Finite-dimensional approximations and semigroup coactions for operator algebras
aa r X i v : . [ m a t h . OA ] J a n FINITE-DIMENSIONAL APPROXIMATIONS AND SEMIGROUPCOACTIONS FOR OPERATOR ALGEBRAS
RAPHA¨EL CLOU ˆATRE AND ADAM DOR-ON
Abstract.
The residual finite-dimensionality of a C ∗ -algebra is known to beencoded in a topological property of its space of representations, stating thatfinite-dimensional representations should be dense therein. We extend thisparadigm to general (possibly non-self-adjoint) operator algebras. While nu-merous subtleties emerge in this greater generality, we exhibit novel tools forconstructing finite-dimensional approximations. One such tool is a notion of aresidually finite-dimensional coaction of a semigroup on an operator algebra,which allows us to construct finite-dimensional approximations for operatoralgebras of functions and operator algebras of semigroups. Our investigationis intimately related to the question of whether residual finite-dimensionalityof an operator algebra is inherited by its maximal C ∗ -cover, which we resolvein many cases of interest. Introduction
In the classification program for certain classes of concrete non-selfadjoint op-erator algebras, a common theme is the analysis of their finite-dimensional rep-resentations; see for instance [31, 33, 40, 54] and the references therein. Althoughthis is a perfectly natural and effective approach in these situations, in general itis not entirely clear to what extent finite-dimensional representations truly capturethe structure of a given operator algebra. Indeed, one has to wonder whether, andto what extent, finite-dimensional representations are sufficient to determine thebehavior of arbitrary representations by approximation. This is case for operatoralgebras that are residually finite-dimensional.An operator algebra is said to be residually finite-dimensional (or RFD) if itadmits a completely norming set of representations on finite-dimensional Hilbertspaces. For C ∗ -algebras, this is a classical notion that has been thoroughly inves-tigated. Yet, the general notion has received comparatively little attention beyond[48] and [3], both of which focus on concrete algebras of functions. However, in [20]a systematic study of RFD operator algebras was undertaken, where several opera-tor algebra were shown to be RFD. Even for concrete classes of operator algebras itis challenging to characterize which of them are RFD, and this provides the impe-tus for developing alternative characterizations of this notion. However, at presentthere are no such tools available for general operator algebras. This stands in sharp Mathematics Subject Classification.
Primary 47L55, 46L07; Secondary 47A20, 20Mxx,47B32.
Key words and phrases.
Finite-dimensional approximation, residual finite-dimensionality,Exel–Loring approximation, semigroup coaction, semigroup algebras, algebras of functions.The first author was partially supported by an NSERC Discovery grant. The second authorwas partially supported by NSF grant DMS-1900916 and by the European Union’s Horizon 2020Marie Sk lodowska-Curie grant No 839412. contrast with the more classical C ∗ -algebraic setting where various conditions havebeen shown to be equivalent to residual finite-dimensionality [6, 24, 38, 39]. In thispaper, we provide new characterizations that apply to general operator algebras.Our approach is inspired by the work of Exel and Loring [38], which we discussnext. Let A be a C ∗ -algebra, and recall that the spectrum of A is the set b A ofunitary equivalence classes of irreducible ∗ -representations, together with the pointstrong operator topology. In [38] it is shown that A is RFD if and only if for every ∗ -representation there exists a net ( π λ ) of finite dimensional ∗ -representations of A that converges to π in the point strong operator topology. Following [38], it wasshown by Archbold [6] that a C ∗ -algebra A is RFD if and only if the set of unitaryequivalence classes of finite-dimensional irreducible ∗ -representations is dense in b A . Hence, the two topological characterizations of residual finite-dimensionality,one in terms of irreducible ∗ -representations, and another in terms of arbitrary ∗ -representations, provides a fair amount of additional flexibility.The main driving force behind the present paper is the investigation of thoserepresentations of general operator algebras that admit a finite-dimensional ap-proximation of the type described in the previous paragraph. As we will see, thereare subtleties intrinsic to working with general operator algebras (as opposed toC ∗ -algebras), wherein there are several different reasonable interpretations of whata finite-dimensional approximation should be. Interestingly, this flexibility allowsone to connect the existence of finite-dimensional approximations to the residualfinite-dimensionality of various C ∗ -covers for the operator algebra. This is emi-nently desirable, as RFD C ∗ -algebras are typically much better understood. Wealso note that the properties of the maximal and minimal C ∗ -covers being RFDhave been explored previously in [20, 21], and the findings of these papers aptlyillustrate the depth of the problem. Therefore, our analysis based on the existenceof finite-dimensional approximations sheds light on some unresolved problems.Next, we describe the organization of the paper. In Section 2, we introducevarious notions of finite-dimensional approximations, which we briefly mention here.In what follows, all representations of operator algebras are completely contractive.Let A be an operator algebra and let π : A → B ( H ) be a representation. Let π λ : A → B ( H ) , λ ∈ Λbe a net of representations with the property that the space C ∗ ( π λ ( A )) H is finite-dimensional for every λ ∈ Λ. We say that the net ( π λ ) is a finite-dimensionalapproximation for π if ( π λ ( a )) converges to π ( a ) in the SOT for every a ∈ A . If, inaddition, we have that ( π λ ( a ) ∗ ) converges to π ( a ) ∗ in the SOT for every a ∈ A , thenwe say that ( π λ ) is an Exel–Loring approximation for π . These two notions coincidewhen A is a C ∗ -algebra. Finite-dimensional approximations do not necessarily arisein geometrically transparent ways (see Example 3.4) and thus, constructing suchapproximations can be highly non-trivial. Consequently, it is beneficial to identifyoperations that preserve the existence of these approximations, and this is the focusof the rest of Section 2.In Section 3, we utilize finite-dimensional approximations to analyze the residualfinite-dimensionality of general operator algebras. Recall that the C ∗ -envelope of A is the smallest C ∗ -algebra generated by a copy A and it is denoted by C ∗ e ( A ).As explained in Subsection 2.1, it can be constructed with the aid of so-called extremal representations of A . Using these ideas, we can now state one of our INITE-DIMENSIONAL APPROXIMATIONS AND SEMIGROUP COACTIONS 3 main results, which is a version of [38, Theorem 2.4] that characterizes residualfinite-dimensionality for general operator algebras (see Theorems 3.1 and 3.2).
Theorem 1.1.
Let A be a separable operator algebra. Then, the following state-ments are equivalent. (i) The algebra A is RFD. (ii) Every extremal representation of A has a finite-dimensional approximation. (iii) Every extremal representation of A has an Exel–Loring approximation. (iv) If π is a ∗ -representation of C ∗ e ( A ) , then π | A admits an Exel–Loring approx-imation. The curious reader may wonder how the statements appearing above relate tohaving RFD C ∗ -envelope. It is easy to construct examples where A is RFD whileC ∗ e ( A ) is not, but a precise condition on A which is equivalent to C ∗ e ( A ) being RFDis unknown; see [21] and [3, Proposition 4.1] for recent related results.At the other extreme in the scale of C ∗ -algebras generated by a copy of A isthe maximal C ∗ -cover, denoted by C ∗ max ( A ). Notably, the existence of Exel–Loringapproximations can be used to characterize the residual finite-dimensionality ofC ∗ max ( A ), as shown in Theorem 3.3. Theorem 1.2.
Let A be an operator algebra. Then C ∗ max ( A ) is RFD if and onlyif every representation of A admits an Exel–Loring approximation. Motivated by these results, let us say that the operator algebra A has property( F ) if all of its representations admit a finite-dimensional approximation. Thequestion of which operator algebras enjoy property ( F ) is an interesting one, andTheorems 1.1 and 1.2 imply thatC ∗ max ( A ) is RFD ⇒ A has property ( F ) ⇒ A is RFD . In trying to understand property ( F ), a natural question emerges. Question 1.
Let A be an RFD operator algebra. Is C ∗ max ( A ) necessarily RFD?This question was already raised in [20, Section 5], where some partial resultssupported an affirmative answer. Our results show that Question 1 is equivalent tothe question of whether or not the existence of a representation admitting a finite-dimensional approximation implies that all representations admit an Exel–Loringapproximation. Our approximation techniques and the resulting property ( F ) thusunearth an intermediate state which may prove useful in answering this question.The rest of the paper is motivated by Question 1, in that we find concrete operatoralgebras whose maximal C ∗ -cover is RFD.While C ∗ -algebras are analogous to groups, general operator algebras are anal-ogous to semigroups. Hence, non-self-adjoint operator algebras arising from semi-groups form a natural class of examples for our study. Semigroup C ∗ -algebras havebeen studied by many authors over the years [4, 5, 16, 25–27, 43, 50], where a unifiedapproach was obtained by Li for the class of independent semigroups in [44–46].An operator algebra analogue of the reduced group C ∗ -algebra is A r ( P ), which isthe operator algebra generated by the left regular representation of P on ℓ ( P ).Suppose P is a cancellative semigroup (both left and right). For p ∈ P we defineits set of right divisors R p = { r ∈ P : p = qr for some q ∈ P } . R. CLOUˆATRE AND A. DOR-ON
We say that P has the finite divisor property (or FDP) if R p is finite for every p ∈ P .It is relatively straightforward to show that A r ( P ) is RFD when P has FDP (seeProposition 4.1). With the aim of answering Question 1 in concrete examples, weintroduce in Section 4 a notion of an RFD semigroup coaction of P on an operatoralgebra A , and prove the following (see Theorem 4.6). Theorem 1.3.
Let P a countable cancellative semigroup with FDP such that thereis a character χ : A r ( P ) → C with χ ( λ p ) = 1 for every p ∈ P . Let A be a separableoperator algebra that admits an RFD coaction of P . Then, C ∗ max ( A ) is RFD. Coactions of groups on general operator algebras were introduced in [34] forthe purpose of showing the existence of C ∗ -algebras satisfying co-universality withrespect to representations of product systems over right LCM monoids. In theC ∗ -algebra literature, coactions by a discrete group G are interpreted as actions ofthe quantum dual group of G on the C ∗ -algebra, and are useful in many instances[17, 37, 53, ?Seh19 ]. Since our semigroups are assumed to be FDP, they will notcontain infinite groups as subsemigroups, and hence our development of coactionsby semigroups differs from the one for group coactions in [34].The class of independent semigroups includes the majority of examples of semi-groups previously studied through their C ∗ -algebras in the literature. In Section 5we use Theorem 1.3 to show that the maximal C*-cover is RFD for semigroup alge-bras of many independent semigroups, as well as for operator algebras of functionswith circular symmetry (see Theorem 5.7 and Theorem 5.12 respectively). Theorem 1.4.
Let P and Q be countable semigroups embedded in some groups G and H respectively. Assume that P and Q are independent, and that Q is left-amenable and has FDP. Assume further that there is a homomorphism ϕ : G → H such that ϕ ( P ) ⊂ Q and ϕ | P is finite-to-one. Then, there is an RFD coaction of Q on A r ( P ) and C ∗ max ( A r ( P )) is RFD. This theorem allows us to show that for many concrete examples of indepen-dent semigroups, the operator algebra A r ( P ) has an RFD maximal C ∗ -cover. Forinstance, this works for N d for any integer d , left-angled Artin monoids (Example5.9) and Braid monoids (Example 5.10). Theorem 1.5.
Let Ω ⊂ C n be a balanced subset and let A be a T -symmetricoperator algebra of functions on Ω . Assume that there is a collection of homogeneouspolynomials spanning a dense subset of A . Then, A admits an RFD coaction by N .In particular, C ∗ max ( A ) is RFD. This theorem is applied to some uniform algebras (Corollary 5.13), to the famousSchur–Agler class of functions (Corollary 5.14) and to some algebras of multipli-ers on well-behaved reproducing kernel Hilbert spaces (Corollary 5.15). Takinginto account Theorem 1.2, these results shed considerable light on the structureof representations of operator algebras that play an important role in multivariateoperator theory; see [12, 18, 19, 22] and the references therein.In particular, it follows from Corollary 5.13 that the maximal C ∗ -cover of thepolydisc algebra A( D n ) is RFD. This result is particularly striking in view of thefollowing observation. It is a consequence of Ando’s theorem [51, Theorem 5.5]that C ∗ max (A( D )) is the universal C ∗ -algebra generated by a pair of commutingcontractions. Hence, this universal C*-algebra is RFD. On the other hand, it followsfrom [23, Theorem 6.11] and the refutation of Connes embedding conjecture [42] INITE-DIMENSIONAL APPROXIMATIONS AND SEMIGROUP COACTIONS 5 that the universal C ∗ -algebra generated by a pair of doubly commuting contractionsis not RFD. 2. Finite-dimensional approximations
In this section, we examine various notions of finite-dimensional approximationsfor representations of operator algebras. Before proceeding however, we need torecall some operator algebraic background.2.1. C ∗ -covers and extremal representations. Let A be an operator algebra.If B is a C ∗ -algebra and ι : A → B is a completely isometric homomorphism with B = C ∗ ( ι ( A )), then we say that the pair ( B , ι ) is a C ∗ -cover of A . Throughoutthe paper, we will be dealing with two special C ∗ -covers, which we introduce next.The maximal C ∗ -cover of A , denoted by (C ∗ max ( A ) , υ ), is the essentially uniqueC ∗ -cover satisfying the following universal property: for any C ∗ -algebra C and anyrepresentation π : A → C , there is a ∗ -representation b π : C ∗ max ( A ) → C such that b π ◦ υ = π . (By a representation of an operator algebra, here and elsewhere we meana completely contractive homomorphism.) For a proof of existence of the maximalC ∗ -cover, the reader should consult [13, Proposition 2.4.2].The other C ∗ -cover that is relevant for our purposes is the “minimal” one. Itis usually referred to as the C ∗ -envelope of A , denoted by (C ∗ e ( A ) , ε ). Its defininguniversal property is the following: for any C ∗ -cover ( ι, B ) of A , there is a ∗ -representation π : B → C ∗ e ( A ) such that π ◦ ι = ε . The existence of this object ishighly non-trivial, see [13, Theorem 4.3.1 and Proposition 4.3.5].Due to important work of Dritschel–McCullough [36] (inspired by previous in-sight of Muhly–Solel [49]), it is known that certain special representations of A areintimately related to its C ∗ -envelope. We recall these important concepts.Let π : A → B ( H π ) and θ : A → B ( H θ ) be representations. We say that θ is a dilation of π if H π ⊂ H θ and π ( a ) = P H π θ ( a ) | H π , a ∈ A . If in addition the space H π is invariant for θ ( A ), then the dilation θ is said to bean extension of π . Similarly, if H π is invariant for θ ( A ) ∗ , then the dilation θ is saidto be a coextension of π .If θ is a dilation of π , then it is said to be a trivial dilation if the space H π isreducing for θ ( A ). In other words, a trivial dilation θ of π can be written as θ ( a ) = π ( a ) ⊕ (cid:0) θ ( a ) | H ⊥ π (cid:1) , a ∈ A . A representation is said to be extension-extremal if all its extensions are trivial.Likewise, it is said to be coextension-extremal if all its coextensions are trivial.Moreover, a representation is simply said to be extremal if all its dilations aretrivial. Clearly, an extremal representation is necessarily extension-extremal andcoextension-extremal. The reader should consult [30] for a thorough study of thesenotions.We now collect some minor variations on known facts that we require later. Weprovide a detailed argument, as some care must be taken when dealing with theunit (or lack thereof).
Theorem 2.1.
Let ( B , ι ) be a C ∗ -cover of some operator algebra A . Let π : A → B ( H π ) be a representation. The following statements hold. R. CLOUˆATRE AND A. DOR-ON (i) If π is extremal, then there is a ∗ -representation σ : B → B ( H π ) such that π = σ ◦ ι . (ii) If π is extension-extremal, then there is an extremal representation σ of A which is a coextension of π . (iii) If π is coextension-extremal, then there is an extremal representation σ of A which is an extension of π . Proof. (i) If A is unital, then necessarily ι and B are both unital and the desiredstatement is an immediate consequence of Stinespring’s dilation theorem [51, The-orem 4.1] applied to the representation π ◦ ι − .If A is not unital, then [47, Section 3] implies the existence of a unital represen-tation π of the unitization A of A that extends π . It is easy to see that π isstill extremal (see for instance [35, Proposition 2.5]). Hence, it suffices to apply theprevious argument to π .(ii) Assume first that A is unital. Then, π ( I ) is a self-adjoint projection withrange M commuting with π ( A ). Thus, the map π ′ : A → B ( M ) defined as π ′ ( a ) = π ( a ) | M , a ∈ A is a unital representation. It is readily verified that π ′ is extension-extremal since π is assumed to be. Invoking [30, Corollary 3.11], we see that there is an extremalunital representation σ ′ : A → B ( H σ ) which is a coextension of π ′ . Define now arepresentation σ : A → B ( H π ) as σ ( a ) = σ ′ ( a ) ⊕ , a ∈ A . Clearly, this is a coextension of π . To see that σ is extremal, we argue as follows.First, we note that the zero map on ι ( A ) has a unique completely positive extensionto B , namely the zero map itself. Invoking [35, Proposition 2.4], we conclude that0 is an extremal representation of A . Once we know this, we may argue as in theproof of [10, Proposition 4.4] (using the version of the Schwarz inequality found in[15, Proposition 1.5.7]) to see that σ ′ being extremal implies that σ is also extremal,by another application of [35, Proposition 2.4].If A is not unital, then as above we may apply [47, Section 3] to find a unitalrepresentation π of the unitization A of A . A trivial modification of the argumentin [35, Proposition 2.5] shows that π is still extension-extremal. An application ofthe previous argument to π completes the proof.(iii) This is completely analogous to (ii).2.2. Finite-dimensional approximations.
Let A be an operator algebra and let π : A → B ( H ) be a representation. Let π λ : A → B ( H π ) , λ ∈ Λbe a net of representations with the property that the space C ∗ ( π λ ( A )) H is finite-dimensional for every λ ∈ Λ. We say that the net ( π λ ) is a finite-dimensionalapproximation for π if ( π λ ( a )) converges to π ( a ) in the SOT for every a ∈ A . If,instead, we have that ( π λ ( a ) ∗ ) converges to π ( a ) ∗ in the SOT for every a ∈ A ,then we say that ( π λ ) is a finite-dimensional ∗ -approximation for π . Finally, we saythat ( π λ ) is an Exel–Loring approximation for π if it is both a finite-dimensionalapproximation and a finite-dimensional ∗ -approximation. Our definitions above aredirectly inspired by work on Exel–Loring [38, Theorem 2.4], according to which all INITE-DIMENSIONAL APPROXIMATIONS AND SEMIGROUP COACTIONS 7 ∗ -representations of an RFD C ∗ -algebra admit what we call Exel–Loring approxi-mations.The rest of this section is devoted to establishing basic facts about finite-dimen-sional approximations that we require throughout the paper. First, we examinedirect sums. Lemma 2.2.
Let A be an operator algebra and let Ω be a set. For each ω ∈ Ω , let π ω : A → B ( H ω ) be a representation that admits an Exel–Loring approximation.Let π : A → B ( ⊕ ω ∈ Ω H ω ) be the representation defined as π ( a ) = M ω ∈ Ω π ω ( a ) , a ∈ A . Then, π admits an Exel–Loring approximation. Proof.
For each ω ∈ Ω, there is a directed set Λ ω and a net π ω,λ : A → B ( H ω ) , λ ∈ Λ ω which is an Exel–Loring approximation for π ω . Let F denote the directed set offinite subsets of Ω. Given F = { ω , ω , . . . , ω n } ∈ F , it is readily verified thatlim λ ∈ Λ ω lim λ ∈ Λ ω · · · lim λ n ∈ Λ ωn n M i =1 π ω i ,λ i ( a ) ⊕ M ω / ∈F ! = M ω ∈ F π ω ( a ) ⊕ M ω / ∈F λ ∈ Λ ω lim λ ∈ Λ ω · · · lim λ n ∈ Λ ωn n M i =1 π ω i ,λ i ( a ) ∗ ⊕ M ω / ∈F ! = M ω ∈ F π ω ( a ) ∗ ⊕ M ω / ∈F a ∈ A . On the other hand, we see thatlim F ∈F M ω ∈ F π ω ( a ) ⊕ M ω / ∈F ! = π ( a ) and lim F ∈F M ω ∈ F π ω ( a ) ∗ ⊕ M ω / ∈F ! = π ( a ) ∗ in the SOT for every a ∈ A . Thus, there is an Exel–Loring approximation for π contained in the set n M i =1 π ω i ,λ i ⊕ M ω = ω ,...,ω n where n ∈ N , ω , ω , . . . , ω n ∈ Ω , λ ∈ Λ ω , λ ∈ Λ ω , . . . , λ n ∈ Λ ω n .Two representations π : A → B ( H π ) and σ : A → B ( H σ ) are said to be approxi-mately unitarily equivalent if there is a sequence of unitary operators U n : H π → H σ , n ∈ N with the property thatlim n →∞ k U n π ( a ) U ∗ n − σ ( a ) k = 0 , a ∈ A . Another elementary fact we need is that the existence of Exel–Loring approxima-tions is preserved under approximate unitary equivalence.
Lemma 2.3.
Let A be an operator algebra and let π : A → B ( H π ) be a represen-tation that admits an Exel–Loring approximation. Let σ : A → B ( H σ ) be anotherrepresentation which is approximately unitarily equivalent to π . Then, σ admits anExel–Loring approximation. R. CLOUˆATRE AND A. DOR-ON
Proof.
By assumption, there is a sequence of unitary operators U n : H π → H σ , n ∈ N with the property that(2.1) lim n →∞ k U n π ( a ) U ∗ n − σ ( a ) k = 0 , a ∈ A . Obviously, we also have that(2.2) lim n →∞ k U n π ( a ) ∗ U ∗ n − σ ( a ) ∗ k = 0 , a ∈ A . Let π λ : A → B ( H ) , λ ∈ Λbe an Exel–Loring approximation for π . Now, let Ω be the directed set consistingof pairs ( n, λ ) where n ∈ N and λ ∈ Λ. For ω = ( n, λ ) ∈ Ω, define a representation σ ω : A → B ( H σ ) as σ ω ( a ) = U n π λ ( a ) U ∗ n , a ∈ A . It then follows easily from Equations (2.1) and (2.2) that the set { σ ω : ω ∈ Ω } contains an Exel–Loring approximation for σ .Next, we show that restrictions to invariant subspaces preserve the existence offinite-dimensional approximations. Lemma 2.4.
Let A be an operator algebra and let π : A → B ( H π ) be a represen-tation. Let H ⊂ H π be a closed subspace. The following statements hold. (i) Assume π admits a finite-dimensional approximation and that H is invariantfor π ( A ) . Then, the representation a π ( a ) | H , a ∈ A admits a finite-dimensional approximation. (ii) Assume π admits an Exel–Loring approximation and that H is reducing for π ( A ) . Then, the representation a π ( a ) | H , a ∈ A admits an Exel–Loring approximation. Proof.
Clearly, we may assume without loss of generality that H is infinite-dimensional.Throughout the proof, we let Λ denote the directed set of triples ( F , V , ε ) where ε > F is a finite subset of the unit ball of A , and V is a finite subset of H .(i) By assumption, there is a net π ω : A → B ( H π ) , ω ∈ Ωof representations such that C ∗ ( π ω ( A )) H π is finite-dimensional and ( π ω ( a )) con-verges in the SOT to π ( a ) for every a ∈ A . Thus, h π ( a ) ξ, π ( b ) η i = lim ω h π ω ( a ) ξ, π ω ( b ) η i for every a, b ∈ A and ξ, η ∈ H π .Next, fix an element λ = ( F , V , ε ) ∈ Λ. Let ε ′ > { x, π ( a ) ξ : a ∈ F , ξ ∈ V } INITE-DIMENSIONAL APPROXIMATIONS AND SEMIGROUP COACTIONS 9 which lies in H , since H is assumed to be invariant for π ( A ). Let ω λ ∈ Ω be chosenlarge enough such that |h π ( a ) ξ, π ( b ) η i − h π ω λ ( a ) ξ, π ω λ ( b ) η i| < ε ′ for every a, b ∈ F , ξ, η ∈ V . Because H is infinite-dimensional, there is an isometry W λ : C ∗ ( π ω λ ( A )) H π → H . Using [32, Lemma 3.5.6], there is a unitary operator U λ ∈ B ( H ) such that k U λ W λ x − x k < ε and k U λ W λ π ω λ ( a ) x − π ( a ) x k < ε for every a ∈ F , x ∈ V . Put V λ = U λ W λ : C ∗ ( π ω λ ( A )) H π → H . For a ∈ F , x ∈ V ,we note that k V ∗ λ x − x k ≤ k x − V λ x k < ε whence k V λ π ω λ ( a ) V ∗ λ x − π ( a ) x k = k V λ π ω λ ( a ) V ∗ λ x − V λ π ω λ ( a ) x k + k V λ π ω λ ( a ) x − π ( a ) x k < k π ω λ ( a ) kk V ∗ λ x − x k + ε ≤ (1 + k a k ) ε ≤ ε. We thus conclude that the net ( V λ π ω λ ( a ) V ∗ λ ) λ converges in the SOT to π ( a ) | H forevery a ∈ A . Moreover, for each λ ∈ Λ we see that C ∗ ( V λ π ω λ ( A ) V ∗ λ ) H is finite-dimensional since it is contained in V λ C ∗ ( π ω λ ( A )) H π . Thus, ( V λ π ω λ ( a ) V ∗ λ ) λ is thedesired finite-dimensional approximation.(ii) The argument is similar to the one above, with some minor modifications.By assumption, there is a net π ω : A → B ( H π ) , ω ∈ Ωof representations such that C ∗ ( π ω ( A )) H π is finite-dimensional. Moreover, for every a ∈ A , in the SOT we have that ( π ω ( a )) converges π ( a ) while ( π ω ( a )) ∗ converges π ( a ) ∗ . Thus, h π ( a ) ξ, π ( b ) η i = lim ω h π ω ( a ) ξ, π ω ( b ) η ih π ( a ) ξ, π ( b ) ∗ η i = lim ω h π ω ( a ) ξ, π ω ( b ) ∗ η ih π ( a ) ∗ ξ, π ( b ) ∗ η i = lim ω h π ω ( a ) ∗ ξ, π ω ( b ) ∗ η i for every a, b ∈ A and ξ, η ∈ H π .Next, fix an element λ = ( F , V , ε ) ∈ Λ. Let ε ′ > { x, π ( a ) ξ, π ( a ) ∗ ξ : a ∈ F , ξ ∈ V } which lies in H , since H is assumed to be reducing for π ( A ). Let ω λ ∈ Ω be chosenlarge enough such that |h π ( a ) ξ, π ( b ) η i − h π ω λ ( a ) ξ, π ω λ ( b ) η i| < ε ′ , |h π ( a ) ξ, π ( b ) ∗ η i − h π ω λ ( a ) ξ, π ω λ ( b ) ∗ η i| < ε ′ , and |h π ( a ) ∗ ξ, π ( b ) ∗ η i − h π ω λ ( a ) ∗ ξ, π ω λ ( b ) ∗ η i| < ε ′ , for every a, b ∈ F , ξ, η ∈ V . Arguing as in (i), we find an isometry V λ : C ∗ ( π ω λ ( A )) H π → H with the property that k V λ x − x k < ε and k V λ π ω λ ( a ) x − π ( a ) x k < ε, k V λ π ω λ ( a ) ∗ x − π ( a ) ∗ x k < ε for every a ∈ F , x ∈ V . As before, the net ( V λ π ω λ ( a ) V ∗ λ ) λ is verified to be thedesired Exel–Loring approximation.An important application of the preceding sequence of lemmas is the next tool. Theorem 2.5.
Let A be a separable operator algebra. Let π : C ∗ ( A ) → B ( H π ) and σ : C ∗ ( A ) → B ( H σ ) be two ∗ -representations. Assume that π is injective and that π | A admits an Exel–Loring approximation. Then, σ | A also admits an Exel–Loringapproximation. Proof.
Because π is injective and A is separable, we may find a separable closedsubspace H ′ π ⊂ H π which is reducing for π (C ∗ ( A )) and such that the restriction π ′ : C ∗ ( A ) → B ( H ′ π ) defined as π ′ ( t ) = π ( t ) | H ′ π , t ∈ C ∗ ( A )is injective and non-degenerate. By Lemma 2.4, we see that π ′ also admits anExel–Loring approximation. Thus, we may assume that π itself is injective, non-degenerate, and that H π is separable.Next, we may write σ = σ ′ ⊕
0, where σ ′ is a non-degenerate ∗ -representationwhich can, in turn, be decomposed as the direct sum of cyclic ∗ -representations.Lemma 2.2 implies that it suffices to establish the desired result for all these cyclic ∗ -representations. Thus, we may simply assume that σ is non-degenerate and that H σ is separable.Consider then the non-degenerate ∗ -representations Π : C ∗ ( A ) → B ( H ( ∞ ) π ) andΣ : C ∗ ( A ) → B ( H ( ∞ ) σ ) defined asΠ( t ) = π ( t ) ( ∞ ) and Σ( t ) = σ ( t ) ( ∞ ) for every t ∈ C ∗ ( A ). In particular, we see that Π is injective, and that Π(C ∗ ( A )) andΣ(C ∗ ( A )) contain no compact operators. The non-degenerate ∗ -representations Πand Σ ⊕ Π are both injective, so that by [29, Corollary II.5.6] they are approximatelyunitarily equivalent. By assumption, π | A admits an Exel–Loring approximation,and thus so does Π | A by Lemma 2.2. In turn, Lemma 2.3 implies that (Σ ⊕ Π) | A admits an Exel–Loring approximation. Finally, Lemma 2.4 implies that Σ | A and σ | A admit an Exel–Loring approximations as well.The final preliminary tool we require is the following, which provides Exel–Loringapproximations for representations that extend to ∗ -representations of some RFDC ∗ -cover. Lemma 2.6.
Let ( B , ι ) be a C ∗ -cover of some operator algebra A such that B isRFD. Let π be a representation of A such that π ◦ ι − extends to a ∗ -representationof B . Then, π admits an Exel–Loring approximation. Proof.
Write π : A → B ( H π ). By assumption, there is a ∗ -representation σ : B → B ( H π ) such that σ ◦ ι = π. Invoking [38, Theorem 2.4], we know that σ admits anExel–Loring approximation σ λ : B → B ( H π ) , λ ∈ Λ . INITE-DIMENSIONAL APPROXIMATIONS AND SEMIGROUP COACTIONS 11
This means that σ λ ( B ) H π is finite-dimensional for every λ ∈ Λ, and ( σ λ ( b )) con-verges to σ ( b ) in SOT for every b ∈ B . For each λ ∈ Λ, let π λ = σ λ ◦ ι . Note firstthat C ∗ ( π λ ( A )) H π = σ λ ( B ) H π is finite-dimensional for every λ ∈ Λ. Next, fix a ∈ A . Then, we find π λ ( a ) = σ λ ( ι ( a )) and π λ ( a ) ∗ = σ λ ( ι ( a ) ∗ )for every λ ∈ Λ. We infer that ( π λ ( a )) converges to π ( a ) and ( π λ ( a ) ∗ ) converges to π ( a ) ∗ in SOT. Consequently, ( π λ ) is an Exel–Loring approximation of π .3. Characterizations of residual finite-dimensionality
In this section, we use finite-dimensional approximations and the related toolsdeveloped in Section 2 to analyze residual finite-dimensionality of general (that is,possibly non-selfadjoint) operator algebras. We start with a characterization ofresidual finite-dimensionality.
Theorem 3.1.
Let A be an operator algebra. The following are equivalent. (i) The algebra A is RFD. (ii) Every extremal representation of A admits a finite-dimensional approxima-tion. (iii) Every extremal representation of A admits an Exel–Loring approximation. (iv) Every coextension-extremal representation of A admits a finite-dimensionalapproximation. Proof. (iii) ⇒ (ii): This is trivial.(ii) ⇒ (i): By virtue of [36, Theorems 1.1 and 1.2] combined with [35, Proposition2.5], we may find a set S of extremal representations of A with the property thatthe map L π ∈S π is completely isometric. Let d ∈ N and A ∈ M d ( A ) with k A k = 1.Let ε >
0. There is a representation π : A → B ( H π ) in S such that k π ( d ) ( A ) k ≥ (1 − ε ) . On the other hand, by assumption we see that π admits a finite-dimensional ap-proximation ( π λ ) λ ∈ Λ . Thus, there is λ ∈ Λ with the property that k π ( d ) λ ( A ) k ≥ (1 − ε ) . Let H λ = C ∗ ( π λ ( A )) H π . This is a finite-dimensional reducing subspace for π λ .Define ρ λ : A → B ( H λ ) to be the representation ρ λ ( a ) = π λ ( a ) | H λ , a ∈ A . We find π λ ( a ) = ρ λ ( a ) ⊕ , a ∈ A whence π ( d ) λ ( A ) is unitarily equivalent to ρ ( d ) λ ( A ) ⊕
0, so that k ρ ( d ) λ ( A ) k ≥ (1 − ε ) . We conclude that A is RFD.(i) ⇒ (iii): By definition of A being RFD, there is a C ∗ -cover ( B , ι ) of A suchthat B is an RFD C ∗ -algebra. Let π be an extremal representation of A . By virtueof Theorem 2.1, we may find a ∗ -representation σ of B such that σ ◦ ι = π . It thenfollows from Lemma 2.6 that π admits an Exel–Loring approximation.(iv) ⇒ (ii): This is trivial. (ii) ⇒ (iv): Let π be a coextension-extremal representation of A . By Theorem2.1, we see that there is an extremal representation σ of A which is an extensionof π . Since σ is assumed to admit a finite-dimensional approximation, Lemma 2.4implies that π admits a finite-dimensional approximation as well.Let A be an operator algebra and let π be an extremal representation of A .Let (C ∗ e ( A ) , ε ) denote the C ∗ -envelope of A . By Theorem 2.1, we see that π ◦ ε − extends to a ∗ -representation of C ∗ e ( A ). Unfortunately, the converse statement failsin general, in the sense that there are ∗ -representations of C ∗ e ( A ) whose restrictionto ε ( A ) is not extremal; in other words, there are operator algebras which are not hyperrigid [10]. We mention that this kind of pathology occurs even in the classicalsetting of uniform algebras on compact metric spaces [52, page 42].In light of this discussion, for non-hyperrigid RFD operator algebras, Theorem3.1 is inadequate to deal with general ∗ -representations of the C ∗ -envelope. Weremedy this shortcoming in the next result, at least in the separable setting. Theorem 3.2.
Let A be a separable operator algebra and let (C ∗ e ( A ) , ε ) denoteits C ∗ -envelope. Then A is RFD if and only if π ◦ ε − admits an Exel–Loringapproximation for every ∗ -representation π of C ∗ e ( A ) . Proof.
Assume first that π ◦ ε − admits an Exel-Loring approximation for every ∗ -representation π of C ∗ e ( A ). As explained before the theorem, this implies inparticular that π ◦ ε − , and hence π , admits an Exel–Loring approximation forevery extremal representation π of A . By virtue of Theorem 3.1, we conclude that A is RFD.Conversely, assume that A is RFD. Let B denote the unitization of A (see [13,Paragraph 2.1.11]), and let (C ∗ e ( B ) , ι ) denote the C ∗ -envelope of B . By [7, Propo-sition 2.2.4],[9, Theorem 7.1], and [10, Proposition 4.4], we may find an injectiveunital ∗ -representation π : C ∗ e ( B ) → B ( H π ) with H π separable such that π ◦ ι is anextremal representation of B . Note now that C ∗ e ( B ) = C ∗ ( ι ( A )) + C I . Thus, asexplained in [8, page 15], we may find two ∗ -representations π , π of C ∗ e ( B ) suchthat π (C ∗ ( ι ( A ))) is non-degenerate, C ∗ ( ι ( A )) ⊂ ker π and π = π ⊕ π . Because π is injective, we see that π must be injective on C ∗ ( ι ( A )). By [13, Paragraph4.3.4], we know that we may choose C ∗ e ( A ) = C ∗ ( ι ( A )) and ε = ι | A . Furthermore,[35, Proposition 2.5] implies that π | ι ( A ) = π | ι ( A ) is extremal since π | ι ( B ) is. Itfollows from Theorem 3.1 that π | ι ( A ) admits an Exel–Loring approximation. Anapplication of Theorem 2.5 now yields the desired statement.As mentioned above, in the case of RFD C ∗ -algebras, it is known that everyrepresentation admits an Exel–Loring approximation [38]. Theorems 3.1 and 3.2make similar claims in the non-selfadjoint context, but only for certain specialrepresentations. It is then natural to wonder what can be said about the existence ofExel–Loring approximations for general representations of RFD operator algebras.As we show next, this property is in fact equivalent to a condition which is a prioristronger, namely the residual finite-dimensionality of the maximal C ∗ -cover. Theorem 3.3.
Let A be an operator algebra and let (C ∗ max ( A ) , υ ) denote its max-imal C ∗ -cover. Then, the following statements are equivalent. (i) The C ∗ -algebra C ∗ max ( A ) is RFD. (ii) Every representation of A admits an Exel–Loring approximation. INITE-DIMENSIONAL APPROXIMATIONS AND SEMIGROUP COACTIONS 13 (iii)
Let H be a Hilbert space and assume that C ∗ max ( A ) ⊂ B ( H ) . Then, the repre-sentation υ : A → B ( H ) admits an Exel–Loring approximation. Proof. (i) ⇒ (ii): Assume that C ∗ max ( A ) is RFD. Let π : A → B ( H π ) be a represen-tation. Then, there is a ∗ -representation b π : C ∗ max ( A ) → B ( H π ) such that b π ◦ υ = π .It thus follows from Lemma 2.6 that π admits an Exel–Loring approximation.(ii) ⇒ (iii): This is trivial.(iii) ⇒ (i): Assume that υ has an Exel–Loring approximation π λ : A → B ( H ) , λ ∈ Λ . Note that linear combinations of words with letters in υ ( A ) ∪ υ ( A ) ∗ are clearlydense in C ∗ max ( A ), since C ∗ max ( A ) = C ∗ ( υ ( A )). Therefore, to show that the latterC ∗ -algebra is RFD it suffices to fix such an element s = 0 and to construct a finite-dimensional ∗ -representation ρ of C ∗ max ( A ) that does not vanish on s , since injective ∗ -representations are automatically completely isometric.For each λ ∈ Λ, we obtain a ∗ -representation b π λ : C ∗ max ( A ) → B ( H ) with theproperty that b π λ ◦ υ = π λ . Because multiplication is jointly continuous in theSOT on bounded sets, it is readily verified that ( b π λ ( s )) converges to s in the SOT.In particular, there exists λ ∈ Λ such that b π λ ( s ) = 0. Note now that the space H λ = b π λ (C ∗ max ( A )) H is finite-dimensional since b π λ (C ∗ max ( A )) H = C ∗ ( b π λ ( υ ( A )) H = C ∗ ( π λ ( A )) H and the latter is finite-dimensional by construction. Thus, we may define a finite-dimensional ∗ -representation ρ : C ∗ max ( A ) → B ( H λ ) as ρ ( t ) = b π λ ( t ) | H λ , t ∈ C ∗ max ( A ) . It follows that b π λ ( t ) = ρ ( t ) ⊕ , t ∈ C ∗ max ( A ) . so in particular ρ has the required property.In light of Theorems 3.1 and 3.3, we see that the existence of finite-dimensionaland Exel–Loring approximations is at the heart of Question 1. Indeed, the questionis equivalent to the one asking whether all representations of an RFD operator al-gebra admit Exel–Loring approximations. Alternatively, one can ask for the weakerproperty ( F ) defined in the introduction. Question 2.
Let A be an RFD operator algebra. Does A have property ( F )? Inother words, do all representations of A admit a finite-dimensional approximation?To illustrate some of the difficulties associated with this question, we remarkthat the approximations from Theorem 3.3 can be rather complicated, even for verytransparent choices of representations. For instance, let A ⊂ B ( H ) be an operatoralgebra such that C ∗ max ( A ) is RFD. Let K ⊂ H be an invariant subspace for A .Consider the representation π : A → B ( K ) defined as π ( a ) = a | K for every a ∈ A .Then, by Theorem 3.3, we know that π admits an Exel–Loring approximation.But this approximation does not necessarily arise as compressions of A to finite-dimensional subspaces of K , as the following concrete example shows. Recall thata closed subspace S ⊂ H is said to be semi-invariant for A if the map a P S a | S , a ∈ A is multiplicative. A standard verification reveals that S is semi-invariant if andonly if S = M ⊖ N for some closed subspaces M , N ⊂ H that are invariant for A . Example 3.4.
Let D ⊂ C denote the open unit disc. Let A( D ) be the classicaldisc algebra acting on the Hardy space H . Let H ∞ denote the Banach algebraof bounded holomorphic functions on D . In this example, we will freely use well-known facts about the structure of these objects; the reader may consult [41] fordetails. Let θ ∈ H ∞ be the singular inner function corresponding to the point massat 1, i.e. θ ( z ) = exp (cid:18) z + 1 z − (cid:19) , z ∈ D . Let K θ = H ⊖ θH and let π : A( D ) → B ( K θ ) be the unital representation definedas π ( a ) = P K θ a | K θ , a ∈ A( D ) . Note that C ∗ max (A( D )) is RFD by [20, Example 3]. Therefore, π admits an Exel–Loring approximation by Theorem 3.3. Nevertheless, as we show next, K θ containsno finite-dimensional subspace which is semi-invariant for π (A( D )).To see this, assume that S ⊂ K θ is such a subspace. We may find closedsubspaces M , N ⊂ K θ which are invariant for π (A( D )) such that S = M ⊖ N .Then, M ⊕ θH and N ⊕ θH are also closed and invariant for A( D ). By Beurling’stheorem, there are distinct inner functions ϕ and ψ in H ∞ such that ϕH = M ⊕ θH , ψH = N ⊕ θH . Because θ ∈ ϕH ∩ ψH , we see that ϕ and ψ must be singular inner functions with ϕH ⊂ ψH or ψH ⊂ ϕH . The first possibility would imply M ⊂ N and hence S = { } contrary to our assumption. Thus, we must have then that ψH ⊂ ϕH and S = M ⊖ N = ϕH ⊖ ψH . We infer that S is infinite-dimensional (this follows for instance from [11, ExerciseIII.1.11]).4. Constructing Exel–Loring approximations via semigroup coactions
Motivated by Question 1, in this section we aim to identify concrete instances ofRFD operator algebras A with the property that C ∗ max ( A ) is RFD. Our approach ispredicated on Theorem 3.3, in that we aim to produce Exel–Loring approximationsfor arbitrary representations. The unifying theme throughout this section will bethat of semigroups and their coactions. For more on semigroups and their C*-algebras, we recommend [28, Chapter 5] and [44–46].Let P be a discrete semigroup. Assume that P is cancellative , so that twoelements p, q ∈ P satisfy p = q whenever there is r ∈ P with rp = rq or pr = qr .For p ∈ P , define R p = { r ∈ P : p = qr for some q ∈ P } and L p = { q ∈ P : p = qr for some r ∈ P } . Let r ∈ R p . By definition, we find q r ∈ L p such that p = q r r . Since P iscancellative, this element q r is uniquely determined. Likewise, given q ∈ L p thereis a unique r q ∈ R p such that p = qr q . We conclude that there is a bijectionbetween R p and L p . We say that the cancellative semigroup P has the finite divisorproperty (or FDP) if for every p ∈ P the sets R p and L p are finite. Clearly thefinite divisor property is inherited by subsemigroups. Semigroups with FDP include INITE-DIMENSIONAL APPROXIMATIONS AND SEMIGROUP COACTIONS 15 important examples such as free monoids, left-angled Artin monoids, Baumslag–Solitar monoids, braid monoids, Thompson’s monoid and more.Next, let λ : P → B ( ℓ ( P )) denote the left regular representation. For each p ∈ P , we denote by e p ∈ ℓ ( P ) the characteristic function of { p } . Thus, { e p } p ∈ P is an orthonormal basis for ℓ ( P ). Because P is left cancellative, it is easily seenthat λ p is an isometry for every p ∈ P . Furthermore, for p ∈ P we see that(4.1) λ ∗ p ( e r ) = ( e q if r = pq r / ∈ pP. Let A r ( P ) ⊂ B ( ℓ ( P )) denote the norm closed operator algebra generated by theimage of λ , and let C ∗ r ( P ) denote the C ∗ -algebra generated by the image of λ . Thedevelopments in this section hinge on the following fact. Proposition 4.1.
Let P be a cancellative semigroup with FDP. For every finitesubset F ⊂ P , there is a representation π F : A r ( P ) → B ( ℓ ( P )) such that P \ (cid:0) ∪ p ∈ F ∪ r ∈R p L r (cid:1) = λ − (ker π F ) . Moreover, the net ( π F ) F is an Exel–Loring approximation for the identity repre-sentation of A r ( P ) . In particular, A r ( P ) is RFD. Proof.
For each p ∈ P , the subset R p is finite by the assumption on the semigroup.Correspondingly, let X p ⊂ ℓ ( P ) denote the finite-dimensional subspace spannedby { e r : r ∈ R p } . Note now that if r ∈ R p , then R r ⊂ R p . Invoking (4.1), we findfor p, s ∈ P and r ∈ R p that(4.2) λ ∗ s e r ∈ { } ∪ { e t : t ∈ R r } ⊂ X p and that(4.3) λ ∗ s e r = 0 if and only if s ∈ L r . We conclude from (4.2) that X p is coinvariant for A r ( P ) for every p ∈ P .Next, for every finite subset F ⊂ P , we let Y F = P p ∈ F X p . Clearly, Y F isa finite-dimensional subspace which is coinvariant for A r ( P ). Let Q F ∈ B ( ℓ ( P ))denote the finite-rank orthogonal projection onto Y F . Thus, the map π F : A r ( P ) → B ( ℓ ( P )) defined as π F ( a ) = Q F aQ F , a ∈ A r ( P )is a representation. It readily follows the fact that ∪ p ∈ P R p = P that the net ( π F ) is an Exel–Loring approximation for the identity representationof A r ( P ), and that A r ( P ) is RFD. Finally, (4.3) implies that π F ( λ s ) = 0 if andonly if s ∈ ∪ p ∈ F ∪ r ∈R p L r , which is equivalent to P \ (cid:0) ∪ p ∈ F ∪ r ∈R p L r (cid:1) = λ − (ker π F ) . Coactions of discrete groups on general operator algebras were introduced in[34] as a generalization of coactions of discrete groups on C*-algebras [53]. Thefollowing is a semigroup variant of [34, Definition 3.1], adapted to our context.
Definition 4.2.
Let A be an operator algebra, P be a cancellative discrete semi-group, and δ : A → A ⊗ min A r ( P ) a completely isometric homomorphism. For each p ∈ P denote by A δp the p -th spectral subspace according to δ , given by A δp = { a ∈ A | δ ( a ) = a ⊗ λ p } . We say that δ is a coaction of P on A if P p ∈ P A δp is norm dense in A .We remark here that in principle, the role of A r ( P ) in the definition of a semi-group coaction can be fulfilled by a norm closed algebra generated by any contrac-tive representation of P . However, we will not need this added generality for ourpurposes. A useful property of semigroup coactions that we require is the following. Lemma 4.3.
Let P be a cancellative discrete semigroup with the property that thereis a character χ : A r ( P ) → C such that χ ( λ p ) = 1 for every p ∈ P . Let A be anoperator algebra and let π : A → B ( H π ) be a representation. If δ is a coaction of P on A , then ( π ⊗ χ ) ◦ δ is unitarily equivalent to π . Proof.
For p ∈ P and a ∈ A δp , we find( π ⊗ χ ) ◦ δ ( a ) = ( π ⊗ χ )( a ⊗ λ p ) = π ( a ) ⊗ . The density of the sum of the spectral subspaces thus implies that ( π ⊗ χ ) ◦ δ isunitarily equivalent to π .Next, let F denote the directed set of finite subsets of P . Proposition 4.1 yieldsthe existence of a net ( π F ) F ∈F which is an Exel–Loring approximation for theidentity representation of A r ( P ). For each F ∈ F , we let Q F be the quotient of A by the norm closure of the ideal generated by P p ∈ λ − (ker π F ) A δp . Definition 4.4.
Let A be an operator algebra, P be a cancellative discrete semi-group, and δ a coaction of P on A . We say that δ is residually finite–dimensional (RFD) if the C ∗ -algebra C ∗ max ( Q F ) is RFD for each F ∈ F .We now take our first crucial step towards answering Question 1 for operatoralgebras with semigroup coactions. Proposition 4.5.
Let A be an operator algebra and let P be a cancellative semi-group with FDP. Assume that there exists an RFD coaction δ of P on A . Let π : A → B ( H π ) be a representation. Then, the representation ( π ⊗ id) ◦ δ admitsan Exel–Loring approximation. Proof.
Let F ⊂ P be a finite subset, and let q F : A → Q F denote the naturalquotient map. If p ∈ λ − (ker π F ) and a ∈ A δp , then( π ⊗ π F ) ◦ δ ( a ) = π ( a ) ⊗ π F ( λ p ) = 0 . We conclude that X p ∈ λ − (ker π F ) A δp ⊂ ker( π ⊗ π F ) ◦ δ. Thus, there is a representation ρ F : Q F → B ( H π ⊗ ℓ ( P ))) with the property that( π ⊗ π F ) ◦ δ = ρ F ◦ q F . Since the coaction δ is assumed to be RFD, we see that C ∗ max ( Q F ) is RFD, so byTheorem 3.3 the representation ρ F admits an Exel–Loring approximation ρ F,µ : Q F → B ( H π ⊗ ℓ ( P )) , µ ∈ Ω F . INITE-DIMENSIONAL APPROXIMATIONS AND SEMIGROUP COACTIONS 17
In particular,C ∗ (( ρ F,µ ◦ q F )( A ))( H π ⊗ ℓ ( P )) = C ∗ ( ρ F,µ ( Q F ))( H π ⊗ ℓ ( P ))is finite-dimensional for every µ ∈ Ω F .For every p ∈ P and every a ∈ A δp , given ξ ∈ H π and η ∈ ℓ ( P ) we obtainlim F lim µ ( ρ F,µ ◦ q F )( a )( ξ ⊗ η ) = lim F ( ρ F ◦ q F )( a )( ξ ⊗ η )= lim F (( π ⊗ π F ) ◦ δ )( a )( ξ ⊗ η )= lim F π ( a ) ξ ⊗ π F ( λ p ) η = π ( a ) ξ ⊗ λ p η = ( π ( a ) ⊗ λ p )( ξ ⊗ η )= (( π ⊗ id) ◦ δ )( a )( ξ ⊗ η )and likewise lim F lim µ ( ρ F,µ ◦ q F )( a ) ∗ ( ξ ⊗ η ) = (( π ⊗ id) ◦ δ )( a ) ∗ ( ξ ⊗ η ) . A standard density argument then reveals that there is a net in { ρ F,µ ◦ q F : F ⊂ P finite , µ ∈ Ω F } which forms an Exel–Loring approximation for ( π ⊗ id) ◦ δ .The following is the connection between coactions of semigroups on RFD oper-ator algebras, and RFD maximal C*-covers. Theorem 4.6.
Let P a countable cancellative semigroup with FDP such that thereis a character χ : A r ( P ) → C with χ ( λ p ) = 1 for every p ∈ P . Let A be a separableoperator algebra that admits an RFD coaction of P . Then, C ∗ max ( A ) is RFD. Proof.
Recall that υ : A → C ∗ max ( A ) is the completely isometric homomorphismsuch that (C ∗ max ( A ) , υ ) is the maximal C ∗ -cover of A . By Theorem 3.3, it sufficesto fix an injective ∗ -representation ρ : C ∗ max ( A ) → B ( H ) and to show that therepresentation ρ ◦ υ : A → B ( H ) admits an Exel–Loring approximation.To see this, let δ be an RFD coaction of P on A . By injectivity of the minimaltensor product, the map γ = υ ⊗ id : A ⊗ min A r ( P ) → C ∗ max ( A ) ⊗ min C ∗ r ( P )is a completely isometric homomorphism. Let b δ : C ∗ max ( A ) → C ∗ max ( A ) ⊗ min C ∗ r ( P )be the ∗ -representation such that b δ ◦ υ = γ ◦ δ .Now, it follows from Lemma 4.3 that ( π ⊗ χ ) ◦ δ is unitarily equivalent to π . Let b χ : C ∗ r ( P ) → C be a state extending χ . Since λ p is an isometry and χ ( λ p ) = 1,it follows that λ p lies in the multiplicative domain of χ for every p ∈ P . Thus, b χ is a character, which in turn implies that ( ρ ⊗ b χ ) ◦ b δ is unitarily equivalent to ρ .In particular, because ρ is injective this forces b δ to be an injective ∗ -representationand thus so is ( ρ ⊗ id) ◦ b δ . Proposition 4.5 now yields an Exel–Loring approximationfor ( π ⊗ id) ◦ δ . But we find( π ⊗ id) ◦ δ = ( ρ ⊗ id) ◦ b δ ◦ υ so that ( ρ ⊗ id) ◦ b δ | υ ( A ) admits an Exel–Loring approximation. Since P is countable,the algebra A r ( P ) is separable and we may invoke Theorem 2.5 to conclude that ρ | υ ( A ) admits an Exel–Loring approximation, and hence the same is true for ρ ◦ υ .The astute reader may notice that the condition that A r ( P ) admit an appro-priate character in the previous theorem is stronger than what is actually neededin the proof. Indeed, therein we only need to know that the coaction δ admitsan extension to an injective ∗ -representation b δ : C ∗ max ( A ) → C ∗ max ( A ) ⊗ min C ∗ r ( P ).Unfortunately, at present we do not know how to verify this condition withoutassuming the existence of a character, or when the coaction is trivial.5. Algebras of semigroups and algebras of functions
In this section, we exhibit several classes examples of operator algebras to whichTheorem 4.6 can be applied. These include algebras arising from semigroups andalgebras of functions.5.1.
Semigroup operator algebras.
We recall some background material onsemigroups. Let P be a cancellative semigroup. By a right ideal of P we mean asubset X ⊂ P which is closed under right multiplication. Given a subset X ⊂ P and an element p ∈ P , we define pX = { px : x ∈ X } and p − X = { x : px ∈ X } . Further, we denote by J P the smallest collection of right ideals that contains ∅ and P , and such that X ∈ J P implies pX, p − X ∈ J P .Assume now that P is embedded in a group G . The full semigroup C ∗ -algebra ,denoted by C ∗ s ( P ), is the universal C ∗ -algebra generated by isometries { v p : p ∈ P } and self-adjoint projections { e X : X ∈ J P } such that(i) e ∅ = 0;(ii) v p v q = v pq for p, q ∈ P ;(iii) whenever p , q , ..., p n , q n ∈ P satisfy p q − ...p − n q n = e in G , then v ∗ p v q ...v ∗ p n v q n = e q − n p n ...q − p P . Furthermore, we let D s ( P ) ⊂ C ∗ s ( P ) denote the C ∗ -subalgebra generated by { e X : X ∈ J P } .The defining relations above can be realized inside of C ∗ r ( P ), as we explain next.For added clarity, thoughout this section we denote the left regular representationof P by λ P . Given a subset X ⊂ P , we let 1 PX ∈ B ( ℓ ( P )) denote the operatorof multiplication by the indicator function of X . Clearly, this is a self-adjointprojection. We denote by D r ( P ) the C*-subalgebra of C ∗ r ( P ) generated { PX : X ∈ J P } . Now, it follows from [44, Lemma 3.1] that the families { λ Pp : p ∈ P } and { PX : X ∈J P } inside B ( ℓ ( P )) satisfy the defining relations above. Hence, by universality weobtain a surjective ∗ -representation λ P ∗ : C ∗ s ( P ) → C ∗ r ( P ) such that λ P ∗ ( p ) = λ Pp , p ∈ P and λ P ∗ ( e X ) = 1 X , X ∈ J P . Now, by [44, Lemma 3.11] there is a faithful conditional expectation E Pr : C ∗ r ( P ) → D r ( P ). INITE-DIMENSIONAL APPROXIMATIONS AND SEMIGROUP COACTIONS 19
A cancellative semigroup P is said to be independent if whenever X ∈ J P canbe written as X = n [ i =1 X i for some X , . . . , X n ∈ J P , then necessarily we have X = X j for some 1 ≤ j ≤ n .This property will play a crucial role below. Fortunately, many semigroups fromthe literature automatically satisfy independence. For instance, all right LCMsemigroups are independent by [46, Lemmas 6.31 and 6.32].Assume now that P embeds into a group and is independent. By [44, Lemma3.13] there is a conditional expectation E Ps : C ∗ s ( P ) → D so that λ P ∗ ◦ E Ps = E Pr ◦ λ P ∗ and(5.1) ker( λ P ∗ ) = { a ∈ C ∗ s ( P ) : E Ps ( a ∗ a ) = 0 } . This fact will be used in the proof of the next result, which shows, roughly speaking,that certain semigroup homomorphisms induce ∗ -representations on the reducedC ∗ -algebras. Proposition 5.1.
Let P and Q be unital semigroups embedded into some groups G and H respectively. Suppose that P is independent and that ψ : G → H is aninjective homomorphism such that ψ ( P ) ⊂ Q . Then, there is a ∗ -representation Ψ r : C ∗ r ( P ) → C ∗ r ( Q ) such that Ψ r ( λ Pp ) = λ Qψ ( p ) , p ∈ P. Proof.
For each p ∈ P , define the isometry w p = λ Qψ ( p ) ∈ B ( ℓ ( Q )). For each X ∈ J P , define the projection f X = 1 Qψ ( X ) Q ∈ B ( ℓ ( Q )). Because ψ ( e ) = e and ψ ( P ) ⊂ Q , we see that ψ ( P ) Q = Q . From this, it readily follows that thatthe collections { w p : p ∈ P } and { f X : X ∈ J P } satisfy properties (i), (ii) and(iii) defining the full semigroup C ∗ -algebra C ∗ s ( P ). Consequently, there is a ∗ -homomorphism Ψ : C ∗ s ( P ) → C ∗ r ( Q ) such thatΨ( v p ) = w p , p ∈ P and Ψ( e X ) = f X , X ∈ J P . Furthermore, given p , . . . , p m , q , . . . , q m ∈ P , we compute using [44, Lemmas 3.1and 3.11] that E Qr ◦ Ψ( v ∗ p v q · · · v ∗ p m v q m ) = E Qr ( λ Q ∗ ψ ( p ) λ Qψ ( q ) · · · λ Q ∗ ψ ( p m ) λ Qψ ( q m ) )= ( Qψ ( q − m ) ψ ( p m ) ··· ψ ( q − ) ψ ( p ) Q if ψ ( p − q · · · p − m q m ) = e in H, ( Qψ ( q − m ) ψ ( p m ) ··· ψ ( q − ) ψ ( p ) Q if p − q · · · p − m q m = e in G, ψ is injective. Using that ψ ( P ) Q = Q again,[44, Lemma 3.13] implies that E Qr ◦ Ψ( v ∗ p v q · · · v ∗ p m v q m ) = Ψ ◦ E Ps ( v ∗ p v q · · · v ∗ p m v q m ) . We conclude from this that E Qr ◦ Ψ = Ψ ◦ E Ps . Combining this equality with (5.1) and recalling that E Qr is faithful, we infer thatker λ P ∗ ⊂ ker Ψ. Since C ∗ r ( P ) ∼ = C ∗ s ( P ) / ker λ P ∗ , we obtain a ∗ -representation Ψ r :C ∗ r ( P ) → C ∗ r ( Q ) with the desired property.We now give an extension [27, Lemma 7.5] that does not require the semigroupsto be Ore. Furthermore, it is not confined to the natural “diagonal” action λ Pp λ Pp ⊗ λ Pp . Proposition 5.2.
Let P and Q be unital semigroups embeddable into some groups G and H respectively. Assume that P is independent and let ϕ : G → H bea homomorphism such that ϕ ( P ) ⊂ Q . Then there is a ∗ -representation ∆ ϕ :C ∗ r ( P ) → C ∗ r ( P ) ⊗ min C ∗ r ( Q ) such that ∆ ϕ ( λ Pp ) = λ Pp ⊗ λ Qϕ ( p ) Proof.
Let ψ : G → G × H be the injective homomorphism given by ψ ( g ) =( g, ϕ ( g )). Hence, by Proposition 5.1 we have a ∗ -homomorphism Ψ r : C ∗ r ( P ) → C ∗ r ( P × Q ) such that Ψ( λ Pp ) = λ P × Q ( p,ϕ ( p )) , p ∈ P. On the other hand, by [44, Lemma 2.16] there is a ∗ -isomorphism Θ : C ∗ r ( P × Q ) → C ∗ r ( P ) ⊗ min C ∗ r ( Q ) such thatΘ( λ P × Qp,q ) = λ Pp ⊗ λ Qq , p ∈ P, q ∈ Q. The map ∆ ϕ = Θ ◦ Ψ r has the required properties.Our next goal is to show that the map ∆ ϕ above restricts to a coaction of Q on A r ( P ). To do this, we need the following simple observation. Lemma 5.3 (Fell’s absorption principle for semigroups) . Let P be a unital cancella-tive semigroup, and suppose V : P → B ( H ) is a representation of P by isometries.Then, there is an isometry W ∈ B ( ℓ ( P ) ⊗ H ) such that W ( λ p ⊗ id H ) = ( λ p ⊗ V p ) W, p ∈ P. Proof.
The map e p ⊗ ξ e p ⊗ V p ( ξ ) , p ∈ P, ξ ∈ H is easily seen to extend to an isometry W : ℓ ( P ) ⊗ H → ℓ ( P ) ⊗ H . But then, for p, q ∈ P and ξ ∈ H we have( λ q ⊗ V q ) W ( e p ⊗ ξ ) = e qp ⊗ V qp ( ξ ) = W ( e qp ⊗ ξ )= W ( λ q ⊗ id H )( e p ⊗ ξ ) . We infer that W ( λ q ⊗ id H ) = ( λ q ⊗ V q ) W as required.We now arrive at a result which will provide us with many examples of coactions. Theorem 5.4.
Let P and Q be unital semigroups embeddable into groups G and H respectively. Assume that P is independent and let ϕ : G → H be a homomorphismsuch that ϕ ( P ) ⊂ Q . Then, there is a coaction δ of Q on A r ( P ) such that δ ( λ Pp ) = λ Pp ⊗ λ Qϕ ( p ) , p ∈ P. INITE-DIMENSIONAL APPROXIMATIONS AND SEMIGROUP COACTIONS 21
Proof.
By virtue of Proposition 5.2, there is a ∗ -representation ∆ ϕ : C ∗ r ( P ) → C ∗ r ( P ) ⊗ min C ∗ r ( Q ) such that ∆ ϕ ( λ Pp ) = λ Pp ⊗ λ Qϕ ( p ) Note that the injectivity of the minimal tensor product we have A r ( P ) ⊗ min A r ( Q ) ⊂ C ∗ r ( P ) ⊗ min C ∗ r ( Q ). Thus, we obtain a representation δ : A r ( P ) → A r ( P ) ⊗ min A r ( Q ) by putting δ = ∆ ϕ | A r ( P ) . In particular, we conclude that δ takes values in A r ( P ) ⊗ min A r ( Q ), as required. By construction, it is clear that the sum of thespectral subspaces for δ is norm dense in A r ( P ).Let V : P → B ( ℓ ( Q )) be given by V p = λ Qϕ ( p ) , which is a representation of P by isometries. By Lemma 5.3 there is an isometry W such that W ( λ Pp ⊗ id H ) = δ ( λ Pp ) W, p ∈ P. Hence, we have W ( a ⊗ id H ) = δ ( a ) W, a ∈ A r ( P ) . Let A = [ a ij ] ∈ M n ( A r ( P )) for some n ∈ N . We find k A k = k [ a ij ⊗ I H ] k = k [ W a ij ⊗ I H ] k = k δ ( n ) ( A )( W ⊕ W ⊕ . . . ⊕ W ) k ≤ k δ ( n ) ( A ) k . We infer that δ is completely isometric, and thus is indeed a coaction of Q on A r ( P ).The following shows that the notion of a semigroup coaction in Definition 4.2 isdeserving of its name, at least for independent semigroups embedded in groups. Corollary 5.5.
Let P be a unital independent semigroup embedded in a group G . Then there is a comultiplication on A r ( P ) , which is the completely isometrichomomorphism ∆ P : A r ( P ) → A r ( P ) ⊗ min A r ( P ) given by ∆ P ( λ Pp ) = λ Pp ⊗ λ Pp .If moreover P acts on some operator algebra A by δ , then we automatically havethe coaction identity ( δ ⊗ id) ◦ δ = (id ⊗ ∆ P ) ◦ δ . Proof.
The first part follows directly from Theorem 5.4 by taking ϕ to be theidentity on G . For the second part, by definition of coaction it suffices to verify thecoaction identity on each A δp , but this is easily deduced.The last piece of the puzzle is to verify that the coaction constructed in Theorem5.4 is in fact RFD. For this purpose, the following elementary fact will be useful. Lemma 5.6.
Let P and Q be semigroups. Assume that Q is cancellative. Let ϕ : P → Q be a homomorphism and let δ : A r ( P ) → A r ( P ) ⊗ min A r ( Q ) be arepresentation such that δ ( λ Pp ) = λ Pp ⊗ λ Qϕ ( p ) , p ∈ P. Then, for each q ∈ Q the set { a ∈ A r ( P ) : δ ( a ) = a ⊗ λ Qq } coincides with the normclosure of P p ∈ ϕ − ( q ) C λ Pp . Proof.
Fix q ∈ Q and a ∈ A r ( P ) such that δ ( a ) = a ⊗ λ Qq . Let ε >
0. Then thereis a finite linear combination a ′ = P p ∈ P c p λ Pp with c p ∈ C such that k a − a ′ k < ε .Let h ∈ ℓ ( P ) be a unit vector and let s ∈ Q . Note that δ ( a ′ )( h ⊗ e s ) = X p ∈ P c p λ Pp h ⊗ e ϕ ( p ) s and δ ( a )( h ⊗ e s ) = ah ⊗ e qs . Since Q is cancellative, we see that the set { e rs : r ∈ Q } is orthonormal, whenceby applying Pythagoras’ theorem we get (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X p ∈ ϕ − ( q ) c p λ Pp − a h (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X p/ ∈ ϕ − ( q ) c p λ Pp h ⊗ e ϕ ( p ) s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X p ∈ ϕ − ( q ) c p λ Pp − a h ⊗ e qs (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = k δ ( a ′ − a )( h ⊗ e s ) k ≤ ε . Since ε > h ∈ ℓ ( P ) were chosen arbitrarily, this implies that a lies in the norm closure of P p ∈ ϕ − ( q ) C λ Pp . The reverse inclusion is obvious.Let P and Q be semigroups embedded in some groups G and H respectively. Ahomomorphism ϕ : G → H is a ( P, Q ) -map if ϕ ( P ) ⊂ Q and the set ϕ − ( q ) is finitefor every q ∈ Q . It is readily verified that in the presence of a ( P, Q )-map with ϕ ( P ) = Q , if P has FDP then so does Q . Surjective ( P, Q )-maps with ϕ ( P ) = Q are sometimes called controlled maps. These were originally introduced in [43] asgeneralizations of length functions. Various kinds of controlled maps were usedsuccessfully in the literature to establish amenability and nuclearity properties forsemigroup C ∗ -algebras [4, 5, 25, 26].We now arrive at one of the main results of the paper. Recall that a cancellativesemigroup P is said to be left-amenable if it admits a left-invariant mean. Moreprecisely, there is a state µ on ℓ ∞ ( P ) such that for every p ∈ P , and f ∈ ℓ ∞ ( P )we have that µ ( p · f ) = µ ( f ) where ( p · f )( q ) = f ( pq ). For instance, all abeliansemigroups are left-amenable. Theorem 5.7.
Let P and Q be countable semigroups embedded in some groups G and H respectively. Assume that P and Q are independent, and that Q is left-amenable and has FDP. Assume also that there exists a ( P, Q ) -map ϕ : G → H .Then, there is an RFD coaction of Q on A r ( P ) and C ∗ max ( A r ( P )) is RFD. Proof.
By Theorem 5.4, there is a coaction δ of Q on A r ( P ) such that δ ( λ Pp ) = λ Pp ⊗ λ Qϕ ( p ) , p ∈ P. For q ∈ Q we let B q = { a ∈ A r ( P ) : δ ( a ) = a ⊗ λ Qq } be the corresponding spectral subspace for δ . For each q ∈ Q , we see by Lemma5.6 that B q is the norm closure of P p ∈ ϕ − ( q ) C λ Pp .Let ( π F ) be the Exel–Loring approximation for the identity representation of A r ( Q ) given by Proposition 4.1. Thus, for each finite subset F ⊂ Q we see that Q \ (cid:0) ∪ q ∈ F ∪ r ∈R q L r (cid:1) = ( λ Q ) − (ker π F ) . By definition of a coaction, we know that A r ( P ) is the norm closure of X q ∈ Q B q = X q ∈ Q X p ∈ ϕ − ( q ) C λ Pp . INITE-DIMENSIONAL APPROXIMATIONS AND SEMIGROUP COACTIONS 23
We conclude that the quotient Q F of A r ( P ) by the norm closure of the idealgenerated by P q ∈ ( λ Q ) − (ker π F ) B q is spanned by the image in the quotient of X q ∈ F X r ∈R q X p ∈ ϕ − ( L r ) C λ Pp . Since Q has FDP and ϕ is a ( P, Q )-map, we get that Q F is finite-dimensional. Weinfer that C ∗ max ( Q F ) is RFD by virtue of [20, Theorem 5.1]. Thus, we see that δ isan RFD coaction.Finally, since Q is left-amenable and independent, by [46, Theorem 6.42] thereis a character χ : C ∗ r ( Q ) → C . Clearly, for every q ∈ Q we have χ ( λ Qq ) = 1 since λ Qq is an isometry. Thus, by Theorem 4.6 we get that C ∗ max ( A r ( P )) is RFD.We now extract some applications of the previous result. Corollary 5.8.
Let P be a countable, independent semigroup with FDP that isembedded in an abelian group. Then C ∗ max ( A r ( P )) is RFD. Proof.
Every abelian semigroup is left-amenable. Hence, in Theorem 5.7 we maytake ϕ to be the identity, so that C ∗ max ( A r ( P )) is RFD.In paricular, the previous result shows that C ∗ max ( A r ( N d )) is RFD for every d ∈ N (including d = ℵ ). We now study some classical examples of semigroups. Example 5.9 (Left-angled Artin monoids) . Let Γ = (
V, E ) be a finite undirectedgraph, where two vertices are connected by at most one edge and no vertex isconnected to itself. We let A Γ be the group generated by V as formal generatorswhere, for v, w ∈ W , we have vw = wv if if there is an edge between v in w . Wethen define the unital semigroup A +Γ as the semigroup generated by the identity and { v : v ∈ V } . It is known that right-angled Artin groups are quasi-lattice ordered bytheir monoids (see [25]), so we get that A +Γ is right LCM, and is hence independent.Next, consider the group Z V . For each v ∈ V , let e v ∈ Z V denote the indicatorfunction of { v } . There is a surjective group homomorphism ϕ : A Γ → Z V such that ϕ ( v ) = e v . Viewing N V as a subsemigroup of Z V , it is clear that ϕ ( A +Γ ) = N V and ϕ − ( f ) is finite for every f ∈ N V . Thus, ϕ is a ( A +Γ , N V )-map. As a consequenceof Theorem 5.7, we get that C ∗ max ( A r ( A +Γ )) is RFD. Example 5.10 (Braid monoids) . For n ≥ B n be the group generated byelements σ , ..., σ n − subject to the relations σ i σ i +1 σ i = σ i +1 σ i σ i +1 for 1 ≤ i ≤ n − σ i σ j = σ j σ i for | i − j | ≥ . We let B + n be the unital semigroup generated by σ , ..., σ n − in B n .For each n ≥
3, it follows from [14, Lemma 2.1] that B + n is right LCM, andhence is independent (see also the discussion preceding [46, Lemma 6.33]). There isa surjective group homomorphism ϕ : B n → Z such that ϕ ( σ i ) = 1 for every i . It isreadily seen that ϕ is a ( B + n , N )-map. By Theorem 5.7, we get that C ∗ max ( A r ( B + n ))is RFD. Operator algebras of functions.
Let Ω ⊂ C n be a subset and let T ⊂ C denote the unit circle. Assume that Ω is balanced , so that ζz ∈ Ω for every z ∈ Ωand every ζ ∈ T . In this case, given a function f : Ω → C and ζ ∈ T , we define f ζ : Ω → C to be the function f ζ ( z ) = f ( ζz ) , z ∈ Ω . Let A be an operator algebra consisting of functions on Ω. We say that A is T -symmetric if for each ζ ∈ T , the map f f ζ , f ∈ A defines a completely isometric isomorphism Φ ζ : A → A . It would certainly benatural here to impose some form of continuity on the map ζ Φ ζ . Fortunately,in our cases of interest this turns out to be automatic. Lemma 5.11.
Let Ω ⊂ C n be a balanced subset and let A be a T -symmetricoperator algebra of functions on Ω . Assume that there is a collection of homogeneouspolynomials spanning a dense subset of A . Then, the map ζ Φ ζ ( f ) , ζ ∈ T is norm continuous for every f ∈ A . Proof.
Given f ∈ A , we define a function δ ( f ) : T → A as δ ( f )( ζ ) = Φ ζ ( f ) , ζ ∈ T . For a homogeneous polynomial f ∈ A of degree n , we see that δ ( f )( ζ ) = ζ n f, ζ ∈ T so in particular δ ( f ) is continuous. A standard ε -argument using our densityassumption shows that in fact δ ( a ) is continuous for every a ∈ A .Next, we show that T –symmetric operator algebras admit RFD coactions by N . Theorem 5.12.
Let Ω ⊂ C n be a balanced subset and let A be a T -symmetricoperator algebra of functions on Ω . Assume that there is a collection of homogeneouspolynomials spanning a dense subset of A . Then, A admits an RFD coaction by N .In particular, C ∗ max ( A ) is RFD. Proof.
Given f ∈ A , we define a function δ ( f ) : T → A as δ ( f )( ζ ) = Φ ζ ( f ) , ζ ∈ T . By Lemma 5.11, we see that δ ( f ) ∈ C( T ; A ) for every f ∈ A . Hence, we obtain acompletely isometric homomorphism δ : A → C( T ; A ). Next, it is a consequenceof [51, Proposition 12.5] and of the injectivity of the minimal tensor product thatthere is a completely isometric isomorphism Θ : A ⊗ min C( T ) → C( T ; A ) such thatΘ( a ⊗ g )( ζ ) = g ( ζ ) a, ζ ∈ T for every a ∈ A and g ∈ C( T ). If f ∈ A is a homogeneous polynomial of degree n ,then we see that Θ − ( δ ( f )) = f ⊗ ζ n . We conclude that (Θ − ◦ δ )( A ) ⊂ A ⊗ min A( D ), where A( D ) denotes the discalgebra. INITE-DIMENSIONAL APPROXIMATIONS AND SEMIGROUP COACTIONS 25
Recall now that there is a surjective ∗ -representation τ : C ∗ r ( N ) → C( T ) suchthat τ ( λ N n ) = ζ n , n ∈ N and whose restriction to A r ( N ) is completely isometric. In particular, this clearlyimplies that there is a character χ : C ∗ r ( N ) → C with χ ( λ N n ) = 1 for every n ∈ N .Putting ρ = ( τ | A r ( N ) ) − , we obtain a completely isometric isomorphism ρ : A( D ) →A r ( N ) such that ρ ( z n ) = λ N n , n ∈ N . Define δ = (id ⊗ ρ ) ◦ Θ − ◦ δ : A → A ⊗ min A r ( N ) , which is a completely isometric representation. We claim that δ is an RFD coactionof N on A .For n ∈ N , we let A n = { a ∈ A : δ ( a ) = a ⊗ λ N n } be the corresponding spectral subspace. By construction, we see that a ∈ A n ifand only if Φ ζ ( a ) = δ ( a )( ζ ) = ζ n a, ζ ∈ T . Hence, A n is the space of all homogeneous polynomials of degree n in A , and inparticular is finite-dimensional. By assumption, we must have that P n ∈ N A n isnorm dense in A . This shows that δ is a coaction of N on A . It remains to showthat this coaction is in fact RFD.Let ( π F ) be the Exel–Loring approximation for the identity representation of A r ( P ) from Proposition 4.1. Given a finite subset F ⊂ N and n ∈ N , we see fromProposition 4.1 that λ Nn ∈ ker π F if n > max F . Thus, the quotient Q F of A r ( N )by the closed ideal generated by X n ∈ λ − (ker π F ) A n has dimension at most that of X n ≤ max F A n and in particular is finite-dimensional. Hence, C ∗ max ( Q F ) is RFD by [20, Theorem5.1]. We conclude that δ is indeed an RFD coaction of N on A . Finally, invokingTheorem 4.6 we get that C ∗ max ( A ) is RFD.Many concrete examples of operator algebras can now be analyzed in light ofthe previous result. Corollary 5.13.
Let Ω ⊂ C n be a balanced bounded subset and let A(Ω) denote theclosure of the polynomials inside the continuous functions
C(Ω) . Then, C ∗ max (A(Ω)) is RFD. Proof.
In view of Theorem 5.12, it suffices to show that A(Ω) is a T -symmetricalgebra. For f ∈ A(Ω) and ζ ∈ T , we see that k f ζ k = max z ∈ Ω | f ( ζz ) | = max w ∈ Ω | f ( w ) | = k f k , f ∈ A(Ω) . This implies that the map f f ζ defines an isometric isomorphism Φ ζ : A(Ω) → A(Ω). The fact that Φ ζ is in fact completely isometric follows then from [51,Theorem 3.9]. By taking Ω above to be the open unit polydisc or the open unit ball, we concludefrom Corollary 5.13 that the maximal C ∗ -covers of the polydisc algebra and of theball algebra are necessarily RFD. This answers a question left open in [20].By definition, the algebra A(Ω) considered in Corollary 5.13 is a uniform algebra ,as it embeds into the commutative C ∗ -algebra C(Ω). Below, we deal with operatoralgebras of functions on Ω which do not necessarily admit such an embedding.Let n be a positive integer. Let Γ denote the set of (unitary equivalence classesof) commuting n -tuples of contractions on some separable Hilbert space. Let p ∈ C [ z , . . . , z n ] be a polynomial in n variables. The quantitysup ( T ,...,T n ) ∈ Γ k p ( T , . . . , T n ) k is easily seen to be finite and to define a norm on C [ z , . . . , z n ]. The correspondingcompletion of C [ z , . . . , z n ] is an operator algebra [51, Chapter 18], usually referredto as the Schur–Agler class [1]. We denote it by S n . It follows from the classicalvon Neumann and Ando inequalities that S ∼ = A( D ) and S ∼ = A( D ). However,for bigger n the Schur–Agler class differs from the polydisc algebra. Nevertheless,we can still prove the following. Corollary 5.14.
Let n be a positive integer and let S n denote the correspondingSchur–Agler class. Then, C ∗ max ( S n ) is RFD. Proof.
By definition we see that | p ( z ) | ≤ k p k S n , z ∈ D n for every polynomial p in n variables, so that there is a contractive inclusion S n ֒ → A( D n ). Next, we note that if ( T , . . . , T n ) is a commuting n -tuple of contractions,then so is ( ζT , ζT , . . . , ζT n ) for each ζ ∈ T . Therefore, there is a completelyisometric isomorphism Φ ζ : S n → S n such thatΦ ζ ( f ) = f ζ , f ∈ S n . Hence, we conclude that S n is a T –symmetric operator algebra of functions on D n ,so that C ∗ max ( S n ) is RFD by Theorem 5.12.For our next application, we will make use of standard terminology from thetheory of reproducing kernel Hilbert spaces; details can be found in [2] and [40].Let Ω ⊂ C n be a balanced open connected subset containing the origin. Let K : Ω × Ω → C be a kernel function normalized at the origin, so that K ( z,
0) = 1for every z ∈ Ω. We say that K is T -invariant if we have that(i) K is holomorphic in the first variable,(ii) for every z, w ∈ Ω the map ζ K ( ζz, w ) , ζ ∈ T is continuous,(iii) for every z, w ∈ Ω and ζ ∈ T we have that K ( z, w ) = K ( ζz, ζw ).Many important classical kernels satisfy these properties, such as the Drury–Arvesonkernel and the Dirichlet kernel; see [40, Example 6.1].Let H be the reproducing kernel Hilbert space corresponding to K and letMult( H ) denote its multiplier algebra. By our assumption on K , we see that H INITE-DIMENSIONAL APPROXIMATIONS AND SEMIGROUP COACTIONS 27 consists of holomorphic functions on Ω. Furthermore, because 1 ∈ H we inferMult( H ) ⊂ H . For each ζ ∈ T , the map Γ ζ : H → H defined as(Γ ζ ( f ))( z ) = f ( ζz ) , f ∈ H , z ∈ Ωis easily checked to be a strongly continuous unitary path (see [40, Section 6]). Foreach ζ ∈ T and ϕ ∈ Mult( H ), we note thatΓ ∗ ζ M ϕ Γ ζ = M Γ ζ ( ϕ ) . Hence, we obtain a completely isometric isomorphism Θ ζ : Mult( H ) → Mult( H )defined as Θ ζ ( M ϕ ) = M Γ ζ ( ϕ ) , ϕ ∈ Mult( H ) . For each n ∈ N , we let A( H ) n ⊂ Mult( H ) be the subspace consisting of thosemultipliers ϕ such that Θ ζ ( M ϕ ) = ζ n M ϕ for every ζ ∈ T . As explained in [40,Example 6.1 (i)], for n ≥ H ) n consists of homogeneous polynomialsof degree n , while A( H ) n = { } for n <
0. We define A( H ) to be the norm closureof P ∞ n =0 A( H ) n inside of Mult( H ).We now arrive at another application of Theorem 5.12. Corollary 5.15.
Let Ω ⊂ C n be a balanced open connected set containing the ori-gin. Let K be a T -invariant kernel on Ω and let H be the corresponding reproducingkernel Hilbert space. Then, C ∗ max (A( H )) is RFD. Proof.
We make a preliminary observation. Let ϕ ∈ A( H ). By definition of A( H ),there is a sequence ( p n ) of polynomials converging to ϕ in the norm topology ofMult( H ). Since the multiplier norm always dominates the supremum norm over Ω,we conclude that the sequence ( p n ) converges to ϕ uniformly on Ω, and thus ϕ isholomorphic on Ω. Hence, A( H ) is an operator algebra of holomorphic functionson Ω.In view of Theorem 5.12, it suffices to show that A( H ) is a T -symmetric algebra.For each ζ ∈ T , we let Φ ζ = Θ ζ | A( H ) . It is clear that Φ ζ (A( H ) n ) ⊂ A( H ) n forevery n ≥
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Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada
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