Residual finite dimensionality and representations of amenable operator algebras
aa r X i v : . [ m a t h . OA ] A ug RESIDUAL FINITE DIMENSIONALITY AND REPRESENTATIONS OFAMENABLE OPERATOR ALGEBRAS
RAPHA¨EL CLOU ˆATRE AND LAURENT W. MARCOUX
Abstract.
We consider a version of a famous open problem formulated by Kadison, ask-ing whether bounded representations of operator algebras are automatically completelybounded. We investigate this question in the context of amenable operator algebras, andwe provide an affirmative answer for representations whose range is residually finite-dimen-sional. Furthermore, we show that weak- ∗ closed, amenable, residually finite-dimensionaloperator algebras are similar to C ∗ -algebras, and in particular have the property that alltheir bounded representations are completely bounded. We prove our results for operatoralgebras having the so-called total reduction property, which is known to be weaker thanamenability. Introduction
Let B ( H ) denote the C ∗ -algebra of bounded linear operators on some complex Hilbertspace H . Given a Banach algebra A , we shall refer to a continuous algebra homomorphism θ : A → B ( H ) as a representation of A . In particular, representations of C ∗ -algebras are notassumed to be self-adjoint. One of the most intriguing questions in the theory of C ∗ -algebrasis the following, which stems from a 1955 paper of R.V. Kadison [13]. Kadison’s similarity problem.
Let A be a C ∗ -algebra and let θ : A → B ( H ) be arepresentation. Does there exist an invertible operator X ∈ B ( H ) so that the representation θ X : A → B ( H ) defined by θ X ( a ) = X − θ ( a ) X, a ∈ A is a ∗ -representation?We say that A has Kadison’s similarity property if the problem above has an affirmativeanswer. It is still unknown to this day whether every C ∗ -algebra has this property, althoughsome deep partial results have emerged throughout the years. For instance, E. Christensen [6]has solved the problem in the affirmative for amenable C ∗ -algebras (we note here that dueto sophisticated results of A. Connes [8] and U. Haagerup [10], the classes of amenable andnuclear C ∗ -algebras are known to coincide). It was shown by Haagerup [11] that Kadison’ssimilarity problem has an affirmative answer for every C ∗ -algebra, provided that one restrictsone’s attention to only those representations which admit a finite cyclic set of vectors. More-over, it is known [11] that a representation of a C ∗ -algebra is similar to a ∗ -representationif and only if it is completely bounded. This allows one to reformulate Kadison’s similarityproblem so that it makes sense in a non self-adjoint context. The following problem appearedin [17]. The generalised similarity problem.
Let A be an operator algebra and let θ : A → B ( H )be a representation. Is θ necessarily completely bounded? The first author was partially supported by an FQRNT postdoctoral fellowship and an NSERC DiscoveryGrant. The second author was partially supported by an NSERC Discovery grant. R. Clouˆatre and L.W. Marcoux
Following [17], we say that the A has the SP property if the problem above has anaffirmative answer. In producing an example of a polynomially bounded operator which isnot similar to a contraction – thereby answering a long-standing open question (i.e. theHalmos Problem) as to the existence of such an operator – Pisier demonstrated that theclassical disc algebra is an example of an operator algebra without the SP property [16].On the other hand, a straightforward verification shows that if
A ⊂ B ( H ) has the SPproperty, then so does X A X − for any invertible operator X ∈ B ( H ). In particular, itfollows from Christensen’s result mentioned above that any operator algebra that is similarto an amenable C ∗ -algebra has the SP property.In the 1980’s there arose the question of determining which operator algebras are similar toamenable C ∗ -algebras. Since amenability is preserved under Banach algebra isomorphisms– of which similarity of operator algebras is an example – an obvious restriction on suchoperator algebras is that they be amenable. It was conjectured by a number of people in theBanach algebra community that this was in fact the only obstruction, and that any amenableoperator algebra was similar to a C ∗ -algebra. This conjecture was verified in some specialcases. Indeed, it was shown to hold for uniform algebras by M.V. ˇSe˘ınberg [20], for abelianamenable subalgebras of finite von Neumann algebras by Y. Choi [4], and very recently forarbitrary abelian amenable operator algebras by the second author and A. Popov [14].However, other recent developments have shown the conjecture to be incorrect in general,and an example of a non-abelian, non-separable, amenable operator algebra which fails tobe similar to a C ∗ -algebra was constructed in [5] by Y. Choi, I. Farah and N. Ozawa. It isnot currently known whether a separable, amenable operator algebra must always be similarto a C ∗ -algebra. Nevertheless, this counterexample shows that a positive solution to thegeneralised similarity problem for amenable operator algebras cannot be achieved throughthis approach by relying on Christensen’s result. Interestingly, the counterexample doeshave the SP property, as shown in the companion paper [7]. Thus, the question of whetherevery amenable operator algebra has the SP property remains unanswered and it motivatedmuch of our efforts. In fact, we will be interested in a slightly more general question.An illuminating insight into Kadison’s similarity problem and the generalised similarityproblem was offered by J.A. Gifford in his PhD thesis [9]. Therein, he obtained a completecharacterization of the C ∗ -algebras having Kadison’s similarity property as those having aremarkably well behaved lattice of invariant subspaces, a feature he called the total reductionproperty (the precise definition is given in Section 2). He also observed that the totalreduction property is strictly weaker than amenability (we point out that the result of [14]mentioned above was in fact proved under this weaker assumption).We can now state the basic question which we wish to address in this paper. Question.
Let A be a norm-closed operator algebra with the total reduction property. Isevery representation θ : A → B ( H ) completely bounded?The reader will notice that this question is a special case of the generalised similarityproblem. Based on the discussion above, we believe it to be a meaningful step towards acomplete understanding of the latter.We now describe the organization of the paper, and state our main results. Section 2deals with preliminaries: we give precise definitions of the concepts we require throughout,gather relevant results from the literature and prove some basic facts in preparation forlater work. In Section 3, we explain how the general problem we are trying to solve can bereduced to one involving matrix algebras, at least in the amenable case. Consequently, intrying to determine whether every amenable operator algebra has the SP property, it is no FD and representations of amenable operator algebras loss of generality to focus on representations whose domains are residually finite-dimensionalalgebras. Motivated by this observation, in Section 4 we examine the structure of subalgebrasof products of matrix algebras that possess the total reduction property. This informationis then leveraged to prove our first main result (see Theorem 4.5 and Corollary 4.6).1.1. Theorem.
Let ( k ( λ )) λ be a net of positive integers and suppose that A ⊂ Q λ M k ( λ ) is a subalgebra with the total reduction property. If A is closed in the weak- ∗ topology, then A is similar to a C ∗ -algebra and in particular it has the SP property. Although the condition of being weak- ∗ closed restricts the range of applicability of theprevious result, we view it as noteworthy partial step towards clarifying the general situation.Finally, Section 5 is devoted to the study of representations whose range is residually finite-dimensional, and our second main result is proved therein (see Corollary 5.5). Roughlyspeaking, it says that one has uniform control on the completely bounded norm of finite-dimensional representations in the presence of the total reduction property.1.2. Theorem.
Let ( k ( λ )) λ be a net of positive integers, let A be an operator algebrawith the total reduction property and let θ : A → Q λ M k ( λ ) be a representation. Then, θ iscompletely bounded. The previous result fails without the total reduction property (Example 5.7).2.
Preliminaries and background material
Operator algebras and completely bounded maps.
Throughout, by an operatoralgebra we mean a subalgebra of some B ( H ) which is closed in the norm topology.If A ⊂ B ( H ) is an operator algebra, then for each integer n ≥ M n ( A )inherits a norm when viewed as a subalgebra of the C ∗ -algebra B ( H ( n ) ). If B is anotheroperator algebra, then a linear map ϕ : A → B induces a sequence of maps ϕ ( n ) : M n ( A ) → M n ( B ) , n ≥ ϕ ( n ) ([ a i,j ]) = [ ϕ ( a i,j )]for all [ a i,j ] ∈ M n ( A ). The map ϕ is completely bounded if the quantity k ϕ k cb = sup n ≥ k ϕ ( n ) k is finite. The map ϕ is said to be completely contractive (respectively, completely isometric )if each ϕ ( n ) is contractive (respectively, isometric). We refer the reader to [15] for a thoroughexposition of the theory of completely bounded maps and of operator algebras.2.2. Amenability and the total reduction property.
We recall the notion of amenabil-ity of a Banach algebra, which was introduced by Johnson in [12]. Let A be a Banach algebraand let X be a Banach A -bimodule. If X ∗ denotes the dual space of X , then X ∗ also carriesthe structure of an A -bimodule under the dual actions defined by( ax ∗ )( x ) = x ∗ ( xa ) and ( x ∗ a )( x ) = x ∗ ( ax )for all a ∈ A , x ∗ ∈ X ∗ and x ∈ X . When it is equipped with this precise module actioninherited from that of A on X , we say that X ∗ is a dual Banach A -bimodule .A derivation δ : A → X is a continuous linear map which satisfies δ ( ab ) = δ ( a ) b + aδ ( b ) R. Clouˆatre and L.W. Marcoux for all a, b ∈ A . The derivation is inner if there exists a fixed element z ∈ X so that δ ( a ) = az − za for all a ∈ A . Finally, we say that A is amenable if every derivation of A into a dual Banach A -bimodule is inner.As mentioned in the introduction, we will be mostly concerned with a notion, introducedin the thesis of J.A. Gifford [9], that is weaker than amenability. An operator algebra A is said to have the total reduction property if, whenever θ : A → B ( H ) is a representationand M ⊂ H is a closed θ ( A )-invariant subspace, there exists another closed θ ( A )-invariantsubspace N which is a topological complement of M , in the sense that H = M + N and M ∩ N = { } . Equivalently, any closed θ ( A )-invariant subspace is the range of some boundedidempotent lying in the commutant θ ( A ) ′ .More precise information about these idempotents is available [9, Proposition 2.2.13].Indeed, for each t ≥ C ( t ) with the property that whenever θ : A → B ( H ) is a representation with k θ k ≤ t and M ⊂ H is a closed θ ( A )-invariantsubspace, there is an idempotent E ∈ θ ( A ) ′ with E H = M and k E k ≤ C ( t ). For each t ≥ κ A ( t ) the minimum of all positive constants C ( t ) satisfying this condition.Clearly, κ A is an increasing function. We single out another one of its basic properties.2.3. Lemma.
Let A be an operator algebra with the total reduction property and let ̺ be arepresentation of A such that ̺ ( A ) is closed. Then, ̺ ( A ) has the total reduction property.Moreover, if θ is a representation of ̺ ( A ) , then κ ̺ ( A ) ( k θ k ) ≤ κ A ( k θ ◦ ̺ k ) . In particular, for every t > we see that κ ̺ ( A ) ( t ) ≤ κ A ( t k ̺ k ) . Proof.
The fact that ̺ ( A ) has the total reduction property is simply [9, Proposition 3.3.1].Next, let θ : ̺ ( A ) → B ( H ) be a representation and let M ⊂ H be a closed θ ( ̺ ( A ))-invariant subspace. Since A has the total reduction property, there is an idempotent E ∈ θ ( ̺ ( A )) ′ such that E H = M and k E k ≤ κ A ( k θ ◦ ̺ k ). We conclude that κ ̺ ( A ) ( k θ k ) ≤ κ A ( k θ ◦ ̺ k ) ≤ κ A ( k θ kk ̺ k ) . (cid:3) We mention in passing that if, in the above result, we do not assume that ̺ ( A ) isclosed, then a similar argument shows that ̺ ( A ) has the total reduction property, andthat κ ̺ ( A ) ( t ) ≤ κ A ( t k ̺ k ) for all t ≥ H is an infinite-dimensional Hilbert space, then B ( H ) has the total reduction property [9, Corollary 2.4.7 ]but it is not amenable since it is not nuclear [22],[8].Before proceeding, we gather here several facts about operator algebras with the totalreduction property that we require numerous times in the sequel. First, we consider ideals. FD and representations of amenable operator algebras Theorem.
Let A be an operator algebra with the total reduction property and let J ⊂ A be a closed two-sided ideal. Then, J has the total reduction property. Moreover, J admits a bounded approximate identity: there exists a bounded net ( e i ) i in J such that lim i k ae i − a k = lim i k e i a − a k = 0 for every a ∈ J .Proof. This follows from [9, Propositions 3.2.7 and 3.3.3]. (cid:3)
Next, we deal with a minor technical detail. Recall that an algebra
A ⊂ B ( H ) acts non-degenerately if AH = H . While operator algebras with the total reduction property do notnecessarily act non-degenerately, they nearly do so.2.5. Lemma.
Let
A ⊂ B ( H ) be an operator algebra with the total reduction property.Then, there exists an invertible operator X ∈ B ( H ) with k X k = k X − k ≤ κ A (1) such that X A X − = A ⊕ { } according to some orthogonal decomposition H = H ⊕ H ⊥ . Moreover, A ⊂ B ( H ) is anon-degenerately acting subalgebra with the total reduction property.Proof. Let M = AH which is a closed A -invariant subspace. There is an idempotent E ∈ A ′ such that E H = M and k E k ≤ κ A (1) . Then, it is well-known that there is an invertibleoperator X ∈ B ( H ) with k X k = k X − k ≤ k E k ≤ κ A (1)and such that XEX − is a self-adjoint projection. The space XM is then reducing for X A X − . Put H = XM and A = ( X A X − ) | H . We note that X A X − H ⊂ XM = H so that X A X − = A ⊕ { } according to the decomposition H = H ⊕ H ⊥ . Thus, A has thetotal reduction property by Lemma 2.3. Moreover, A has a bounded approximate identity( e i ) i by Theorem 2.4. Then, if ξ ∈ H and a ∈ A we have that aξ = lim i e i aξ ∈ A M whence M = A M . Thus A H = X A M = XM = H which shows that A acts non-degenerately on H . (cid:3) We can now extract more information about ideals.2.6.
Theorem.
Let
A ⊂ B ( H ) be a weak- ∗ closed operator algebra with the total reductionproperty and let J ⊂ A be a weak- ∗ closed two-sided ideal. Then, there is a central idempotent e ∈ J ∩ A ′ such that J = e A and k e k ≤ κ A (1) .Proof. The existence of an idempotent e ∈ J ∩ A ′ such that J = e A follows from Lemma2.5 and [9, Proposition 3.2.2 and Corollary 3.1.5]. To get the announced norm estimate,we proceed as follows. Consider the closed subspace M = e H . Since e ∈ A ′ , we see that M is A -invariant. Hence, there is another idempotent f ∈ A ′ with k f k ≤ κ A (1) such that M = f H . Note that ef = f e because e ∈ J ⊆ A . Commuting idempotents with identicalranges must be equal, so that indeed k e k ≤ κ A (1). (cid:3) R. Clouˆatre and L.W. Marcoux
As a consequence of the previous theorem, we see that if
A ⊂ B ( H ) is a weak- ∗ closedoperator algebra with the total reduction property, then A has a unit u with k u k ≤ κ A (1).In particular, if J ⊂ A is a weak- ∗ closed two-sided ideal, then we have a topological directsum decomposition A = J + ( u − e ) A where J = e A .We now state the result mentioned in the introduction relating the total reduction propertyto Kadison’s similarity property.2.7. Theorem.
Let A be a C ∗ -algebra. (1) If A has Kadison’s similarity property, then A has the total reduction property. (2) If A has the total reduction property, then A has Kadison’s similarity property. Moreprecisely, if θ : A → B ( H ) is a representation, then there is an invertible operator X ∈ B ( H ) such that a Xθ ( a ) X − , a ∈ A is a ∗ -homomorphism of A and k X kk X − k ≤ κ A ( k θ k ) . In particular, we see that k θ k cb ≤ κ A ( k θ k ) . Proof.
This is a combination of Lemmas 2.4.1, 2.4.3 and Proposition 2.4.4 in [9]. (cid:3)
We close this section with one of the main results of [9], which states that if an operatoralgebra consisting of compact operators has the total reduction property, then it is similarto a C ∗ -algebra. We require a refined form of a special case of this theorem, which we provebelow.2.8. Theorem.
Let H be a finite-dimensional Hilbert space and suppose that A ⊂ B ( H ) is an operator algebra with the total reduction property. Then, there exists an invertibleoperator X ∈ B ( H ) with the property that X A X − is a C ∗ -algebra, and such that k X kk X − k ≤ (1 + κ A (1)) κ A (1)) κ A (1 + 2 κ A (1)) . Proof.
First, we note that by virtue of Lemma 2.5, we may assume without loss of general-ity that A acts non-degenerately upon conjugating with an invertible operator V ∈ B ( H )satisfying k V k = k V − k ≤ κ A (1) . It is this potential initial conjugation that accounts for the first term of (1 + κ A (1)) on theright-hand side of the inequality appearing in the statement.The proof then consists of a combination of results scattered throughout [9]. We see that A consists of compact operators on H , and so by [9, Lemma 4.3.12] there exist finitely manyminimal idempotents E , . . . , E n ∈ A ′′ ∩ A ′ such that A = E A E + . . . + E n A E n and P nk =1 E k = I. Necessarily we have that these idempotents are pairwise orthogonal andthat for each k the algebra ( E k A E k ) ′′ = E k A ′′ E k contains no proper central idempotent. FD and representations of amenable operator algebras By [9, Lemmas 1.0.3 and 3.2.3], we know that there exists an invertible operator Y ∈ B ( H )such that P k = Y E k Y − is a self-adjoint projection for every 1 ≤ k ≤ n , and moreover k Y kk Y − k ≤ κ A (1) . Note that P k Y A Y − P k = Y E k A E k Y − for each 1 ≤ k ≤ n , so we find Y A Y − = Y E A E Y − + . . . + Y E n A E n Y − = n M k =1 P k Y A Y − P k . Moreover, we see that ( P k Y A Y − P k ) ′′ = Y ( E k A E k ) ′′ Y − contains no proper central idempotent for each 1 ≤ k ≤ n . Using [9, Lemma 4.3.11], for each k we find an invertible operator Z k such that the algebra Z k P k Y A Y − P k Z − k is self-adjointand k Z k k = k Z − k k ≤ √ κ P k Y A Y − P k (1) . Now, by Lemma 2.3 we see that κ P k Y A Y − P k (1) ≤ κ A ( k Y kk Y − k ) ≤ κ A (1 + 2 κ A (1))so that k Z k k = k Z − k k ≤ √ κ A (1 + 2 κ A (1)) . Since the orthogonal projections P , . . . , P n are pairwise orthogonal and satisfy P nk =1 P k = I ,if we set Z = L nk =1 P k Z k P k then Z is invertible with Z − = L nk =1 P k Z − k P k . Moreover, wesee that k Z kk Z − k ≤ κ A (1 + 2 κ A (1)) . Finally, by setting X = ZY we obtain X A X − = n M k =1 P k Z k P k Y A Y − P k Z − k P k which is a C ∗ -algebra, and k X kk X − k ≤ (1 + 2 κ A (1))128 κ A (1 + 2 κ A (1)) . (cid:3) A reduction to residually finite-dimensional operator algebras
This section is meant as motivation for the rest of the paper. The goal here is to showthat for amenable operator algebras, the generalized similarity problem can be transplantedto the concrete setting of products of matrix algebras without loss of generality. We willaccomplish this by considering cones of operator algebras. Recall that if A is a Banachalgebra, then the cone of A is the Banach algebra C ( A ) = { f : [0 , → A : f is continuous and f (0) = 0 } where if f ∈ C ( A ) then k f k = sup t ∈ [0 , k f ( t ) k . R. Clouˆatre and L.W. Marcoux
Alternatively, we see that C ( A ) = C ((0 , ⊗ A , equipped with the injective tensor norm.Interestingly, taking the cone of an algebra preserves amenability [19, Exercise 2.3.6]. Weneed the following routine fact.3.1. Proposition.
Let A and B be operator algebras and θ : A → B be a representation.Then θ induces a representation Φ θ : C ( A ) → C ( B ) defined by the formula (Φ θ f )( t ) = θ ( f ( t )) , t ∈ [0 , . Furthermore, the map θ is completely bounded if and only if Φ θ is completely bounded, andwe have k θ k cb = k Φ θ k cb .Proof. Let n ∈ N . We see that if f ∈ M n ( C ( A )), then k (Φ ( n ) θ f )( t ) k M n ( C ( B )) ≤ k θ ( n ) kk f ( t ) k M n ( A ) for every t ∈ [0 , k Φ ( n ) θ k ≤ k θ ( n ) k . For the reverse inequality, let a ∈ M n ( A ).Define f a ∈ M n ( C ( A )) as f a ( t ) = ta, ≤ t ≤ . Then, we see that k f a k M n ( C ( A )) = k a k M n ( A ) andΦ ( n ) θ ( f a ) = f θ ( n ) ( a ) . Hence k θ ( n ) ( a ) k M n ( B ) = k f θ ( n ) ( a ) k M n ( C ( B )) = k Φ ( n ) θ ( f a ) k M n ( C ( B )) ≤ k Φ ( n ) θ kk f a k M n ( C ( A ) = k Φ ( n ) θ kk a k M n ( A ) which shows that k θ ( n ) k ≤ k Φ ( n ) θ k . (cid:3) We thus see that to verify whether an operator algebra A has the SP property, it issufficient to check that the cone C ( A ) has it.Before we proceed further, we introduce some notation which will be used throughout theremainder of the paper. Let Λ = ∅ be a set and let k : Λ → N be a function. We associatewith k the following C ∗ -algebra: M k = Y λ M k ( λ ) = { ( a λ ) λ : a λ ∈ M k ( λ ) for all λ ∈ Λ and sup λ k a λ k < ∞} . Let us also define for each λ ∈ Λ the component map q k λ : M k → M k ( λ ) via q k λ (( a α ) α ) = a λ . When Λ is a directed set, we shall use the notation L k instead of M k in order to emphasizethis distinction. The direction on Λ allows us to define the closed, two-sided ideal J k = { ( a λ ) λ ∈ L k : lim λ k a λ k = 0 } . It is easily verified that J k is nuclear, and hence amenable [10]. We may now construct thequotient C ∗ -algebra Q k = L k / J k . We let π k : L k → Q k FD and representations of amenable operator algebras denote the quotient map. The main observation of this section is the following, which isa combination of classical facts. It is a direct adaptation of the discussion found after [5,Theorem 1]. We present the proof here for the convenience of the reader.3.2. Theorem.
The following statements are equivalent. (i)
Every amenable operator algebra has the SP property. (ii)
Let k : Λ → N and k ′ : Λ ′ → N be nets, let D ⊂ L k be an amenable operator algebra,and let θ : D → Q k ′ be a representation. Then θ is completely bounded.Proof. We need only prove that (ii) implies (i). Assume therefore that (ii) holds and that
A ⊂ B ( H ) is an amenable operator algebra. Let θ : A → B ( H θ ) be a representation. Weproceed to show that θ is completely bounded.For this purpose, let A = C ∗ ( A ) ⊂ B ( H ). Then, C ( A ) ⊂ C ( A ). It is easy to see that C ( A ) is homotopic to zero, so that C ( A ) is quasidiagonal by [23, Theorem 5] (alternatively,see [3, Corollary 7.3.7] for the precise statement we need). In particular, by a straightforwardadaptation of [3, Exercise 7.1.3] we may view C ( A ) as a C ∗ -subalgebra of Q k for some net k : Λ → N . We conclude that C ( A ) ⊂ Q k . An identical argument shows that C ( B ) ⊂ Q k ′ for some net k ′ : Λ ′ → N , where B = θ ( A ).By Proposition 3.1, there is a representationΦ θ : C ( A ) → C ( B )which is completely bounded if and only if θ is. In turn, it is easily verified that Φ θ iscompletely bounded if and only ifΦ θ ◦ π k : π − k ( C ( A )) → C ( B ) ⊂ Q k ′ is completely bounded. We know that the cone C ( A ) is amenable. If we let D = π − k ( C ( A )),then we may conclude from [19, Theorem 2.3.10] that D is an amenable subalgebra of L k .Hence (ii) implies that Φ θ ◦ π k , and thus θ , is completely bounded. (cid:3) We remark here that it is plausible that a version of this theorem holds for algebraswhich merely have the total reduction property. However, a direct adaptation of the proofwould require some technology which is unavailable at present, and as such we postpone thisinteresting issue to future work.We also emphasize that Theorem 3.2 shows that from the point of view of attempting tosolve the generalised similarity problem for amenable operator algebras, it is very meaningfulto study amenable subalgebras of products of matrix algebras. We undertake this task forthe larger class of operator algebras with the total reduction property, and accordingly weintroduce the following convenient terminology.A subalgebra
A ⊂ B ( H ) is said to be residually finite-dimensional if there exists a family offinite-dimensional Hilbert spaces H λ and a family of completely contractive representations ̺ λ : A → B ( H λ )such that the map a M λ ̺ λ ( a ) , a ∈ A is completely isometric. We mention that this definition is consistent with common usage ofthe term within the realm of C ∗ -algebras. R. Clouˆatre and L.W. Marcoux
It is clear that for any function k : Λ → N , the algebra M k considered above is residuallyfinite-dimensional. Furthermore, subalgebras of residually finite-dimensional operator alge-bras are residually finite-dimensional as well. We now exhibit a less trivial example, whichwe will revisit later in the paper.3.3. Example.
Let H be a Hilbert space and let ( P λ ) λ be a net of finite rank projectionsincreasing strongly to I . Let T ⊂ B ( H ) denote the collection of operators which are trian-gular with respect to these projections, that is T ∈ T if and only if P λ T P λ = T P λ for every λ . A standard verification shows that T is a weak- ∗ closed algebra. For each λ , the map ̺ λ : T → B ( P λ H )defined by ̺ λ ( T ) = P λ T P λ , T ∈ T is a completely contractive homomorphism. Moreover, since ( P λ ) λ increases to I , it is easyto see that L λ ̺ λ is completely isometric, so that T is residually finite-dimensional.For future use, we also point that each ̺ λ is clearly weak- ∗ continuous. By a standardapplication of the Krein-Smulian theorem [1, Theorem A.2.5], we see that L λ ̺ λ is a com-pletely isometric weak- ∗ homeomorphic algebra homomorphism. (cid:3) Interestingly, residual finite dimensionality of an operator algebra
A ⊂ B ( H ) is not equiv-alent to that of C ∗ ( A ), as the following examples show.3.4. Example.
Let H be an infinite-dimensional, separable Hilbert space with orthonor-mal basis { e m } ∞ m =1 . For each n ∈ N , denote by P n the orthogonal projection of H ontospan { e , e , . . . , e n } . Let T ⊂ B ( H ) be the algebra of triangular operators with respect to( P n ) n . By Example 3.3, we see that T is residually finite-dimensional. On the other hand,it is easy to verify that the ideal of compact operators K ( H ) belongs to C ∗ ( T ). To showthat C ∗ ( T ) is not residually finite-dimensional, it suffices to show that K ( H ) is not. Butany completely contractive homomorphism of K ( H ) is a ∗ -homomorphism, and there are nonon-zero ∗ -homomorphisms from K ( H ) into a finite-dimensional C ∗ -algebra, in view of H being infinite-dimensional. (cid:3) We close this section by exhibiting a class of residually finite-dimensional C ∗ -algebraswhich is of particular interest to us, in light of Theorem 3.2. We suspect that the followingstatement is well-known to experts, but we provide a proof for the reader’s convenience.3.5. Proposition.
Let k : Λ → N be a bounded net. Then, the C ∗ -algebra Q k is residuallyfinite-dimensional. More precisely, there is another set Λ ′ = ∅ , a constant function r : Λ ′ → N and a completely isometric ∗ -homomorphism Γ : Q k → M r .Proof. Let r ∈ N such that k ( λ ) ≤ r for every λ ∈ Λ. We note that if m ∈ N and m ≤ r ,then there is a completely isometric ∗ -homomorphism ε m : M m → M r defined via ε m ( a ) = a ⊕ r − m , a ∈ M m . Next, given b = ( b λ ) λ + J k ∈ Q k , it is easily verified that k b k = inf µ ∈ Λ sup λ ≥ µ k b λ k . FD and representations of amenable operator algebras In particular, we see that k b k is a cluster point of {k b λ k : λ ∈ Λ } . Thus, there exists a cofinalultrafilter F b on Λ for which k b k = lim λ →F b k b λ k . Closed balls in M r are compact, so thatgiven ( d λ ) λ ∈ L k the limit lim λ →F b ε k ( λ ) ( d λ )exists in M r . Note also that since F b is cofinal, we have thatlim λ →F b ε k ( λ ) ( d λ ) = 0whenever ( d λ ) λ ∈ J k . We may therefore define a map γ b : Q k → M r such that if d = ( d λ ) λ + J k then γ b ( d ) = lim λ →F b ε k ( λ ) ( d λ ) . A routine verification establishes that γ b is a ∗ -homomorphism. For any b ∈ Q k we see that k γ b ( b ) k = lim λ →F b k ε k ( λ ) ( b λ ) k = lim λ →F b k b λ k = k b k by choice of the ultrafilter F b .Finally, let Λ ′ denote the unit sphere of Q k . Define r : Λ ′ → N as r ( b ) = r for every b ∈ Λ ′ . Then, the map Γ : Q k → M r defined by Γ( d ) = ( γ b ( d )) b ∈ Λ ′ is an isometric ∗ -homomorphism, and is thus completely isometric. (cid:3) Residually finite-dimensional operator algebras with the totalreduction property
Motivated by Theorem 3.2, in this section we examine in detail the structure of subalgebrasof M k with the total reduction property, where Λ = ∅ is a set and k : Λ → N is a function.We establish one of our main results based partly on this detailed analysis.Our first goal is to show that if A ⊂ M k has the total reduction property, then up tocompletely bounded isomorphism, we may assume that each component map q k λ is surjectiveon A .4.1. Theorem.
Let Λ = ∅ be a set and let k : Λ → N be a function. Suppose that A ⊂ M k is a subalgebra with the total reduction property. Then, there exist a set Λ ′ , afunction m : Λ ′ → N and a subalgebra B ⊂ M m which is completely boundedly isomorphicto A and such that for every α ∈ Λ ′ we have that q m α ( B ) = M m ( α ) . Furthermore, B is weak- ∗ closed if A is.Proof. For each λ ∈ Λ, let A λ = q k λ ( A ). Note that A λ ⊂ M k ( λ ) so that A λ is necessarilyclosed and hence has the total reduction property by Lemma 2.3. Moreover, we see fromLemma 2.3 that κ A λ (1) ≤ κ A ( k q k λ k ) ≤ κ A (1) . By Theorem 2.8, for each λ ∈ Λ there exists an invertible operator X λ ∈ M k ( λ ) such that X λ A λ X − λ ⊂ M k ( λ ) is a C ∗ -algebra and k X λ k k X − λ k ≤ ∆ , R. Clouˆatre and L.W. Marcoux where ∆ is a positive constant depending only on κ A (1). Upon rescaling, we may assumethat k X λ k = k X − λ k ≤ ∆ / for each λ ∈ Λ. Then, the operator X = L λ ∈ Λ X λ ∈ M k is bounded and invertible, and wehave X A X − ⊂ M λ X λ A λ X − λ . For each λ , there exist a natural number r λ and non-negative integers d ( λ, , d ( λ, , . . . , d ( λ, r λ )along with an orthogonal decomposition C k ( λ ) = C d ( λ, ⊕ C d ( λ, ⊕ C d ( λ, ⊕ · · · ⊕ C d ( λ,r λ ) . With respect to this decomposition we must have X λ A λ X − λ ⊂ { } ⊕ M d ( λ, ⊕ M d ( λ, ⊕ · · · ⊕ M d ( λ,r λ ) and q d ( λ,j ) ◦ p λ ( X λ A λ X − λ ) = M d ( λ,j ) for all 1 ≤ j ≤ r λ , where p λ : { } ⊕ M d ( λ, ⊕ M d ( λ, ⊕ · · · ⊕ M d ( λ,r λ ) → M d ( λ, ⊕ M d ( λ, ⊕ · · · ⊕ M d ( λ,r λ ) and q d ( λ,j ) : M d ( λ, ⊕ M d ( λ, ⊕ · · · ⊕ M d ( λ,r λ ) → M d ( λ,j ) denote the natural projections. Define p = ⊕ λ p λ which is completely isometric X A X − andweak- ∗ homeomorphic on the weak- ∗ closure of X A X − . Put B = p ( X A X − ). It remainsto define the set Λ ′ and the function m : Λ ′ → N . For each λ ∈ Λ, we defineΣ λ = { ( λ, j ) : 1 ≤ j ≤ r λ , d ( λ, j ) = 0 } . We put Λ ′ = ∪ λ ∈ Λ Σ λ and m (( λ, j )) = d ( λ, j ) for all ( λ, j ) ∈ Λ ′ . (cid:3) Next, we make an important observation: simple subalgebras of M k with the total re-duction property are similar to finite-dimensional C ∗ -algebras.4.2. Corollary.
Let Λ = ∅ be a set, let k : Λ → N be a function and let A ⊂ M k bea simple subalgebra which has the total reduction property. Then, A is similar to a finite-dimensional C ∗ -algebra.Proof. Invoking [11, Theorem 1.10], we see that it is sufficient to prove the statement fora completely boundedly isomorphic image of A . Hence, by virtue of Theorem 4.1 we mayassume that q k λ ( A ) = M k ( λ ) for every λ ∈ Λ. Fix λ ∈ Λ, and consider the surjective,contractive homomorphism q k λ | A : A → M k ( λ ) . Since A is simple, we see that q k λ | A isinvertible, and hence that A is boundedly isomorphic to the finite-dimensional C ∗ -algebra M k ( λ ) . We conclude that A must be similar to a ∗ -isomorphic image of M k ( λ ) [24]. (cid:3) We mention in passing that Corollary 4.2 can be extended to cover the case where thealgebra possesses finitely many ideals. We leave the details to the interested reader.Before proceeding with the main result of this section, we require two preliminary facts.We first establish a useful estimate.
FD and representations of amenable operator algebras Lemma.
Let Λ = ∅ be a set and let k : Λ → N be the constant function k ( λ ) = k, λ ∈ Λ for some fixed k ∈ N . Let ( X λ ) λ ∈ Λ be a collection of invertible operators in M k and let A = (M λ ∈ Λ X − λ T X λ : T ∈ M k ) . Suppose that
A ⊂ M k , that A has the total reduction property and that there exists λ ∈ Λ for which X λ = I . Then sup λ k X λ k k X − λ k ≤ κ A (1) . Proof.
Upon rescaling if necessary, we may assume that k X λ k = k X − λ k = k X λ k / k X − λ k / for all λ ∈ Λ. For each ν ∈ Λ with ν = λ , letΓ ν : A → M k be defined by Γ ν M λ ∈ Λ X − λ T X λ ! = T ⊕ X − ν T X ν for every T ∈ M k . Observe that since X λ = I , each such map Γ ν is a contractive homo-morphism. Therefore Γ ν ( A ) has the total reduction property by Lemma 2.3. Now considerthe subspace W ν = { ξ ⊕ X − ν ξ : ξ ∈ C k } . Clearly, W ν is invariant for Γ ν ( A ), so there exists an idempotent E ν ∈ Γ ν ( A ) ′ with k E ν k ≤ κ A (1) and whose range is W ν . Routine calculations show thatΓ ν ( A ) ′ = (cid:26)(cid:20) αI k βX ν γX − ν δI k (cid:21) : α, β, γ, δ ∈ C (cid:27) ⊂ M k . Using the fact that the range of E ν is W ν , we infer that E ν = (cid:20) α ν I k β ν X ν α ν X − ν β ν I k (cid:21) for an appropriate choice of α ν , β ν ∈ C . Moreover, since E ν is idempotent and dim W ν = k ,we deduce that k = tr( E ν ). On the other hand, we have that tr( E ν ) = ( α ν + β ν ) k , so that α ν + β ν = 1. In particular max {| α ν | , | β ν |} ≥ / k E ν k ≥ max {k α ν X − ν k , k β ν X ν k} ≥ k X ν k / . Thus k X ν k ≤ k E ν k ≤ κ A (1). We conclude that k X ν k k X − ν k = k X ν k ≤ κ A (1) for every ν ∈ Λ \ { λ } . Since X λ = I k and κ A (1) ≥
1, we are done. (cid:3)
We also need a property of central idempotents in weak- ∗ closed algebras having the totalreduction property. R. Clouˆatre and L.W. Marcoux
Lemma.
Let
A ⊂ B ( H ) be a weak- ∗ closed algebra with the total reduction propertyand with unit u ∈ A . Let ( e i ) i ∈ I be a family of central idempotents in A such that ∩ i ∈ I ( u − e i ) A = { } . Let
S ⊂ B ( H ) denote the smallest weak- ∗ closed subspace containing e i A for every i ∈ I .Then, A = S .Proof. We note that e i A ⊂ A for every i ∈ I , so that S ⊂ A since A is weak- ∗ closed. It isreadily checked that S is a weak- ∗ closed two-sided ideal, and thus by Theorem 2.6 there isa central idempotent f ∈ A with f A = S . Note that e i = e i u ∈ e i A ⊂ S , so that e i f = e i and e i ( u − f ) = 0 for every i ∈ I . We claim that f A = A . Indeed, if a ∈ A then we seethat ( u − f ) a = ( u − e i )( u − f ) a for every i ∈ I , so that ( u − f ) a ∈ ∩ i ∈ I ( u − e i ) A = { } whence f a = a . We conclude that A = f A = S . (cid:3) We can now prove one of the main results of the paper.4.5.
Theorem.
Let Λ = ∅ be a set and let k : Λ → N be a function. Suppose that A ⊂ M k is a subalgebra which has the total reduction property. If A is weak- ∗ closed, then A is similarto a C ∗ -algebra.Proof. We start by noting once again that it is sufficient to prove the statement for a com-pletely boundedly isomorphic image of A . Thus, by virtue of Theorem 4.1 we may assumethat A is a weak- ∗ closed subalgebra of M k such that q k λ ( A ) = M k ( λ ) for every λ ∈ Λ.For each λ ∈ Λ, consider the weak- ∗ closed ideal J λ = ker q k λ . Then A / J λ ≃ q k λ ( A ) = M k ( λ ) . Moreover, by Theorem 2.6 there exists a central idempotent f λ ∈ J λ ∩ A ′ with J λ = f λ A , k f λ k ≤ κ A (1) and such that we have a topological direct sum decomposition A = J λ + ( u − f λ ) A where u denotes the unit of A and satisfies k u k ≤ κ A (1) (see the discussion following Theorem2.6). Set now e λ = ( u − f λ ) for each λ ∈ Λ, and put K λ = e λ A so that A = J λ + K λ . By Lemma 2.3 we see that K λ has the total reduction property and that κ K λ (1) ≤ κ A ( k e λ k ) ≤ κ A (2 κ A (1)) . Moreover, we note that K λ ≃ A / J λ ≃ M k ( λ ) and consequently K λ is simple and a minimal ideal of A . Hence, for each α ∈ Λ we must havethat q k α ( K λ ) is a two-sided ideal of M k ( α ) , and therefore is either { } or M k ( α ) . Equivalently, q k α ( e λ ) = 0 or q k α ( e λ ) = I .For each λ ∈ Λ let ∆ λ = { α ∈ Λ : q k α ( e λ ) = I } . Note that for every λ ∈ Λ we have q k λ ( u ) = I since q k λ ( A ) = M k ( λ ) . Recall that f λ A = J λ = ker q k λ FD and representations of amenable operator algebras thus 0 = q k λ ( f λ ) = I − q k λ ( e λ )so that λ ∈ ∆ λ for every λ ∈ Λ. In particular, we see thatΛ = ∪ λ ∈ Λ ∆ λ . If λ , λ are distinct elements of Λ, then by minimality we must have either K λ = K λ or K λ ∩ K λ = { } . Note also that K λ ∩ K λ = { } if and only if e λ e λ = 0, and K λ = K λ if and only if e λ = e λ . We conclude that if λ , λ ∈ Λ, then either ∆ λ = ∆ λ , or∆ λ ∩ ∆ λ = ∅ . The equivalence relation λ ∼ λ if and only if e λ = e λ partitions Λ into a set Ω of disjoint subsets. For each ω ∈ Ω, choose λ ω ∈ ω . Then, we havethe following disjoint union ∪ ω ∈ Ω ∆ λ ω = Λ . If ω , ω ∈ Ω are distinct, then e λ ω e λ ω = 0. Let now a ∈ A and assume that e λ ω a = 0 forevery ω ∈ Ω. Then, we see that q k α ( a ) = 0 for every α ∈ ∪ ω ∈ Ω ∆ λ ω = Λ, so that a = 0. ByLemma 4.4, we conclude that A is the smallest weak- ∗ closed subspace containing e λ ω A forevery ω ∈ Ω. On the other hand, if ω , ω ∈ Ω are distinct, then ∆ λ ω ∩ ∆ λ ω = ∅ . Hence,we find that A = M ω ∈ Ω e λ ω A = M ω ∈ Ω K λ ω . Now, for each λ ∈ Λ we have that q k α ( K λ ) = M k ( α ) for all α ∈ ∆ λ . Since K λ is simple,we see that q k α | K λ is a bounded isomorphism between K λ and M k ( α ) for every α ∈ ∆ λ . Inparticular, k is constant on ∆ λ . In addition, if we let µ λ be a fixed element of ∆ λ , then wesee that for each α ∈ ∆ λ there is an invertible operator X λ,α ∈ M k ( α ) such that X λ,α q k µ λ ( a ) X − λ,α = q k α ( a )for every a ∈ K λ . Since K λ = (M α ∈ Λ q k α ( a ) : a ∈ K λ ) , we see that up to a (unitary) reordering of the components K λ = M α ∈ Λ \ ∆ λ { } ⊕ M α ∈ ∆ λ X λ,α bX − λ,α : b ∈ M k ( µ λ ) . By Lemmas 2.3 and 4.3, we find that k X − λ,α k k X λ,α k ≤ κ K λ (1) ≤ κ A (2 κ A (1)) for every α ∈ ∆ λ . Upon rescaling we may assume that k X λ,α k = k X − λ,α k ≤ κ A (2 κ A (1)) . for every α ∈ ∆ λ .Hence, the operator X = M ω ∈ Ω M α ∈ Λ \ ∆ λω I ⊕ M α ∈ ∆ λω X λ ω ,α R. Clouˆatre and L.W. Marcoux is invertible with bounded inverse. Finally, we find X − A X = X − M ω ∈ Ω K λ ω ! X = M ω ∈ Ω M α ∈ Λ \ ∆ λω { } ⊕ M α ∈ ∆ λω b : b ∈ M k ( µ λω ) ≃ M ω ∈ Ω M k ( µ λω ) which is a C ∗ -algebra. (cid:3) The reader will notice that the reason we require the algebra above to be weak- ∗ closedis to avail ourselves of Theorem 2.6. We now record a straightforward consequence of theprevious result.4.6. Corollary.
Let Λ = ∅ be a set and let k : Λ → N be a function. Suppose that A ⊂ M k is a subalgebra which has the total reduction property. If A is weak- ∗ closed, then it has theSP property .Proof. We know that A is similar to a C ∗ -algebra by Theorem 4.5. Next, recall that C ∗ -algebras with the total reduction property have Kadison’s similarity property by Theorem2.7, so that A must have the SP property. (cid:3) Finally, we close this section with an application to triangular algebras (see Example 3.3for the definition).4.7.
Corollary.
Let
A ⊂ B ( H ) be a weak- ∗ closed triangular operator algebra with the totalreduction property. Then, A is similar to a C ∗ -algebra.Proof. By Example 3.3, we know that there is a completely isometric weak- ∗ homeomorphichomomorphism Φ : A → L k , for some net k : Λ → N . Thus, by Lemma 2.3 and Theorem4.5 we see there is an invertible operator X such that X Φ( A ) X − is a C ∗ -algebra. The map X Φ( a ) X − a, a ∈ A is completely bounded on a C ∗ -algebra, and thus is similar to a ∗ -homomorphism [11].Consequently, A is similar to the image of a C ∗ -algebra under a ∗ -homomorphism, and so issimilar to a C ∗ -algebra. (cid:3) Representations with residually finite-dimensional range
In the previous section, we proved that a weak- ∗ closed residually finite-dimensional op-erator algebra with the total reduction property has the SP property: all its representationsare automatically completely bounded. In other words, we restricted our attention to thecase where the domains of the representations are contained in a product of matrix al-gebras. In this section, we shift our focus to the range and assume that it is residuallyfinite-dimensional, while the domain is allowed to be an arbitrary operator algebra with thetotal reduction property.The driving force behind our efforts in this section is the following simple observation,which we record for ease of reference. FD and representations of amenable operator algebras Lemma.
Let A and B be operator algebras. Assume that B is residually finite-dimensional and let ( ̺ λ ) λ be the corresponding family of completely contractive represen-tations of B such that L λ ̺ λ is completely isometric. Let θ : A → B be a representation.Then θ is completely bounded if and only if sup λ k ̺ λ ◦ θ k cb < ∞ . Although completely elementary, this lemma shows that to determine whether a repre-sentation with residually finite-dimensional range is completely bounded, we may restrictour attention to finite-dimensional representations. In particular, we have the followingconsequence which the reader may want to compare with Theorem 3.2.5.2.
Theorem.
Let A be an operator algebra and let k : Λ → N be a bounded net. If θ : A → Q k is a bounded representation, then θ is completely bounded.Proof. By Proposition 3.5 we can find another set Λ ′ = ∅ and a constant function r : Λ ′ → N for which there exists a completely isometric ∗ -homomorphismΓ : Q k → M r . It suffices to show that Γ ◦ θ is completely bounded. To see this, use Lemma 5.1 to concludethat θ is completely bounded if and only ifsup λ k q k λ ◦ θ k cb < ∞ . Now, a classical result of R.R. Smith [21] shows that for any λ ∈ Λ we have k q k λ ◦ θ k cb ≤ k ( λ ) k q k λ ◦ θ k ≤ (cid:18) sup λ ∈ Λ k ( λ ) (cid:19) k θ k . The proof is complete. (cid:3)
The remainder of this section is devoted to establishing another one of our main resultsdealing with representations of operator algebras with the total reduction property whoserange are residually finite-dimensional. The key technical tool is the following observation,which generalizes the fact that finite-dimensional C ∗ -algebras can be faithfully representedon finite-dimensional Hilbert spaces. We suspect it is well-known. The idea is reminiscent ofthat found in the proof of [18, Theorem 6.3], which apparently has its origins in the work ofDixon. We also note that the classical Blecher-Ruan-Sinclair theorem [2] may not producea finite-dimensional Hilbert space and thus does not meet our specific needs.5.3. Proposition.
Let A be a finite-dimensional operator algebra, and fix ε > and d ∈ N . Then, there exist a finite-dimensional Hilbert space H and a completely contractivehomomorphism ϕ : A → B ( H ) with k ϕ ( n ) ( A ) k M n ( B ( H )) ≥ (1 − ε ) k A k M n ( A ) for every A ∈ M n ( A ) and every ≤ n ≤ d .Proof. By considering the unitization of A (which is also finite-dimensional) if necessary,we may assume that A is unital [1]. Since A is finite-dimensional, its closed unit ball iscompact. Hence, we may choose α , . . . , α N ∈ A such that k α k k ≤ ≤ k ≤ N ,and with the property that for every a ∈ A with k a k ≤ ≤ k ≤ N such that k a − α k k < ε/d. R. Clouˆatre and L.W. Marcoux
In particular, given A ∈ M n ( A ) with k A k M n ( A ) = 1 where 1 ≤ n ≤ d , we can find A ′ =( a ′ ij ) ij ∈ M n ( A ) with the property that each a ′ ij belongs to the set { α , . . . , α N } and k A − A ′ k M n ( A ) < ε. Next, note that there is a finite-dimensional subspace H with the property that k P ( n ) H A ∗ AP ( n ) H k ≥ (1 − ε ) k A k M n ( A ) whenever A = ( a ij ) ij ∈ M n ( A ) for some 1 ≤ n ≤ d and each a ij belongs to the set { α , . . . , α N } . Define a unital completely positive map ω : C ∗ ( A ) → B ( H )as ω ( t ) = P H t | H for every t ∈ C ∗ ( A ). Next, we carefully analyze the Stinespring dilationof ω , and for that purpose we briefly recall the details of its construction.Define a positive semi-definite bilinear form on the vector space C ∗ ( A ) ⊗ H asΨ X i t i ⊗ h i , X j s j ⊗ k j = X i,j h ω ( s ∗ j t i ) h i , k j i H . Let Z = { v ∈ C ∗ ( A ) ⊗ H : Ψ( v, v ) = 0 } , which is a subspace of C ∗ ( A ) ⊗ H . The quotient ( C ∗ ( A ) ⊗ H ) / Z is an inner product space,and we denote its completion by E . Given v ∈ C ∗ ( A ) ⊗ H we denote its image in E by [ v ].Define π : C ∗ ( A ) → B ( E )via π ( t ) X j s j ⊗ h j = X j ts j ⊗ h j for every t ∈ C ∗ ( A ). The complete positivity of ω implies that π is well-defined. Further-more, it is readily verified that π is a unital ∗ -homomorphism. Now, here is the key point:we denote by H the finite-dimensional space spanned by the elements of the form [ a ⊗ h ] for a ∈ A and h ∈ H . Since A is an algebra, the space H is invariant for π ( A ). Hence, we maydefine a unital completely contractive homomorphism ϕ : A → B ( H )as ϕ ( a ) = π ( a ) | H for every a ∈ A . It only remains to establish the announced lower bound.Notice that for h, k ∈ H we haveΨ(1 ⊗ h, ⊗ k ) = h ω (1) h, k i H = h h, k i H and thus k [1 ⊗ h ] k H = k h k H for every h ∈ H . In particular, if ξ = ( ξ , . . . , ξ n ) t ∈ H ( n )1 satisfies k ξ k H ( n )1 = 1 then the vectorΞ = (1 ⊗ ξ , . . . , ⊗ ξ n ) t ∈ H ( n ) also satisfies k Ξ k H ( n ) = 1. Furthermore, if A ∈ M n ( A ) then a routine calculation shows that h ω ( n ) ( A ∗ A ) ξ, ξ i H ( n )1 = k π ( n ) ( A )Ξ k E ( n ) = k ϕ ( n ) ( A )Ξ k H ( n ) . Hence, h ω ( n ) ( A ∗ A ) ξ, ξ i H ( n )1 ≤ k ϕ ( n ) ( A ) k M n ( B ( H )) FD and representations of amenable operator algebras and we conclude that k ϕ ( n ) ( A ) k M n ( B ( H )) ≥ k ω ( n ) ( A ∗ A ) k M n ( B ( H )) = k P ( n ) H A ∗ AP ( n ) H k ≥ (1 − ε ) k A k M n ( A ) whenever A = ( a ij ) ij ∈ M n ( A ) where 1 ≤ n ≤ d and each a ij belongs to the set { α , . . . , α N } .As noted above, given a general element A ∈ M n ( A ) with 1 ≤ n ≤ d and k A k M n ( A ) = 1, wecan find A ′ = ( a ′ ij ) ij ∈ M n ( A ) where each a ′ ij belongs to the set { α , . . . , α N } and k A − A ′ k M n ( A ) < ε. In particular, k A ′ k M n ( A ) ≥ − ε . Since ϕ is completely contractive, we find k ϕ ( n ) ( A ) k M n ( B ( H )) ≥ k ϕ ( n ) ( A ′ ) k M n ( B ( H )) − ε ≥ (1 − ε ) / − ε and the proof is complete. (cid:3) This theorem allows us to prove a uniform estimate for certain representations of operatoralgebras with the total reduction property.5.4.
Theorem.
Let A be an operator algebra with the total reduction property and let θ : A → B ( H ) be a representation. Assume that A / ker θ is finite-dimensional. Then, thereis a positive constant ∆ depending only on κ A and k θ k such that k θ k cb ≤ ∆ . Proof.
Let b θ : A / ker θ → B ( H ) be the representation induced by θ . Then, we have k θ k cb = k b θ k cb . By Proposition 5.3, there is a finite-dimensional Hilbert space H ′ , an operator algebra B ⊂ B ( H ′ ) and completely contractive isomorphism ϕ : A / ker θ → B with k ϕ − k ≤
2. Therefore, B has the total reduction property and κ B ( t ) ≤ κ A / ker θ ( k ϕ k t ) ≤ κ A ( t )for every t ≥
0, by Lemma 2.3. Then, an application of Theorem 2.8 yields an invertibleoperator X ∈ B ( H ′ ) such that X B X − is a C ∗ -algebra and k X kk X − k ≤ ∆ for some positive constant ∆ depending only on κ A (1). We will use the following notationAd X ( T ) = XT X − , T ∈ B ( H ′ ) . We see that k Ad X k cb ≤ ∆ , k Ad − X k cb ≤ ∆ . Then, X B X − = Ad X ( B ) has the total reduction property and κ X B X − ( t ) ≤ κ B ( k Ad X k t ) ≤ κ A (∆ t )for every t ≥
0, again by Lemma 2.3.Now, by virtue of Theorem 2.7, we see that the map b θ ◦ ϕ − ◦ Ad − X : X B X − → B ( H ) R. Clouˆatre and L.W. Marcoux is completely bounded with k b θ ◦ ϕ − ◦ Ad − X k cb ≤ κ X B X − ( k b θ ◦ ϕ − ◦ Ad − X k ) ≤ κ A (∆ k b θ ◦ ϕ − ◦ Ad − X k ) ≤ κ A (∆ k ϕ − kk Ad − X kk θ k ) ≤ κ A (2∆ k θ k ) . Finally, note that b θ = b θ ◦ ϕ − ◦ Ad − X ◦ Ad X ◦ ϕ. so that k b θ k cb ≤ k Ad X k cb k ϕ k cb k b θ ◦ ϕ − ◦ Ad − X k cb ≤ κ A (2∆ k θ k ) so we may take ∆ = 128∆ κ A (2∆ k θ k ) . (cid:3) We can now establish the main result of this section, which says that for operator algebraswith the total reduction property, representations with residually finite-dimensional rangesare necessarily completely bounded.5.5.
Corollary.
Let Λ = ∅ be a set and let k : Λ → N be a function. Let A be an operatoralgebra with the total reduction property and let θ : A → M k be a representation. Then, θ is completely bounded.Proof. Combine Lemma 5.1 and Theorem 5.4. (cid:3)
We give an application to triangular algebras (see Example 3.3 for the definition).5.6.
Corollary.
Let A be an operator algebra with the total reduction property and let θ : A → B ( H ) be a representation such that θ ( A ) is a triangular algebra. Then, θ iscompletely bounded.Proof. By Example 3.3, we know that there is a completely isometric homomorphismΦ : θ ( A ) → L k , for some net k : Λ → N . Thus, Φ ◦ θ is completely bounded byvirtue of Corollary 5.5. Since Φ is completely isometric, we conclude that θ is completelybounded as well. (cid:3) In closing, we exhibit an example showing that the total reduction property cannot simplybe removed from the assumptions of Corollary 5.5.5.7.
Example.
For each n ∈ N , let t n : M n → M n denote the transpose map. Then, it iswell-known that k t n k = 1 and k t ( n ) n k = n. Next, let A n ⊂ M n denote the unital subalgebra consisting of elements of the form (cid:20) λI n A λI n (cid:21) where λ ∈ C , A ∈ M n . The map θ n : A n → M n defined as θ n (cid:18)(cid:20) λI n A λI n (cid:21)(cid:19) = (cid:20) λI n t n ( A )0 λI n (cid:21) , λ ∈ C , A ∈ M n FD and representations of amenable operator algebras is easily verified to be a unital homomorphism with the property that k θ n k = 1 and k θ ( n ) n k ≥ k t ( n ) n k = n. Consequently, we see that the map Θ : Q n A n → Q n M n defined asΘ( a n ) n = ( θ n ( a n )) n , ( a n ) n ∈ Y n A n is a unital bounded representation which is not completely bounded. Finally, fix m ∈ N and note that the subspace C m ⊕ { } is invariant for A m yet it does not admit an invarianttopological complement as a straightforward calculation establishes. Therefore, A m does nothave the total reduction property. Since A m is a closed two-sided ideal of Q n A n , we concludefrom Theorem 2.4 that Q n A n does not have the total reduction property either. (cid:3) References [1] David P. Blecher and Christian Le Merdy.
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