Reduced Crossed Products of Pro-Banach Algebras?

Lizhong HUANG Yu GUO

Abstract This paper gives the concept of the reduced pro-Banach algebra crossed product associated with inversely pro-Banach algebra dynamical system, and shows that the reduced crossed product is an inverse limit of an inverse system of Banach algebra crossed products.Also, the authors show that if the locally compact group is amenable, then the crossed product and the reduced crossed product are isometrically isomorphic.

Keywords Pro-Banach algebras dynamical systems, Reduced crossed products,Crossed products, Representations

1 Introduction and Preliminaries

The crossed product algebras is on the main pillars of the theory of operator algebras, and it has many applications in mathematics and quantum physics.Over the past few decades, the crossed product algebras, including von Neumann algebras, C?-algebras, Banach algebras etc,have attracted a great deal of attention, and a large part of the literature is concerned with crossed products(see[1–3,12–15]and the literature therein).For a given C?-algebra dynamical system, in addition to the full crossed products of C?-algebra, there is another very important crossed product C?-algebra, the reduced crossed products.A lot of work has been done on the connection between the crossed products and the reduced crossed products.Zeller-Meier[16] has firstly proved that the crossed product and the associated reduced crossed product are equal in the C?-algebra dynamical system with the discrete group, and whereafter Takai[13] generalized this result.Huang and Lu [5] generalized it to the general Banach algebra setting, and proved that the crossed product Banach algebra, which was defined by Dirkson[2], is isometrically isomorphic to the reduced crossed product Banach algebra associated with a Banach algebra dynamical system with the amenable group.

As a prominent generalization of the idea of C?-algebra crossed product, Joita introduced the concept of pro-C?-algebra (also called locally C?-algebra)crossed products and Hilbert pro-C?-modules, and gave a detailed systematic exposition of the recent development in it, see for example [6–9].

Motivated by the study of crossed product of pro-C?-algebras(see[6]),in the present paper,we define the reduced crossed product of pro-Banach algebra associated with a Banach algebra dynamical system,and show that a pro-Banach reduced crossed product is an inverse limit of an inverse system of Banach algebra crossed products, which generalize the corresponding result of the pro-C?-algebra reduced crossed products.More importantly, we establish the equality between the crossed product and the reduced crossed product with the amenable group in the pro-Banach algebra dynamical system setting.

We now recollect some basic definitions, notations and results.The details on pro-Banach algebras are available in [11].

A pro-Banach algebra is a complete Hausdorfftopological complex algebra A whose topology is given by a directed family S(A)of continuous submultiplicative seminorms.For p ∈S(A),ker(p)is a closed bilateral ideal of A, and the quotient algebra A/ker(p)is a normed algebra in the norm induced by p, and its completion is denoted by Ap.The canonical map from A to A/ker(p)is denoted by ?p.For p,q ∈S(A)with p ≥q, there is a unique continuous morphism ?pq: Ap→Aqwith dense range such that ?pq??p= ?q, and {Ap,?pq}p,q∈S(A)is an inverse system of Banach algebras and its inverse limit is a pro-Banach algebra which is topologically isomorphic to A (see [11, Chapter III, Theorem 3.1]).

2 Pro-Banach Algebra Dynamical Systems

Let A be a pro-Banach algebra and G be a locally compact group.An action G on A is a strongly continuous representation α : G →Aut(A), where Aut(A)is the group of bounded automorphism of A.An action α of G on A is an inverse limit action, if p(αr(a))= p(a)for all p ∈S(A), r ∈G, a ∈A.If α is an inverse limit action, then, for all p ∈S(A), there is an action α(p)of G on Ap, such that??p=?p?αsfor all s ∈G, and(Ap,G,α(p))is a Banach algebra dynamical system for each p ∈S(A).

Definition 2.1A triple(A,G,α)is called a pro-Banach algebra dynamical system, ifAis a pro-Banach algebra,Gis a locally compact group, andα:G →Aut(A)is a strongly continuous action ofGonA.Ifαis an inverse limit action ofGonA, then(A,G,α)is an inversely pro-Banach algebra dynamical system.

Definition 2.2Let(A,G,α)be a pro-Banach algebra dynamical system.A pair(π,U)is called a covariant representation of(A,G,α)on a Banach spaceX, ifπis a representation ofAonX,Uis a representation ofGonX, such that for alla ∈A,s ∈G,

Ifπis non-degenerate, then(π,U)is called non-degenerate;ifπis continuous andUis stronglycontinuous, then(π,U)is called a continuous covariant representation.

Suppose that (A,G,α)is an inversely pro-Banach algebra dynamical system.We define convolution on the linear space Cc(G,A)of a continuous function from G to A with compact supports by

for all f,g ∈Cc(G,A).Straightforward computations show that Cc(G,A)becomes an associative algebra with convolution.Also assume that X is a Banach space, for each f ∈Cc(G,X),define1 ≤p < ∞.Then Cc(G,X)is a normed algebra, and we denote by Lp(G,X)its completion.To avoid confusion between symbols, we will be concerned only with L1(G,X).For a given Banach algebra dynamical system(A,G,α), the crossed product (A ?αG)Ris the completion of Cc(G,A)corresponding to the uniformly bounded class R of continuous covariant representations of (A,G,α).More details on Banach algebra crossed products are available in (see [1–2, 10]).

Definition 2.3Let(A,G,α)be a pro-Banach algebra dynamical system, andRbe a family of continuous covariant representations of(A,G,α).ThenRis called uniformly semi-bounded,if there exists a functionζ : G →[0,∞), which is bounded on compact subsets ofG, such that‖Ur‖≤ζ(r)for all(π,U)∈Randr ∈G.

Now let R be a non-empty uniformly semi-bounded class of continuous covariant representations of inversely pro-Banach algebra dynamical system (A,G,α)on a Banach space X.For(π,U)∈R, we define the reduced algebra representationand the left regular representation Λ on L1(G,X)by the formula

Lemma 2.1Let(A,G,α)be an inversely pro-Banach algebra dynamical system,Rbe a family of continuous representations of(A,G,α)on a Banach spaceX, and the representationsandΛbe defined as above.

(1)There existsp ∈S(A), such that‖(a)‖≤p(a)for anya ∈A.

(2)For allr ∈G,Λris an invertible isometry onL1(G,X).

(4)Suppose(π,U)∈R,whereUsatisfiesSpecially, ifUis an isometric representation ofGon a Banach spaceX, we have‖‖≤‖π‖.

Proof(1)If π is a continuous representation of A, then there is p ∈S(A), such that‖π(a)‖ ≤p(a), a ∈A.Since α is an inverse limit action of G on A, for all a ∈A,h ∈L1(G,X),s ∈G, we have

(2)and (3)are straightforward verifications.

(4)From the convariance relation π(αr(a))= Urπ(a), it follows that ‖π(αr(a))‖ ≤M2‖π‖‖a‖ for all a ∈A and r ∈G, hence

Proposition 2.1Let(A,G,α)be an inversely pro-Banach algebra dynamical system, and letRbe the uniformly semi-bounded class of continuous covariant representations of(A,G,α)on a Banach spaceX.Also assume thatare the associated regular covariant representations of(A,G,α)onL1(G,X).For allf ∈Cc(G,A), defineThenis a finite sub-multiplicative semi-norm onCc(G,A).

ProofLet f,g ∈Cc(G,A), ξ ∈Cc(G,X)and s ∈G, we have

And then,

where

3 Reduced Crossed Products

In this section,we mainly study the reduced crossed products associated with pro-Banach algebra dynamical systems.As we know if R is a non-empty uniformly semi-bounded class of continuous covariant representations of inversely pro-Banach algebra dynamical system (A,G,α),then for all

is a sub-multiplicative semi-norm on Cc(G,A).Crossed product(A?αG)Ris thus the completion of Cc(G,A)in {ρRp: p ∈S(A)}.We define the reduced crossed products of pro-Banach algebras analogously.

Definition 3.1Let(A,G,α)be an inversely pro-Banach algebra dynamical system,Rbe a non-empty uniformly semi-bounded class of continuous covariant representations of(A,G,α),andbe the associated class of regular covariant representations of(A,G,α).By Proposition2.1,{: p ∈S(A)}is a set of sub-multiplicative semi-norms onCc(G,A).Then the completion ofCc(G,A)in{: p ∈S(A)}is called the reduced crossed product of(A,G,α)associated todenoted by

3.1 Inverse limit of Banach algebra crossed products

Let π be a continuous representation of pro-Banach algebra A on a Banach space X, andbe the associated regular representation of A on L1(G,X).From the continuity of, there exists p ∈S(A), such that ‖‖ ≤p(a).Define a mapby a+ker(p)(a), thenis a bounded homomorphism from A/ker(p)to B(L1(G,X)), and thus it can be extended to a bounded homomorphism from Apto B(L1(G,X)), still denoted by.It is easy to verify that the following diagram

Note that A/ker(p)is dense in Ap, we get

To differentiate, we denote by(p)the uniformly bounded set of continuous covariant representations of Banach algebra dynamical system (Ap,G,α(p)).

If R is a non-empty uniformly semi-bounded class of continuous covariant representations of inversely pro-Banach algebra dynamical system (A,G,α), by Arens-Michael decomposition theorem (see [11]), we may show that pro-Banach algebra crossed product (A ?αG)Ris an inverse limit of a family of Banach algebra crossed products (Ap?α(p)G)R(p), that is,

The proof for reduced crossed product (A ?αG)~Ris similar.To make our exposition selfcontained, we will give a detailed proof.

Theorem 3.1Let(A,G,α)be an inversely pro-Banach algebra dynamical system,andRbe a non-empty uniformly semi-bounded class of continuous covariant representations of(A,G,α)on a Banach spaceX.Suppose thatis the corresponding semi-bounded class of regular covariant representations of(A,G,α)onL1(G,X).Then for allp ∈S(A), we have

up to a topological algebraic isomorphism.

ProofNote thatis a complete pro-Banach algebra, using Arens-Michael deposition theorem (see [11]), we have

So it is enough to prove thatis topologically isomorphic toDefine a map T :

for all f ∈Cc(G,A,α), where ?pis the canonical map from A to Ap.

So the definition of T is unambiguous.To show that T is a topological isomorphism, we divide our proof in three steps.

First, we need to verify that T is a linear homomorphism from Cc(G,A,α)/ker()to Cc(G,Ap,α(p))/ker().Clearly,T is linear,so it suffices to show that T is a homomorphism.Note that for all f,g ∈Cc(G,A),

It follows that

The next is to show that T is isometric.To do it, let f ∈Cc(G,A,α).Then

Now, we have already proved that T is an isometric homomorphism.Therefore, it can be extended to an isometric homomorphism fromdenoted by

Finally, we have to show thatis surjective.Let ξ ∈Cc(G), a ∈A; then

It follows from the surjection of ?p:A →A/ker(p)that the range ofcontains the set

Since A/ker(p)is dense in Ap, Y is also dense in(see [2, Corollary 3.6]).This implies thatis surjective.The proof is complete.

3.2 Reduced crossed products with amenable groups

Takai [13] proved that the crossed products and the associated reduced crossed products of C?-algebra dynamical system are equal in the amenable group condition.The same holds for a Banach algebra dynamical system with the amenable group(see[5]).To establish this theorem in the pro-Banach algebra dynamical system setting, we introduce the following lemma.

Lemma 3.1(see [4])LetGbe a locally compact group with Haar measureμ.IfGis amenable, then, for everyε>0and compact setK ?Gcontaining the identity element ofG,there exists a compact setUwith

HereKU △U =(KU U)∪(U KU)is the symmetric difference of the setsKUandU.

Theorem 3.2Let(A,G,α)be an inversely pro-Banach algebra dynamical system,Rbe a non-empty uniformly semi-bounded class of continuous covariant representations of(A,G,α),and suppose thatsup{‖Ur‖ : (π,U)∈Rp,r ∈G} < ∞for somep ∈S(A).Letbe the class of regular covariant representations of(A,G,α)associated withR.IfGis amenable, thenandare topologically isomorphic.

ProofWe will prove the theorem directly without using the lemma.Let (π,U)be in R,then there exists p ∈S(A)such that ‖π(a)‖ ≤p(a)for all a ∈A.Suppose that (,Λ)is the regular representation associated with (π,U).By Lemma 2.1, (,Λ)is a continuous covariant representation of(A,G,α)satisfying‖(a)‖≤p(a).For convenience,we write M =sup{‖Ur‖:(π,U)∈Rp,r ∈G}.First, we will show that

Define a bounded map h:L1(G,X)→L1(G,X)as follows:

It is easy to check that h is a linear bijection bounded by M.In fact, for all ξ ∈L1(G,X), we have

Similarly, it follows from the equality [h?1(ξ)](s):=Us(ξ(s))that h?1is also M-bounded.

Let f ∈Cc(G,A).Notice the fact that

So the proof will be finished if one shows, given ε>0, that

Without loss of generality, we assume that (π×U)(f)≠ 0 and ε < ‖(π×U)(f)‖.Choose x0∈X0such that ‖x0‖=1 and ‖(π×U)(f)x0‖>‖(π×U)(f)‖?.Let

then δ > 0.Write S = supp(f)∪{e}.Since f ∈Cc(G,A), S and S?1are both compact.Therefore, there is a compact subset K ?G such that 0 < μ(K)< ∞and μ(S?1K △K)<δμ(K)by Lemma 3.1, and thus μ(S?1K)< (1+δ)μ(K)which is due to the latter inequality.From Uryshon’s lemma, we may define η ∈Cc(G,X)by

Then

For all s ∈G, we have (hη)(s)= Us?1(η(s))= Us?1(x0), it follows from the strong continuity of U that hη ∈c(G,X).For any r ∈K, noting η(s?1r)=x0if s ∈S with f(s)≠0, we have

Thus,

We conclude from the inequalityhence that, and this yields that

Using the fact that

we conclude that the semi-normsandare equivalent on Cc(G,A).This implies that the locally convex topologies determined by {: p ∈S(A)} and {: p ∈S(A)}are equal.Thus it follows from the definitions of the crossed product and the reduced crossed product thatis equal toup to a topological isomorphism.The proof is completed.

As we know that the crossed product and the associated reduced crossed product are equal for a Banach algebra dynamical system with amenable group (see [5]).Inspired by it, we will give a similar result in case of pro-Banach algebra dynamical system setting.

Theorem 3.3Let(A,G,α)be an inversely pro-Banach algebra dynamical system with the amenable groupG,Rbe a uniformly semi-bounded class of continuous covariant representations on a Banach space such thatUsis an isometry for alls ∈G, andRbe the class of regular representations of(A,G,α)associated withR.Then the crossed product(A ?αG)Rand the reduced crossed productare isometrically isomorphic.

ProofSince Usis isometrical for each s ∈G, we obtain?Rpfor some p ∈S(A).It follows from the first part of the proof of Theorem 3.2 that

for all f ∈Cc(G,A).This shows that (A ?αG)Randare isometrically isomorphic.

Since all abelian topological groups are amenable(see[15]),and as a consequence of Theorem 3.3, we naturally get the following result.

Corollary 3.1Let(A,G,α)be an inversely pro-Banach algebra dynamical system whereGis abelian, andRandbe given as in Theorem3.3.Then(A?αG)Randare equal.In particular,, whereZandRrespectivelystand for the group of the integer numbers and the group of the real numbers.

AcknowledgementThe authors thank sincerely the anonymous referees for the very thorough reading of the paper and valuable comments on improving the paper.