Banach Algebra Dynamical Systems?

Dong LI Shengzhi XU

1 Introduction

In quantum physics,time evolution or spacial translation of the observables is described by a(non-commutative)-dynamical system.A-dynamical system is a locally compact groupGacting by automorphisms on a-algebraA.It is a triple(A,G,α),whereAis aC?-algebra,Gis a locally compact group,andαis a strongly continuous action ofGonAas involutive automorphisms.A crossed product is a-algebra built out of a-dynamical system.The theory of crossed products of-algebras started with the papers by Turumaru[1]and Zeller-Meier[2].Given a-dynamical system(A,G,α),the corresponding crossed productAGis a-algebra.The crossed product construction provides a means to construct new examples of-algebras.One of the basic facts for a crossed product-algebraAGis that the nondegenerate representations of this algebra on Hilbert spaces are one-to-one correspondences with the non-degenerate involutive covariant representation of(A,G,α),i.e.,with the pair(π,U),whereπis a non-degenerate involutive representation ofAon a Hilbert space,andUis a unitary strongly continuous representation ofGon the same space,such that the covariance conditionπ((a))=is satisfied fora∈Aands∈G.

For a given-dynamical system(A,G,α),besides the “full” crossed product-algebraAG,there is another important crossed product-algebra,the reduced crossed productAG,which was defined by Zeller-Meyer for discrete groups in[2]and generalized by Takai in[3].They correspond to different completions of(G,A).The former corresponds to the universal representation,and the latter corresponds to the so-called regular representation which can be regarded as a subrepresentation of the universal one.In general,the two crossed productC?-algebras are different,but Landstad[4]proved that if(A,G,α)is a-dynamical system withGamenable,thenAGis equal toAG.This is an important theorem inthe-dynamical systems theory,since the reduced crossed product is more concrete and many familiar groups are amenable,such as the abelian group.It serves as a major step towards the duality theorem for the crossed product,and it is also a key point in Connes’Thom isomorphism forAαR(see[5])and Pimsner and Voiculescu’s results(see[6]).

This paper generalizes-dynamical systems to the general Banach algebra setting.We define a Banach algebra dynamical system(A,G,α),whereAis a Banach algebra,Gis a locally compact group,andαis a strongly continuous action ofGonAas isometric automorphisms.We construct the Banach algebra of the crossed productAGfrom these data.We also study that the representations ofAGare all the representations of the integrated form.There are some differences,since the representations of Banach algebras may not be contractive,unlike the case of-algebras.But in some sense,roughly speaking,we can still prove that the non-degenerate covariant representations of(A,G,α)are in bijection with the non-degenerate representation of this crossed product Banach algebraAG.To do it,we generalize the methods in[7].We notice that,Sjoerd Dirksen,Marcel De Jeu and Marten Wortel also constructed a kind of crossed product Banach algebra theory(see[8]).They started from a semi-norm on the algebraCc(G,A),and it is difficult to supply a non-trivial example under their definition.But in our construction,there are many interesting examples and we can prove that the semi-norm which we define onCc(G,A)is in fact a norm,and this will bring us much convenience for further discussion.

We also construct the reduced crossed product Banach algebraAG.Different from the-algebra case,the completion of(G,A)depends on the faithful regular representation.To avoid this trouble,we use the supreme form to define the reduced norm.Then a natural question will be asked:When doesAGcoincide withAGfor a Banach algebra dynamical(A,G,α)It is natural to think that a more strict condition may be required for the groupG,sinceAis weakened to be a Banach algebra.We find a sufficient condition,and thus whenGis a compact group,AGcoincides withAGfor a Banach algebra dynamical system(A,G,α).

2 Preliminaries

In this section,we introduce the basic definitions and notations,and establish some preliminary results.IfXis a normed space,we denote byB(X)the normed algebra of bounded linear operators onX.We letB(X)×denote the group of the invertible operators inB(X).IfAis a normed algebra,we write Aut(A)for the group of bounded automorphisms.A representationUof a groupGon a normed spaceXis a group homomorphismU:G→B(X)×.A representationπof a normed algebraAon a normed spaceXis an algebra homomorphismπ:A→B(X).The representationπis non-degenerate ifπ(A)X:=span{π(a)x:a∈A,x∈X}is dense inX.

In order to make sense of the integrals,where the integrand is a function taking values in a Banach algebra or a Banach space,we need a workable theory of what is referred to in the literature as vector-valued integration.Fortunately,the theory simplifies significantly when it is possible to restrict to the Haar measure on a locally compact groupG,and to integrands which are continuous with compact support onGtaking values in a Banach space.We base ourselves on an integral defined by duality.The definition,as well as the existence is contained in the next result,for the proof,we refer to[7,Lemma 1.91].

Lemma 2.1Suppose that X is a Banach space and G is a locally compact group with the left Haar measure.Then there exists a linear mapto X,whichis characterized by

The integral from Lemma 2.1 enables us to integrate compactly supported and strongly continuous operator-valued functions.We summarize the results in the next proposition without any proofs.

Proposition 2.1Let X be a Banach space,G be a locally compact group,and ψ:G→B(X)be compactly supported and strongly continuous.Define

where the integral on the right-hand side is the integral from Lemma2.1.ThenB(X),and

If T,R∈B(X),then

Definition 2.1(Banach Algebra Dynamical System)A Banach algebra dynamical system is a triple(A,G,α),where A is a Banach algebra,G is a locally compact group,and α:G→Aut(A)is a strongly continuous representation of G on A with each αtbeing an isometric automorphism on A.

Example 2.1LetAbe a Banach algebra,σ:A→Abe an isometric automorphism andα:Z→Aut(A)be defined byThen(A,Z,α)is a Banach algebra dynamical system.

Example 2.2LetA=Kbe the compact operator algebra on(1

andα:G→Aut(A)be defined by

Then(A,G,α)becomes a Banach algebra dynamical system.

In fact,fixf∈Cc(G)and letK:=suppf.If>0,there exists a neighbourhoodBofeinGsuch thatBand|f(s)?f(t)|<∈B.Then for fixed∈B,

Therefore,Uis strongly continuous.

To shows→(a)is norm continuous,we can assume thata∈K((G))is a finite-rank operator,since(G)has an approximate property.Now we assumea=|k)(θ|,and thus

forl,k∈Lp(G);θ∈=1.Sincewe have

ifs→s0by the strong continuity ofU.Then we are done.

Definition 2.2(Covariant Representation)Let(A,G,α)be a Banach algebra dynamical system,and let X be a Banach space.Then a covariant representation of(A,G,α)on X is a pair(π,U),where πis a representation of A on X and U is a representation of G on X,such that for all a∈A and s∈G,

The covariant representation(π,U)is called non-degenerate if(π,X)is a non-degenerate representation of A.

Example 2.3Give a Banach algebra dynamical system(A,G,α),whereAis a Banach algebra,unital or nonunital.LetbeA+,the unitalization,ifAis nonunital;and letbeAitself ifAis unital.Letπ:A→B()be the natural representation defined by

ifAis nonunital;and

ifAis unital,wherea,b∈A,t∈C.

Define:A→B(Lp(G,)(p≥1)by

andλ:G→B(Lp(G,))by

Then(,λ,Lp(G,))is a covariant representation of the Banach algebra dynamical system(A,G,α).

In fact,it is easy to check thatis a homomorphism.To see the contraction proposition,we have

by the isometry ofαt,so(x)≤x.Therefore,is a contractive representation,and similarly we can check thatλis also a representation.For the covariant proposition,

and

Therefore,

3 Crossed Product Banach Algebras

Suppose that(A,G,α)is a Banach algebra dynamical system.We define convolution on the linear spaceCc(G,A)of a continuous function fromGtoAwith compact supports by

for allf,g∈(G,A).It is well-defined,as shown in[7 Lemma 1.102].Straightforward computations show thatG,A)becomes an algebra with convolution as a product.For eachf∈Cc(G,A),define

ThenCc(G,A)is a normed algebra,and we denote byL1(G,A)its completion.

If(π,U,X)is a covariant representation of a given Banach algebra dynamical system(A,G,α),forf∈Cc(G,A),the functionsπ(f(s))is strongly continuous fromGtoB(X)by continuity of multiplication in the strong operator topology on uniformly bounded subsets.Therefore we can define

where the integral on the right-hand side is as in Proposition 2.1.We callπUthe integrated form of(π,U).

Differently from the case of-algebra,the norms ofπandUtmay be bigger than 1,which will pose difficulties for our definition of crossed products.To solve this problem,we may restrict the covariant representation in a proper way.

Definition 3.1Given a Banach algebra dynamical system(A,G,α),letdenote the set of covariant representation(π,U,X)of(A,G,α),such thatπ≤1and≤1for each t∈G.

Remark 3.1is non-empty since the covariant representationgiven by Example 2.3 is inFor the homomorphismU:G→B(x)×,since≤1 for eacht∈G,thenx=≤≤and henceUtis an isometric automorphism for eacht∈G.

Proposition 3.1Let(A,G,α)be a Banach algebra dynamical system and(π,U,X)∈Then the integrated form πU is a contractive representation of(G,A)on X.

ProofForf,g∈(G,A),we have

and

Therefore,πUis a contractive representation ofCc(G,A)onX.

By virtue of Proposition 3.1,we get a bounded set of non-negative numbers

Then we can define a semi-norm onCc(G,A)by

In fact,it is a norm on(G,A),and to prove that,it is necessary to find a faithful integrated representation.Example 2.3 will give a perfect one.Letbe the covariant representation of(A,G,α)given by Example 2.3.Forf∈Cc(G,A),if(f)==0,then for any

Hence,=0.Then for any?∈A?,we have

LetThenη∈(G),sincefis compactly supported.Then by(3.2),we haveis an approximate unit of(G),then 0=η ?→η,and henceη0.Thenη=0 by the continuity ofη.Therefore,f=0 sinceseparates the points ofA.It follows thatis a faithful representation of(G,A),and hence(3.1)defines a norm ofCc(G,A).

Definition 3.2(Crossed Product Banach Algebra)Given a Banach algebra dynamical system(A,G,α),the crossed product Banach algebra of this dynamical system is the completion of Cc(G,A)with respect to the norm defined by(3.1),and denoted by AG.

4 Representations of the Crossed Product Banach Algebras

In this section,by extending the representation of a given Banach algebra to its left multiplier algebra,we generalize the method in[7].Hence we show that,roughly speaking,all boundedrepresentationsof the crossed product are integrated forms of the given covariant representations of the original dynamical system.Thus we totally describe the representations of the crossed productAG.

Proposition 4.1Given a Banach algebra dynamical system(A,G,α),suppose that A has an m-bounded left approximate identity(ui).LetΛbe a neighbourhood basis of e,of which all elements are contained in a fixed compact set K.For each V∈Λ,take a positivewith support contained in V and an integral equal to one.Then the setwhere

directed by(V,i)≤(W,j)if and only if W?V and i≤j,is a left approximate identity of AG.

ProofWe first showfor eachf∈Aands∈G,whereCc(G)?Ais an algebraic tensor product.

For this,it is sufficient to consider elementary tensors,so letf=z?awith∈Cc(G)anda∈A.

Givenε>0,by the uniform continuity ofz,there exists a neighbourhoodU1ofe,such thatfor allr∈U1ands∈G.By the strong continuity ofα,there exists a neighbourhoode,such that(a)?a<ε(3mz∞)for allr∈U2.There exists an indexi0such that?a

Since(G)?Ais dense inCc(G,A)in the inductive limit topology(see[7]),we havef(s)→f(s)for eachf∈Cc(G,A),s∈G.It follows that→fin the norm defined by(3.1).To show it,let(π,U)be any covariant representation inandI=Ksupp(f),so then

Since→funiformly onI,we have thatin the norm defined by(3.1).

If(π,U)is any covariant representation inwe have

It follows thatis uniformly bounded byminSince(G,A)is dense inG,a 3ε-argument shows thatis indeed a bounded left approximate identity ofG.

Next,we will extend the representation(π,X)of Banach algebraAto the left multiplier algebraMl(A),whereAhas a left approximate identity.It is the key step to construct the one-to-one correspondences between the covariant representations and the integrated representations.Recall that the left multiplier algebra(A)for the Banach algebraAis defined by

There exists a canonical contractive homomorphismλ:A→(A),defined byλ(a)b=abfora,b∈A.SinceL?λ(a)=λ(L(a)),fora∈AandL∈(A),λ(A)is a left ideal in(A).

We begin with the general case,and for this we collect some results from[9].It is not hard to prove these results.

Theorem 4.1(see[9,Theorem 4.1])Let A be a Banach algebra with an m-bounded approximate left identity.If(π,X)is a non-degenerate representation of A,then there exists a unique representation:(A)→B(X)such that

Besides,is non-degenerate,unital,and≤mπ.Moreover,for a∈A and L∈Ml(A),

Definition 4.1(Integrable Covariant Representation)Let(π,U,X)be a covariant representation of a given Banach algebra dynamical system(A,G,α),and call(π,U,X)an integrable covariant representation if(U,X)is a bounded representation of(G,A)with respect to the crossed product norm.

Lemma 4.1Suppose that(A,G,α)is a Banach algebra dynamical system.Then there exists a non-degenerate bounded homomorphism

and a bounded strongly continuous homomorphism

such that for f∈Cc(G,A),r,s∈G and a∈A,

Moreover,is covariant,and thus

If,in addition,A has a bounded left approximate identity,thenis non-degenerate,and moreover if(π,U)is a non-degenerate integrable covariant representation,then

ProofFirstly we consider:A→End((G,A))and:G→End((G,A)),defined by(4.1).It is easy to check that the two maps are homomorphisms.If(π,U)is a covariant representation of(A,G,α),then fora∈A,r∈G,f∈(G,A),

Therefore

Similarly,

Then

Hence(a)≤afora∈A.Similarly,for allr∈G,f∈(G,A),sinceis isometric,we have

Thus we can extend(a)and(r)to the maps ofAGto itself,and forf,g∈(G,A),we have

HenceiG(r)is a well-defined element ofand the case of(a)is similar.

To check thatis covariant,we only have to check the actions on(G,A)as the boundedness of(a)andr).Forf(G,A),r,s∈Ganda∈A,

To show the last assertion,we claim that if(π,U,X)is a non-degenerate integrable covariant representation,thenUis a non-degenerate representation ofAG.

In fact,givexXwhich is of the formx(a)y,and≥0.Then there exists a neighbourhoodVofesuch that fors∈V,y?yLet(G)be nonnegative with compact support contained inVand with an integral equal to 1.Then a simple computation shows that

and it suffices to show the non-degenerateness ofU.

By Proposition 4.1,we can use Theorem 4.1 to the Banach algebraAG,since

andUis non-degenerate,so we have

Similarly,

To show thatis strongly continuous,fixf∈(G,A),a compact neighbourhoodWofeinG,and letK=suppf.Notice that as long asW,

If>0,then the uniform continuity offimplies that we can chooseV?W,such thatr∈Vimplies

Sincefhas compact support,we can shrinkVif needed so thatr∈Valso implies

Since

it follows that

Thereforer→(r)is strongly continuous.

If,in addition,Ahas a bounded left approximate identity(),thenis non-degenerate asconverges toIstrongly.

Theorem 4.2Suppose that(A,G,α)is a Banach algebra dynamical system,where A has a bounded left approximate identity.Then(π,u,X)→(πu,X)is a one-to-one correspondence between the non-degenerate integrable covariant representations of(A,G,α)and the non-degenerate bounded representations of AG.

ProofIn Lemma 4.1,we have already proved that a non-degenerate integrable covariant representation(π,U)of(A,G,α)induces a non-degenerate integrated representationπUofAG,and that

To prove the one-to-one correspondence,suppose that(L,X)is a bounded non-degenerate representation ofAG,and let

We first show thatπis non-degenerate.Givenf∈(G,A)andx∈X,ifis a bounded left approximate identity inA,then by Theorem 4.1,

Then by the non-degenerateness ofLand as(G,A)is dense inAG,we have thatπis non-degenerate.Sinceis strongly continuous,it is similar to check thatUis strongly continuous.Since the covariance condition is straightforward to check,it follows that(π,U)is a non-degenerate covariant representation.Forf,g∈(G,A),x∈X,

Let us computeas follows:

Thus(g)=f?g.It follows that

Therefore,Since(G,A)is dense inAGandLis nondegenerate,we have thatπU=Lon

In Theorem 4.2,ifAhas anm-bounded left approximate identity,thenLis a non-degenerate bounded representation ofAG,and

Then by Theorem 4.1,

and

On the other hand,if(π,U)thenUis contractive.Therefore,we get the following corresponding theorem,which is more similar to the-algebra case.

Theorem 4.3Suppose that(A,G,α)is a Banach algebra dynamical system,where A has a1-bounded left approximate identity.Then(π,u,X)→is a one-to-one correspondence between the non-degenerate covariant representations inand the non-degenerate contractive representations of AG.

5 Regular Representations

In the case of-algebra,if(A,G,α)is a-dynamical system,then a representation(π,H)ofAdetermines a regular representation of(G,A).This representation induces a reduced norm on(G,A),and the completion of this norm,denoted byG,is called the reduced crossed product,which is independent of the choice of the faithful representation(π,H).The regular representations are very important,since they are concrete and very easy to get.Moreover,Landstad[4]constructed an important theorem which implies thatAGis equal toAGwhenGis amenable.This theorem ensures the further development of the theory of crossed products.For example,it is a crucial step to construct the duality theory for crossed products,Connes’Thom isomorphism forA(see[5]),and Pimsner and Voiculescu’s six-term exact sequence for theK-group of certain crossed product-algebras(see[6]).

We will generalize this theory to the case of Banach algebra dynamical systems.In this case,things are very different.Let us start with the definition of the regular representation.

Give a Banach algebra dynamical system(A,G,α)and a representation(π,X)ofA.Since the representation space of a Banach algebra is a Banach space,we will let the covariant representation space beLp(G,X)withp≥1,not just asL2(G,X).We define a covariant representationby

for everyxinA,sinGandinLp(G,X),

and henceSois bounded byπ.λsis a natural isometry for anys∈G,and it is easy to check thatandλare homomorphisms and that

Therefore,is a well-defined covariant representation of(A,G,α).Then we can define the integral representation ofCc(G,A)onLp(G,X).To determine the reduced norm,let us consider the semi-normFor theC?-dynamical system,wherep=2 andXis a Hilbert space,it is a norm if(π,H)is a faithful representation,and the norm is independent of the choice of the faithful representation(π,H).But in the case of Banach algebra dynamical system,things are different.We give some examples to show that.

We firstly fixp,and let(A,G,α)be a Banach algebra dynamical system,whereG={e}andAbe any Banach algebra.Then

Hence the semi-norm depends on the choice of the representation of(π,X).But ifAis aC?-algebra,the semi-norm is independent of the choice of the representation(π,H)since the monomorphism betweenC?-algebras is an isometry(see[10,Chapter VIII,Theorem 4.8]).But in the case of Banach algebras,this property does not hold.

Next,let(A,G,α)be a Banach algebra dynamical system,whereA=C andαtis trivial.Define a representationπ:C→Then

We see that this semi-norm depends on the numberp.

Summing up the two examples above,we see thatdepends on the choice of bothpand(π,X).But we can use the supreme to define the reduced norm.We give the semi-norm onCc(G,A)below,forf∈Cc(G,A),and let

It is easy to check that this gives a semi-norm onCc(G,A),but in fact,this is a norm.To check it,we only have to find a representation(π,X),such that(λ,Lp(G,X))is a faithful representation onCc(G,A),and Example 2.3 gives a wonderful example(see the procedure of the definition of crossed products in Section 3).Then we can give the definition of the reduced crossed products Banach algebras.

Definition 5.1(Reduced Crossed Product Banach Algebras)Given a Banach algebra dynamical system(A,G,α),the reduced crossed product Banach algebra of this system is the completion of Cc(G,A)by the norm of·and is denoted by A

In the following,we will discuss whenAGequalsAGfor a given Banach algebra dynamical system(A,G,α).We give a result as a theorem.

Theorem 5.1If(A,G,α)is a Banach algebra dynamical system,then AG equals Ais a compact group.

ProofLet(π,U,X)be a covariant representation of(A,G,α),and(π,U,X)lie inSince the Haar measure of a compact group is finite,μ(G)is a finite positive number.Define a mapν:X→(G,X)byν(x)(t)=Then we have

Therefore,νis well-defined and is an isometry,soνis an isometric embedding.Recall that the representation:A→B((G,X))is defined by

fora∈A,t∈Gandξ∈(G,A).

For eachx∈X,t∈G,νπ(a)x(t)=and

Then by the covariant property of(π,U),we have thatνπ(a)x(t)=(a)ν(x)(t).It follows that

for eacha∈A.Then we get the commutative diagram

On the other hand,for eachx∈Xandt∈G,

and

Thusν=and the diagram

is commutative.

Then for eachf∈(G,A),

Since the diagrams above are commutative,is well-defined as it only acts on Ran(ν).Therefore,

This may not be an equation,sinceνmay not be an isometric isomorphism,so we can not get

but that is enough to get our result.By the definitions of crossed products and reduced crossed products,the reduced norm is equal to the universal norm on(G,A),and we have thatAGis equal toAG.

AcknowledgementsThe authors would like to thank all the members in their functional analysis seminar for the inspiring discussions,and they would also like to thank the referees sincerely for the suggestions on improving the paper.

[1]Turumaru,T.,Crossed products of operator algebras,T?ohoku Math.J.,10,1958,355–365.

[2]Zeller-Meyer,G.,Produits croiss d’uneC?-algbre par un groupe d’automorphismes,J.Math.Pures Appl.,47,1968,101–239.

[3]Takai,H.,On a duality for crossed products ofC?-algebras,J.Funct.Anal.,19,1975,25–39.

[4]Landstad,M.B.,Duality theory of covariant systems,Trans.Amer.Math.Soc.,248,1979,223–267.

[5]Connes,A.,An analogue of the Thom isomorphism for crossed products of aC?-algebra by an action of R,Adv.Math.,39,1981,31–55.

[6]Pimsner,M.and Voiculescu,D.,Exact sequences forK-groups and EXT-groups of certain cross-productC?-algebras,J.Operator Theory,4,1980,93–118.

[7]Williams,D.P.,Crossed products ofC?-algebras,Mathematical Surveys and Monographs,Amer.Math.Soc.,134,2007.

[8]Dirksen,S.,de Jeu,M.and Wortel,M.,Crossed products of Banach algebras I,Dissertationes Math.,2011,to appear.arXiv:math.FA/1104.5151v2

[9]Dirksen,S.,de Jeu,M.and Wortel,M.,Extending representations of normed algebras in Banach spaces,2009.arXiv:math.FA/0904.3268v2

[10]Conway,J.B.,A course in functional analysis,Graduate Texts in Mathematics,96,2nd edition,Springer-Verlag,New York,1990.