined by this relationup to unitary equivalence.

3i.e. πφ (A )ξφ is dense in Hφ .

Proof of GNS and Stinespring (Sketch)

1 Consider the linear space A .

2 Define semi-scalar product 〈a,b〉φ

:= φ(b∗a).

If A = C (X ) then 〈a,b〉φ

=∫X a(x)b(x)dµ(x).

3 Since φ is positive, 〈a,a〉φ

= φ(a∗a)≥ 0.By Cauchy-Schwarz the set N := {u ∈A : 〈u,u〉

φ= 0} is a

linear space.

4 Define K0 := A /N and complete with respect to

‖[u]‖φ

:=√〈[u], [u]〉

φto get the Hilbert space K

(here [u] = u+N ).

Proof of GNS II

5 A acts on A via π0(a)(b) = ab.

6 Now π0(a)(N )⊆N so π0(a) induces π1(a) on K0.

7 Prove that ‖π1(a)([u])‖φ≤ ‖a‖‖[u]‖

φ.

[For A = C (X ), ‖au‖2 ≤ ‖a‖∞‖u‖2 .]

Hence π1(a) extends to a bdd operator π(a) on K .

Easy: π : a→ π(a) : A →B(K ) is a *-representation.

8 Set ξφ = [1A ]. Then⟨π(a)ξφ ,ξφ

⟩K

= 〈π(a)[1], [1]〉K= 〈a,1〉K = φ(1∗a) = φ(a) . �

Stinespring: Operator-valued GNS

Theorem (Stinespring)

For every unital cp map φ from a C*-algebra A to B(H ) there isa triple (π,K ,V ) where π is a *-representation of A on K andV : H →K is an isometry such that

φ(a) = V ∗π(a)V for all a ∈A .

Proof of Stinespring (Sketch)

1 Consider the linear spaceA ⊗H = span{a⊗ξ : a ∈A ,ξ ∈H }(see the Appendix [30]).When H = C then A ⊗H 'A .

2 Define semi-inner product 〈a⊗ξ ,b⊗η〉φ

:= 〈φ(b∗a)ξ ,η〉H(extend linearly [30]).When H = C then 〈a,b〉

φ= φ(b∗a).

3 Since φ is cp prove 〈∑n an⊗ξn,∑m am⊗ξm〉φ ≥ 0.By Cauchy-Schwarz the set N := {u ∈A ⊗H : 〈u,u〉

φ= 0}

is a linear space.

4 Define K0 := (A ⊗H )/N and complete with respect to

‖[u]‖φ

:=√〈[u], [u]〉

φto get the Hilbert space K

(here [u] = u+N ).

Proof of Stinespring II

5 A acts on A ⊗H via π0(a)(b⊗ξ ) = ab⊗ξ

(π0(a)(b) = ab).

6 Now π0(a)(N )⊆N so π0(a) induces a map π1(a) on K0.

7 Prove that ‖π1(a)([u])‖φ≤ ‖a‖‖[u]‖

φ. Hence π1(a) extends

to a bdd operator π(a) on K .

Easy: π : a→ π(a) : A →B(K ) is a *-representation.

8 Define V : H →K : ξ → 1A ⊗ξ → [1A ⊗ξ ]. V satisfies

‖V ξ‖2φ

= 〈[1⊗ξ ], [1⊗ξ ]〉φ

= 〈φ(1∗1)ξ ,ξ 〉H = ‖ξ‖2Hhence is an isometry V : H →K and for all ξ ,η ∈H ,

〈V ∗π(a)V ξ ,η〉H = 〈π(a)V ξ ,Vη〉H = 〈π(a)[1⊗ξ ], [1⊗η]〉K= 〈[a⊗ξ ], [1⊗η]〉K = 〈φ(1∗a)ξ ,η〉H

so V ∗π(a)V = φ(a) . �

The (algebraic) tensor product

Consider the linear space A as a space of complex-valuedfunctions on a set S and the linear space H as a space ofcomplex-valued functions on a set T .

If a ∈A and ξ ∈H define a⊗ξ : S×T → C by(a⊗ξ )(s, t) := a(s)ξ (t).

The (algebraic) tensor product A ⊗H is defined to be the linearspan of such functions; it consists of all functions u : S×T → C ofthe form u(s, t) = ∑

ni=1 ai (s)ξi (t), where ai ∈A and ξi ∈H .

Thus the semi-inner product 〈·, ·〉φ

is defined on A ⊗H by⟨n

∑i=1

ai ⊗ξi ,n

∑i=1

bi ⊗ηi

⟩φ

:=n

∑i ,j=1

⟨φ(b∗j ai )ξi ,ηj

⟩H

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What is an Operator Algebra?

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