Finite-dimensionality in the non-commutative Choquet boundary: peaking phenomena and C ∗ -liminality
aa r X i v : . [ m a t h . OA ] S e p FINITE-DIMENSIONALITY IN THE NON-COMMUTATIVECHOQUET BOUNDARY: PEAKING PHENOMENA AND C ∗ -LIMINALITY RAPHA¨EL CLOU ˆATRE AND IAN THOMPSON
Abstract.
We explore the finite-dimensional part of the non-commutativeChoquet boundary of an operator algebra. In other words, we seek finite-dimensional boundary representations. Such representations may fail to existeven when the underlying operator algebra is finite-dimensional. Nevertheless,we exhibit mechanisms that detect when a given finite-dimensional representa-tion lies in the Choquet boundary. Broadly speaking, our approach is topologi-cal and requires identifying isolated points in the spectrum of the C ∗ -envelope.This is accomplished by analyzing peaking representations and peaking pro-jections, both of which being non-commutative versions of the classical notionof a peak point for a function algebra. We also connect this question with theresidual finite-dimensionality of the C ∗ -envelope and to a stronger propertythat we call C ∗ -liminality. Recent developments in matrix convexity allowus to identify a pivotal intermediate property, whereby every matrix state islocally finite-dimensional. Introduction
In unraveling the structure of C ∗ -algebras, a fruitful paradigm is to model theseobjects, insofar as possible, by finite-dimensional ones. The idea that finer struc-tural properties of a large class of C ∗ -algebras can be detected upon approximationby matrix algebras has become a major trend in the field. Nuclearity, a notion atthe center of recent capstone results in the classification program for some simpleC ∗ -algebras (see [30],[51] and references therein), is an important example of an“internal” finite-dimensional approximation property. Closely related to the themeof this paper is a different, “external” finite-dimensional approximation propertywhich we now describe.A C ∗ -algebra is said to be residually finite-dimensional (or RFD) if it admits aseparating set of finite-dimensional ∗ -representations. Roughly speaking, RFD C ∗ -algebras should be thought of as being diagonal with finite-dimensional blocks. Inparticular, they can be completely understood by examining their representationsas matrices. Interestingly, various characterizations of this important property haveemerged over the years [26],[5],[31],[18].In principle, similar ideas have the potential to unlock the structure of moregeneral, possibly non-selfadjoint, operator algebras. This possibility was exploredin [41] for some concrete operator algebras of functions. In a related context, ananalysis of finer forms of residual finite-dimensionality was performed recently in [4]. Date : September 29, 2020.R.C. was partially supported by an NSERC Discovery Grant. I.T. was partially supported bya Manitoba Graduate Scholarship.
Predating this last work, a thorough investigation of non-selfadjoint residual finite-dimensionality was initiated in [17]. Therein, consistently with the more familiarself-adjoint setting, an operator algebra is said to be RFD if it admits a completelynorming collection of completely contractive homomorphisms into matrix algebras.Several basic properties of such algebras were established, such as the non-obviousfact that finite-dimensional algebras are RFD. Furthermore an attempt was made toconnect back with the more classical C ∗ -algebra setting, via the following procedure.Let A be a unital operator algebra, which we assume is concretely representedon some Hilbert space H , so that A ⊂ B ( H ). Then one can consider C ∗ ( A ),the C ∗ -algebra generated by A . Assuming that A is RFD, it is then natural to askwhether this property is inherited by C ∗ ( A ). To make this discussion more efficient,the language of C ∗ -covers is useful. Recall that a C ∗ -cover for A is a pair ( A , ι ),where A is a C ∗ -algebra and ι : A → A is a completely isometric homomorphismsuch that A = C ∗ ( ι ( A )). If A is indeed RFD, then essentially by definition there issome C ∗ -cover ( A , ι ) such that A is RFD. The question raised above only becomesinteresting, then, if the C ∗ -cover is fixed in advance. Two natural candidates arethe minimal and the maximal C ∗ -covers.In some special cases, it is known that the maximal C ∗ -cover of an RFD operatoralgebra does inherit the property of being RFD (see [17, Section 5]). As of thiswriting, it is still unknown whether this phenomenon always occurs; we will notaddress this problem here. The corresponding question for the minimal C ∗ -cover isthe main driving force of the current paper, so we discuss it in more details.Due to seminal work of Arveson [6] and Hamana [32], any unital operator algebraadmits an essentially unique minimal C ∗ -cover, which is called its C ∗ -envelope . Thequestion we are interested in then asks whether an RFD unital operator algebraalways admits an RFD C ∗ -envelope. As shown in [17, Example 4], the answer isnegative even for finite-dimensional operator algebras. Nevertheless, some specialcases where the answer is affirmative were identified in that same paper. The presentpaper can be viewed as a natural continuation, aiming to clarify this questionfurther.Our approach is predicated on Arveson’s insight for constructing the C ∗ -envelopeof a concretely represented unital operator algebra A ⊂ B ( H ). By considering theC ∗ -envelope C ∗ e ( A ) as being determined by a non-commutative analogue of theShilov boundary of a function algebra, Arveson’s vision was that it could be con-structed via a non-commutative analogue of the Choquet boundary. This originalvision has now been fully realized thanks to non-trivial contributions from manyresearchers [42],[25],[8],[20].In Arveson’s analogy, points are identified with characters in the classical com-mutative world, and correspond to irreducible ∗ -representations in the non-commu-tative realm. Thus, the C ∗ -envelope is determined by the boundary representations (see Subsection 2.5 for details.) Therefore, the residual finite-dimensionality ofthe C ∗ -envelope should be detectable through the lens of these boundary repre-sentations. For instance, an abundance of finite-dimensional boundary represen-tations is known to force the C ∗ -envelope to be RFD. Conversely, the residualfinite-dimensionality of the C ∗ -envelope implies that, in an appropriate sense, boththe finite-dimensional irreducible ∗ -representations and the boundary representa-tions are dense. It is unclear, however, whether these two dense sets have any INITE-DIMENSIONALITY IN THE NC CHOQUET BOUNDARY 3 overlap; even the existence of a single finite-dimensional boundary representationappears to be difficult to ascertain. Our main question is thus the following.
Question 1.
Let A be a unital operator algebra with an RFD C ∗ -envelope. Mustthe envelope admit a finite-dimensional boundary representation for A ?To give the reader a sense for why this question may be non-trivial, we mentionthat there are examples of finite-dimensional unital operator algebras for whichthe C ∗ -envelope admits no finite-dimensional boundary representations (Example4). In fact, it can even happen that the C ∗ -envelope of a unital finite-dimensionaloperator algebra admits simply no finite-dimensional ∗ -representations whatsoever(Example 5). This occurs despite the fact that finite-dimensional operator algebrassatisfy a strong form a residual finite-dimensionality: they are in fact normed infinite dimensions [17, Theorem 3.5] (see Subsection 2.2 below for a definition).Consequently, if the answer to Question 1 is to be affirmative, it must be so fordeeper reasons than the mere residual finite-dimensionality of A . We also mentionthat a boundary representation being finite-dimensional is a much stronger notionthan it being “accessible” in the sense of [40, Definition 6.32].In view of this difficulty, a strategy for exhibiting finite-dimensional boundaryrepresentations must be formulated. We opt for the following topological approachto establish most our main results. As mentioned above, the set of (unitary equiv-alence classes of) boundary representations is dense in the spectrum of the C ∗ -envelope. Thus, if a finite-dimensional irreducible ∗ -representation is known to bean isolated point, it would necessarily be a boundary representation.In keeping with the paradigm suggested by Arveson that the C ∗ -envelope shouldbe analyzed using non-commutative analogues of ideas from function theory, torealize the aforementioned strategy we view operator algebras as comprising non-commutative functions. Accordingly, we aim to apply tools from what one couldcall non-commutative uniform algebra theory. More precisely, we verify that certainfinite-dimensional representations are indeed isolated points by showing that theyare “non-commutative peak points”. Classically, given a uniform algebra A ofcontinuous functions on a compact metric space X , a point ξ ∈ X is said to be a peak point for A if there is ϕ ∈ A such that ϕ ( ξ ) = 1 > | ϕ ( x ) | , x ∈ X, x = ξ. A theorem of Bishop implies that the set of peak points coincides with the Choquetboundary of A , which is in turn dense in the Shilov boundary of A [45, Section 8].We will exploit two rather different non-commutative versions of peak points.The first one replaces points by irreducible ∗ -representations of the C ∗ -envelope;this is based on the point of view that points in X should be interpreted as char-acters. The second version identifies points with their characteristic functions, andthus replaces them by projections lying in the bidual. We will utilize both inter-pretations.We now describe the organization of the paper and state our main original con-tributions more precisely.Section 2 collects various prerequisite material on operator algebras, and provessome elementary facts used throughout.In Section 3, we tackle Question 1. Central to our approach and results is thefollowing definition. Let A is a unital operator algebra and let π be an irreducible ∗ -representation of its C ∗ -envelope C ∗ e ( A ). Let n ∈ N and let T ∈ M n (C ∗ e ( A )). RAPHA¨EL CLOUˆATRE AND IAN THOMPSON
Then, T is said to peak at π if k π ( n ) ( T ) k > k σ ( n ) ( T ) k for every irreducible ∗ -representation σ of C ∗ e ( A ) which is not unitarily equivalentto π . In this case, we also say that π is a peaking representation for A .We introduce a more flexible, “local” version of this notion. We show that alocally peaking representation for A must necessarily be a boundary representation(Theorem 3.1). It is thus highly desirable to be able to recognize locally peakingrepresentations. To do this, we leverage the theory of peaking projections developedin a series of papers starting with work of Hay [34],[11],[46],[10],[15].Let a ∈ A be a contraction and consider the support projection s π of the ir-reducible ∗ -representation π (see Subsection 2.4). We say that a peaks at s π if a s π = s π and k ap k < p orthogonal to s π . Upon spe-cializing some known deep results to the separable setting, we show in Corollary3.3 that this occurs if and only if s π lies in the weak- ∗ closure of A inside the bidualof C ∗ e ( A ). We also relate this notion of peaking to the previous one (Theorem 3.4).We summarize these results. Theorem 1.1.
Let A be a separable unital operator algebra and let π be an ir-reducible finite-dimensional ∗ -representation of C ∗ e ( A ) . If s π ∈ A ⊥⊥ ⊂ C ∗ e ( A ) ∗∗ ,then π is a locally peaking representation for A . In particular, π is a boundaryrepresentation for A . We complement this theorem with some concrete examples where it applies (Ex-ample 1). For the rest of Section 3, we turn to a certain uniform version of peakingrepresentations called strongly peaking representations; these have been consideredby other authors [9],[43],[13],[22] in different contexts. Motivated by the foregoingdiscussion, we are interested in strongly peaking representations for A . Interest-ingly, this requirement is in fact equivalent to the a priori weaker stipulation of beinga strongly peaking representation for the larger algebra C ∗ e ( A ). Furthermore, whenC ∗ e ( A ) is assumed to be RFD, strongly peaking representations are automaticallyfinite-dimensional. The following is Corollary 3.7. Theorem 1.2.
Let A be a unital operator algebra such that C ∗ e ( A ) is RFD. Let π be a strongly peaking ∗ -representation for C ∗ e ( A ) . Then, π is a finite-dimensionalboundary representation for A . Much like for locally peaking representations, support projections can be usedto guarantee that a ∗ -representation is strongly peaking. Indeed, we show inLemma 3.8 that a finite-dimensional irreducible ∗ -representation is strongly peak-ing if its support projection is both closed and open in the sense of Akemann’snon-commutative topology (see Subsection 2.4). We also identify some sufficientconditions for this property to hold (Theorem 3.10, Example 3).Finally, in Section 4, in an attempt to better understand the subtleties inherentto Question 1, we analyze unital operator algebras whose C ∗ -envelope are knownto admit many finite-dimensional boundary representations. We start by obtaininga characterization of unital operator algebras admitting an RFD C ∗ -envelope; thisis accomplished in Theorem 4.1 with the aid of finite-dimensional approximationsof representations, in the spirit of work of Exel–Loring [26]. We then study unitaloperator algebras for which all boundary representations are finite-dimensional.We call such operator algebras C ∗ -liminal . The following is Corollary 4.7 and it INITE-DIMENSIONALITY IN THE NC CHOQUET BOUNDARY 5 summarizes our main results on this topic. Notably, the proof uses some recentdevelopments in matrix convexity [33].
Theorem 1.3.
Let A be a unital operator algebra. Consider the following state-ments. (i) The algebra A is C ∗ -liminal. (ii) Every matrix state of A can be dilated locally to a finite-dimensional ∗ -repre-sentation of C ∗ e ( A ) . (iii) The algebra C ∗ e ( A ) is RFD.Then, we have (i) ⇒ (ii) ⇒ (iii) . While we know that (iii) (ii), we do not know if the other implication can bereversed. 2. Operator algebraic preliminaries
Representations of C ∗ -algebras. The focal point of this paper is the ex-istence of certain ∗ -representations of C ∗ -algebras. In this subsection, we brieflyrecall various equivalence relations and topologies related to these objects. Thereader should consult [23, Chapter 3] for greater detail.Let A be a C ∗ -algebra. Recall that a closed two-sided ideal J ⊂ A is said tobe primitive if it is the kernel of an irreducible ∗ -representation. The collection ofprimitive ideals of A , denoted by Prim( A ), is known as its primitive ideal space .It can be endowed with the Jacobson topology. Given a subset S ⊂
Prim( A ), theclosure of S in this topology is the set of primitive ideals of A containing T s ∈S s .It follows that S is dense in Prim( A ) if and only if ∩ s ∈S s = { } . If for each s ∈ S we choose an irreducible ∗ -representation π s of A with s = ker π s , then we see that S is dense if and only if L s ∈S π s is injective.Given two irreducible ∗ -representations π and σ of A , we say that σ is weaklycontained in π and write σ ≺ π if ker π ⊂ ker σ . This is equivalent to the existenceof a ∗ -homomorphism θ : π ( A ) → σ ( A ) such that θ ◦ π = σ . We say that π and σ are weakly equivalent and write π ∼ σ if they are weakly contained in one another.It is well known that finite-dimensional irreducible ∗ -representations are minimal inthe partial order given by weak containment and in fact display significant rigidity. Lemma 2.1.
Let π and σ be two irreducible ∗ -representations of a C ∗ -algebra A .Assume that π is finite-dimensional and that σ ≺ π . Then, σ is finite-dimensionaland unitarily equivalent to π .Proof. By assumption, there is a ∗ -homomorphism θ : π ( A ) → σ ( A ) such that θ ◦ π = σ . Now, since π is irreducible and finite-dimensional, we see that π ( A ) is asimple C ∗ -algebra, so θ is a ∗ -isomorphism and σ ( A ) is a finite-dimensional simpleC ∗ -algebra. This forces θ to be implemented by some some unitary equivalence. (cid:3) The spectrum of A , denoted by b A , is the set of unitary equivalence classes ofirreducible ∗ -representations. Given an irreducible ∗ -representation π , we let [ π ] ∈ b A denote its unitary equivalence class. We can define a topology on b A by requiringthe natural map b A → Prim A , [ π ] ker π RAPHA¨EL CLOUˆATRE AND IAN THOMPSON to be continuous. It follows from this that the closure of the singleton [ π ] in b A isthe set { [ σ ] ∈ b A : σ ≺ π } . If A is a unital C ∗ -algebra, then S ( A ) denotes the state space. The pure statesare the extreme points of the convex set S ( A ). Given ψ ∈ S ( A ), we let ( σ ψ , H ψ , ξ ψ )be the corresponding GNS representation . That is to say, H ψ is a Hilbert spaceand σ ψ : A → B ( H ψ ) is a ∗ -representation with cyclic vector ξ ψ such that ψ ( t ) = h σ ψ ( t ) ξ ψ , ξ ψ i , t ∈ A . The state ψ is pure if and only if σ ψ is irreducible [23, Proposition 2.5.4]. If ∆ ⊂ b A ,then we let S ∆ ( A ) denote the set of (necessarily pure) states ψ with the propertythat [ σ ψ ] ∈ ∆. This is readily seen to coincide with the collection of states of theform t
7→ h π ( t ) η, η i , t ∈ A for some irreducible ∗ -representation π : A → B ( H π ) with [ π ] ∈ ∆ and some unitvector η ∈ H π . Lemma 2.2. [23, Theorem 3.4.10]
Let A be a unital C ∗ -algebra and let ∆ ⊂ b A bea subset. Then, ∆ is dense in b A if and only if S ∆ ( A ) is weak- ∗ dense in the set ofpure states. Residually finite-dimensional operator algebras.
Let A be an operatoralgebra. For a positive integer n , we let M n denote the n × n complex matrices.Likewise, M n ( A ) denotes the algebra of n × n matrices with entries in A . By a representation of A , we will always mean a completely contractive homomorphism π : A → B ( H π ) for some Hilbert space H π . For each n ∈ N , we let π ( n ) : M n ( A ) → B ( H ( n ) ) denote the natural ampliation of π .Let S be a set of representations of A and let M ⊂ A be a subspace. Then, S is said to be(a) separating for M if L π ∈S π is injective on M ;(b) completely norming for M if L π ∈S π is completely isometric on M .Note that if M is a C ∗ -algebra, then S is separating for M if and only if it iscompletely norming for M .We say that A is residually finite-dimensional (RFD) if there is a set of represen-tations of A on finite-dimensional Hilbert spaces that is completely norming for A .We wish to record some known characterizations of RFD C ∗ -algebras. For this pur-pose, we introduce some terminology. Let A be a C ∗ -algebra and let π : A → B ( H π )be a ∗ -representation. A net of ∗ -representations π λ : A → B ( H π ) , λ ∈ Λis said to be an approximation for π iflim λ π λ ( a ) ξ = π ( a ) ξ, a ∈ A , ξ ∈ H π . Here, the limit is taken with respect to the norm topology on H π . If in addition thespace π λ ( A ) H π is finite-dimensional for each λ ∈ Λ, then the net ( π λ ) is said to be a finite-dimensional approximation for π . Note that in this case, because π λ ( A ) H π isfinite-dimensional, it follows that π λ ( A ) is a finite-dimensional C ∗ -algebra for every λ ∈ Λ. INITE-DIMENSIONALITY IN THE NC CHOQUET BOUNDARY 7
Theorem 2.3.
Let A be a C ∗ -algebra. Then, the following statements are equiva-lent. (i) The algebra A is RFD. (ii) Every ∗ -representation of A admits a finite-dimensional approximation. (iii) Every irreducible ∗ -representation of A admits a finite-dimensional approxi-mation. (iv) The set of unitary equivalence classes of finite-dimensional irreducible ∗ -rep-resentations is dense in b A .Proof. (i) ⇔ (ii): This is [26, Theorem 2.4].(ii) ⇒ (iii): This is trivial.(iii) ⇒ (i): There is always a set of irreducible ∗ -representations of A whichis separating for A . It thus follows that there is a set of finite-dimensional ∗ -representations of A which is also separating for A , whence A is RFD.(i) ⇔ (iv): By definition of the topologies involved, we see that the set of unitaryequivalence classes of finite-dimensional irreducible ∗ -representations of A is densein b A if and only if there is a set of finite-dimensional irreducible ∗ -representationsof A which is separating for A . The latter is clearly equivalent to A being RFD. (cid:3) Another useful characterization that we require is the following.
Theorem 2.4.
Let A be a C ∗ -algebra. Let N ⊂ A be the subset consisting of thoseelements a ∈ A for which there is a finite-dimensional ∗ -representation π of A suchthat k a k = k π ( a ) k . Then, the following statements hold. (i) The algebra A is RFD if and only if N is dense in A . (ii) All irreducible ∗ -representations of A are finite-dimensional if and only if N = A .Proof. This is [18, Theorems 3.2 and 4.4]. (cid:3)
In [18], the authors call a C ∗ -algebra FDI if it satisfies (ii) above. In the unitalcase, this is the same class as the liminal (or CCR ) C ∗ -algebras [23, Section 4.2].There is yet another related notion that we require. Let A be an operatoralgebra. Following [17], we say that A is normed in finite-dimensions (NFD) iffor every n ∈ N and every A ∈ M n ( A ) there is a finite-dimensional Hilbert space H π and a representation π : A → B ( H π ) such that k π ( n ) ( A ) k = k A k . If A is aunital C ∗ -algebra, then it is well known that the irreducible ∗ -representations of theC ∗ -algebra of M n ( A ) are all unitarily equivalent to one of the form π ( n ) for someirreducible ∗ -representation π of A (see the last paragraph of [36, page 485]). Inview of this observation, it is an easy consequence of Theorem 2.4(ii) that a unitalC ∗ -algebra is NFD if and only if it is FDI.Finally, we point out that the full group C ∗ -algebra of the free group F is RFDbut not liminal, as it is infinite-dimensional and primitive [14].2.3. Biduals of operator algebras.
Some of our techniques will require us towork within biduals of operator algebras, so we recall some relevant basic factshere. Let A be a C ∗ -algebra. Then, the Banach space A ∗∗ can be given thestructure of a von Neumann algebra (see for instance [48, Theorem III.2.4] or [12,Theorem A.5.6]). Furthermore, if A ⊂ A is a subalgebra, then A ∗∗ can be viewed assubalgebra of A ∗∗ (see the proof of [12, Corollary 2.5.6]). Throughout, we identify RAPHA¨EL CLOUˆATRE AND IAN THOMPSON A with its canonical image in A ∗∗ , so in particular we may view A as a subalgebraof A ∗∗ .Let N be a von Neumann algebra with predual N ∗ . Let ι : N ∗ → N ∗ be thecanonical embedding. Let ψ : A → N be a bounded linear map. We let b ψ = ι ∗ ◦ ψ ∗∗ : A ∗∗ → N . Then, b ψ is a weak- ∗ continuous extension of ψ . In general, ψ ∗∗ and b ψ do notcoincide and it is important to distinguish between them; we will be consistentwith our notations to prevent confusion. Note however that ψ ∗∗ = b ψ when N isfinite-dimensional.Let H be a Hilbert space. Multiplication on the left or on the right by a fixedoperator is weak- ∗ continuous on B ( H ). Therefore, ( B ( H ) ∗ ) ⊥ ⊂ B ( H ) ∗∗ is a weak- ∗ closed two-sided ideal. Thus, by [47, Proposition I.10.5] there is a central projection z ∈ B ( H ) ∗∗ such that z B ( H ) ∗∗ = ( B ( H ) ∗ ) ⊥ . In particular, a functional ϕ ∈ B ( H ) ∗ lies in B ( H ) ∗ if and only if b ϕ ( ξ ) = b ϕ (( I − z ) ξ ) , ξ ∈ B ( H ) ∗∗ . Let K denote the ideal of compact operators on H . The following is well known,but we lack a precise reference. Lemma 2.5.
We have that K ⊥⊥ = ( I − z ) B ( H ) ∗∗ . In particular, if ( a n ) is a sequence of compact operators converging to in the weak- ∗ topology of B ( H ) , then ( a n ) converges to in the weak- ∗ topology of B ( H ) ∗∗ .Proof. Let ϕ ∈ K ⊥ . Now, K ⊥⊥ is a weak- ∗ closed two-sided ideal in B ( H ) ∗∗ , sothat ( I − z ) K ⊂ K ⊥⊥ . We conclude that the linear functional t b ϕ (( I − z ) t ) , t ∈ B ( H )is weak- ∗ continuous and annihilates K , and thus is identically zero. This meansthat ϕ ∈ (( I − z ) B ( H ) ∗∗ ) ⊥ . Hence, K ⊥ ⊂ (( I − z ) B ( H ) ∗∗ ) ⊥ or ( I − z ) B ( H ) ∗∗ ⊂ K ⊥⊥ . Conversely, given ϕ ∈ B ( H ) ∗ we may use [44, Theorem 8.4] to find a ∗ -representation π : B ( H ) → B ( H ′ ) and unit vectors v, w ∈ H ′ such that ϕ ( t ) = h π ( t ) v, w i , t ∈ B ( H ) . Splitting π into a direct sum according to the ideal K (see [7, page 15]), we see thatthere is ω ∈ B ( H ) ∗ and τ ∈ K ⊥ such that ϕ = ω + τ . If we assume further than ϕ ∈ ((1 − z ) B ( H ) ∗∗ ) ⊥ , then we see for k ∈ K that ϕ ( k ) = b ϕ ( z k ) = b ω ( z k ) + b τ ( z k ) = b τ ( z k ) = 0since zK ⊂ K ⊥⊥ . In other words, we have shown that(( I − z ) B ( H ) ∗∗ ) ⊥ ⊂ K ⊥ which implies K ⊥⊥ ⊂ ( I − z ) B ( H ) ∗∗ . (cid:3) INITE-DIMENSIONALITY IN THE NC CHOQUET BOUNDARY 9
Support projections of representations and non-commutative topol-ogy.
Let A be a C ∗ -algebra and let π : A → B ( H π ) be a ∗ -representation. Then, π ∗∗ : A ∗∗ → B ( H π ) ∗∗ is a weak- ∗ continuous ∗ -representation, and so is b π . Uponidentifying elements with their canonical images inside biduals, we see that π ∗∗ ( t ) = π ( t ) for every t ∈ A . Basic functional analytic arguments reveal that ker π ∗∗ =(ker π ) ⊥⊥ ; indeed, both of these objects coincide with (ran π ∗ ) ⊥ . Because A ∗∗ isa von Neumann algebra and ker π ∗∗ is a weak- ∗ closed two-sided ideal, there is acentral projection e π ∈ A ∗∗ such that e π A ∗∗ = ker π ∗∗ = (ker π ) ⊥⊥ . We call the projection s π = I − e π the support projection of π . It has the propertythat π ∗∗ is injective on s π A ∗∗ . In some simple situations, this projection can bereadily identified. Lemma 2.6.
Let A be a C ∗ -algebra and let p ∈ A be a central projection. Let π : A → A be the ∗ -representation defined as π ( a ) = ap, a ∈ A . Then, p is the support projection of π .Proof. A density argument shows that p is also central in A ∗∗ . The map ξ ξp, ξ ∈ A ∗∗ is thus a weak- ∗ continuous ∗ -representation agreeing with π on A , so we inferthat it coincides with π ∗∗ . Thus, ( I − p ) A ∗∗ ⊂ ker π ∗∗ = e π A ∗∗ and I − p ≤ e π .Conversely, we find e π p = π ∗∗ ( e π ) = 0so that e π ≤ I − p . This shows that s π = p . (cid:3) We recall here Akemann’s so-called hull-kernel structure , which can be thought ofas a noncommutative topology [2]. Let A be a C ∗ -algebra. A projection p ∈ A ∗∗ issaid to be open if there is an increasing net of positive contractions in A convergingto p in the weak- ∗ topology of A ∗∗ . A projection q ∈ A ∗∗ is closed if I − q is openin the previous sense.Elementary arguments show that p is open if and only if there is a closed leftideal J ⊂ A with J ⊥⊥ = A ∗∗ p [2],[3]. In particular, if π is a ∗ -representation of A , then its support projection s π is always closed. We will use this observationimplicitly throughout the paper.Next, we give a well-known two-sided version of a deep theorem of Hay [34,Theorem 4.2] (see also [10, Theorem 3.3] for a streamlined proof). Lemma 2.7.
Let A be a unital C ∗ -algebra and let A ⊂ A be a unital subalgebra.Let q ∈ A ∗∗ be a closed projection lying in A ⊥⊥ . Then, the two-sided ideal ( I − q ) A ∗∗ ( I − q ) ∩ A admits a contractive approximate unit.Proof. Let J = ( I − q ) A ∗∗ ∩ A . By [10, Theorem 3.3], J admits a left contractiveapproximate unit ( e i ) and J ⊥⊥ = ( I − q ) A ∗∗ . In turn, by [12, Proposition 2.5.8]we see that ( e i ) converges to ( I − q ) in the weak- ∗ topology of A ∗∗ . Thus, by[12, Proposition 2.5.8] and its proof we obtain that ( I − q ) A ∗∗ ( I − q ) ∩ A has both aleft and a right contractive approximate unit, and hence a contractive approximateunit. (cid:3) Purity, boundary representations and the C ∗ -envelope. In this paperwe are interested in unital operator algebras and their C ∗ -envelopes, the latterbeing defined herein. However, for technical reasons that will become apparent inSection 4, we need to introduce C ∗ -envelopes for the more general class of unitaloperator spaces.Let M ⊂ B ( H ) be a unital operator space and let ψ : M → B ( K ) be a unitalcompletely contractive map. Consider the operator system S = { a + b ∗ : a, b ∈ M} ⊂ B ( H ) . We note that C ∗ ( M ) = C ∗ ( S ). By [44, Proposition 3.5], there is a unique com-pletely positive map ˜ ψ : S → B ( K ) extending ψ . This basic fact is very useful, andshows that unital completely contractive maps on M correspond, in a natural way,to unital completely positive maps on S . We use this well-known correspondencetacitly in what follows. In particular, we import many important results from thesetting of operator systems and unital completely positive maps to that of unitaloperator spaces and unital completely contractive maps.Given a completely positive map ϕ : S → B ( K ), we write ϕ ≤ ˜ ψ if ˜ ψ − ϕ is alsocompletely positive. We say that ψ is pure if whenever ϕ ≤ ˜ ψ , there exists a scalar0 ≤ λ ≤ ϕ = λ ˜ ψ .If ψ : C ∗ ( M ) → B ( K ) is a unital completely positive map, then it is pure if andonly if its minimal Stinespring representation ( σ ψ , K ψ ) is irreducible [6, Corollary1.4.3]. Here, K ψ is a Hilbert space containing K and σ ψ : C ∗ ( M ) → B ( K ψ ) is aunital ∗ -representation such that ψ ( t ) = P K σ ψ ( t ) | K , t ∈ C ∗ ( M ) . See [44, Theorem 4.1] for more details about the existence of the Stinespring rep-resentation.Following [12], we say that a C ∗ -extension of M is a pair ( A , ι ) consisting ofa unital C ∗ -algebra A and a unital completely isometric linear map ι : M → A such that A = C ∗ ( ι ( M )). The C ∗ -envelope of M is a C ∗ -extension ( E , u ) withthe property that given any other C ∗ -extension ( A , ι ), there exists a surjective ∗ -homomorphism π : A → E with π ◦ ι = u . We usually abuse notation slightly andconsider just the C ∗ -algebra E to be the C ∗ -envelope, and thus denote it by C ∗ e ( M );in this case, we identify M with its image u ( M ) in E . It is readily seen that theC ∗ -envelope is essentially unique. The fact that it exists is a non-trivial result ofHamana [32]. Its universal property implies that the C ∗ -envelope is a quotient ofevery other C ∗ -extension through a quotient map that is completely isometric on M . For this reason, we think of the C ∗ -envelope as the minimal C ∗ -extension.Remarkably, when we start with a unital operator algebra A , then the map u : A → C ∗ e ( A ) above is actually multiplicative [12, Proposition 4.3.5]. In particular, u ( A ) is also a unital operator algebra in this case.To determine the C ∗ -envelope, the most commonly used technique is due toArveson and consists of finding enough ∗ -representations of C ∗ ( M ) with a certainuniqueness property. Let us be more precise. Let π : C ∗ ( M ) → B ( H π ) be a unital ∗ -representation. We say that it has the unique extension property with respectto M if the only unital completely positive extension of π | M to C ∗ ( M ) is π itself.Before proceeding, we record some elementary facts related to the unique extensionproperty. INITE-DIMENSIONALITY IN THE NC CHOQUET BOUNDARY 11
Lemma 2.8.
Let
M ⊂ B ( H ) be a unital operator space. The following statementshold. (i) Let π : C ∗ ( M ) → B ( H π ) be a unital ∗ -representation with the unique exten-sion property with respect to M . Let X ⊂ H π be a closed reducing subspacefor π (C ∗ ( M )) . Then, the unital ∗ -representation t π ( t ) | X , t ∈ C ∗ ( M ) also has the unique extension property with respect to M . (ii) Let π λ : C ∗ ( M ) → B ( H λ ) , λ ∈ Λ be a set of unital ∗ -representations with the unique extension property withrespect to M . Then, the unital ∗ -representation t M λ ∈ Λ π λ ( t ) , t ∈ C ∗ ( M ) also has the unique extension property with respect to M .Proof. (i) Let ψ : C ∗ ( M ) → B ( X ) be a unital completely positive map that satisfies ψ ( a ) = π ( a ) | X , a ∈ M . Define Ψ : C ∗ ( M ) → B ( H π ) asΨ( t ) = ψ ( t ) ⊕ ( π ( t ) | X ⊥ ) , t ∈ C ∗ ( M ) . Then, we see that Ψ is a unital completely positive map that agrees with π on M .By assumption, this means that Ψ = π and in particular ψ ( t ) = π ( t ) | X , t ∈ C ∗ ( M ) . (ii) This is [9, Proposition 4.4]. (cid:3) Let π : C ∗ ( M ) → B ( H π ) be a ∗ -representation. Consider its essential subspace E π = π (C ∗ ( M )) H π . Then, π is said to have the essential unique extension property with respect to M if the unital ∗ -representation ρ : C ∗ ( M ) → B ( E π ) defined as ρ ( t ) = π ( t ) | E π , t ∈ C ∗ ( M )has the usual unique extension property with respect to M . Lemma 2.9.
Let
M ⊂ B ( H ) be a unital operator space and let π : C ∗ ( M ) → B ( H π ) be a unital ∗ -representation with the unique extension property with respectto M . Let K be another Hilbert space and let V : H π → K be an isometry. Define π ′ : C ∗ ( M ) → B ( K ) as π ′ ( t ) = V π ( t ) V ∗ , t ∈ C ∗ ( M ) . Then, π ′ is a ∗ -representation with the essential unique extension property withrespect to M .Proof. Since V is an isometry, it is readily verified that π ′ is a ∗ -representation.Let U = P V H π V : H π → V H π . Then, U is unitary and with respect to thedecomposition K = V H π ⊕ ( V H π ) ⊥ we find π ′ ( t ) = U π ( t ) U ∗ ⊕ , t ∈ C ∗ ( M ) . It is easily verified that
U π ( · ) U ∗ is a unital ∗ -representation of C ∗ ( M ) with theunique extension property with respect to M , which immediately implies the de-sired statement. (cid:3) Due to important contributions of Muhly–Solel [42] and Dritschel–McCullough[25], it is known that unital ∗ -representations of C ∗ ( M ) with the unique extensionproperty with respect to M are always plentiful. In fact, more is true. As the nextresult shows, we can also require these ∗ -representations to be irreducible.If an irreducible ∗ -representation π of C ∗ ( M ) has the unique extension withrespect to M , then we say that π is a boundary representation for M on C ∗ ( M ).Recall that a unital completely contractive map ψ : M → B ( F ) is called a matrixstate whenever F is finite-dimensional. The following improves on [39, Theorem3.1] slightly. Theorem 2.10.
Let
M ⊂ B ( H ) be a unital operator space, let n ∈ N and let A ∈ M n ( M ) . Then, there is a boundary representation β : C ∗ ( M ) → B ( H β ) for M such that k A k = k β ( n ) ( A ) k . Furthermore, there is a finite-dimensional subspace
F ⊂ H β such that if we definea matrix state ψ : M → B ( F ) as ψ ( b ) = P F β ( b ) | F , b ∈ M then we have k A k = k ψ ( n ) ( A ) k . Proof.
By the proof of [39, Theorem 2.5], there is a two-dimensional Hilbert space E and a pure matrix state ϕ : M n ( M ) → B ( E ) such that k ϕ ( A ) k = k A k . In turn,by [20, Theorem 2.4] we find a boundary representation γ : C ∗ ( M n ( M )) → B ( H γ )for M n ( M ) with the property that ϕ ( B ) = P E γ ( B ) | E , B ∈ M n ( M ) . Note now that C ∗ ( M n ( M )) = M n (C ∗ ( M )). It follows from the last paragraph in[36, page 485] that there is an irreducible ∗ -representation β : C ∗ ( M ) → B ( H β )and a unitary operator U : H ( n ) β → H γ with the property that γ ( T ) = U β ( n ) ( T ) U ∗ , T ∈ M n (C ∗ ( M )) . By the theorem on [36, page 486], we conclude that β is a boundary representationfor M . Moreover, k β ( n ) ( A ) k = k γ ( A ) k ≥ k ϕ ( A ) k = k A k so in fact k β ( n ) ( A ) k = k A k .Next, we see that P U ∗ E β ( n ) ( A ) | U ∗ E = U ∗ P E U β ( n ) ( A ) | U ∗ E = U ∗ P E γ ( A ) U | U ∗ E = V ∗ ( P E γ ( A ) | E ) V where V = P E U | U ∗ E : U ∗ E → E is a unitary operator. Thus, we conclude that k P U ∗ E β ( n ) ( A ) | U ∗ E k = k ϕ ( A ) k = k A k . INITE-DIMENSIONALITY IN THE NC CHOQUET BOUNDARY 13
Finally, choose any finite-dimensional subspace
F ⊂ H β with the property that F ( n ) contains the two-dimensional subspace U ∗ E and define ψ : M → B ( F ) as ψ ( b ) = P F β ( b ) | F , b ∈ M . Then, we find k ψ ( n ) ( A ) k = k P F ( n ) β ( n ) ( A ) | F ( n ) k ≥ k P U ∗ E β ( n ) ( A ) | U ∗ E k = k A k as desired. (cid:3) By Theorem 2.10, there always exists a set ∆ of boundary representations for M on C ∗ ( M ) with the property that L β ∈ ∆ β is completely isometric on M (seealso [8, Theorem 7.1] and [20, Theorem 3.3]). Let Σ ∆ ⊂ C ∗ ( M ) denote the kernelof L β ∈ ∆ β . We may invoke [6, Theorem 2.2.3] to see that Σ ∆ coincides with theintersection of the kernels of all boundary representations for M on C ∗ ( M ), andis thus independent of ∆. This ideal is called the Shilov ideal of M in C ∗ ( M ). By[6, Proposition 2.2.4 and Theorem 2.2.5] we conclude that C ∗ e ( M ) is ∗ -isomorphicto C ∗ ( M ) / Σ ∆ . For this reason, we usually think of the collection of boundaryrepresentations of M as the non-commutative Choquet boundary of M . We alsonote here that the Shilov ideal of M in C ∗ e ( M ) is trivial [6, Proposition 2.2.4].If β is a boundary representation for M on C ∗ ( M ), then ker β contains the Shilovideal by [6, Proposition 2.2.3]. Therefore, there is an irreducible ∗ -representation ˜ β on C ∗ e ( M ) such that β = ˜ β ◦ q , where q is the quotient map on C ∗ ( M ) correspondingto the Shilov ideal. In other words, a boundary representation always “factorsthrough the C ∗ -envelope”.The next observation is simple but useful. Lemma 2.11.
Let M be a unital operator space and let ∆ be a set of irreducible ∗ -representations of C ∗ e ( M ) . Then, ∆ is completely norming for M if and only ifit is separating for C ∗ e ( M ) . In particular, there is a set of boundary representationsfor M on C ∗ e ( M ) which is separating for C ∗ e ( M ) .Proof. Let σ = L π ∈ ∆ π . If ∆ is completely norming for M , then ker σ is the Shilovideal of M in C ∗ e ( M ), which is trivial as mentioned above. Thus, σ is injective andhence ∆ is separating for C ∗ e ( M ). Conversely, if ∆ is separating for C ∗ e ( M ) then σ is injective and hence completely isometric. The second statement follows fromthe first and from Theorem 2.10. (cid:3) In the classical situation of a unital subspace M of continuous functions ona compact Hausdorff space, if M separates the points then it is known that theChoquet boundary is always dense in the Shilov boundary [45, Proposition 6.4].There are several reasonable candidates for a non-commutative analogue of theShilov boundary, and for all of them we have a similar density property. Theorem 2.12.
Let M be a unital operator space and let B ⊂ \ C ∗ e ( M ) denote theset of unitary equivalence classes of boundary representations for M . Then, thefollowing statements hold. (i) Let S B (C ∗ e ( M )) denote the set of states on C ∗ e ( M ) whose GNS representationis a boundary representation for M . Then, S B (C ∗ e ( M )) is weak- ∗ dense inthe set of pure states on C ∗ e ( M ) . (ii) The set { ker β : [ β ] ∈ B} is dense in Prim(C ∗ e ( M )) . (iii) The set B is dense in \ C ∗ e ( M ) . Proof.
Statement (ii) follows from the definition of the Jacobson topology on theprimitive ideal space along with Lemma 2.11. Furthermore, by definition of thetopology on \ C ∗ e ( M ), (ii) immediately implies (iii). Finally, (i) is always equivalentto (iii) by Lemma 2.2. (cid:3) Non-commutative peak points
The goal of this section is to introduce an important tool to detect “isolated”irreducible ∗ -representations, and ultimately to exhibit finite-dimensional boundaryrepresentations. The rough idea is to view certain irreducible ∗ -representations asanalogues of classical peak points. Let us be more precise.Let A be a C ∗ -algebra and let π : A → B ( H π ) be an irreducible ∗ -representation.Then, π is said to be a peaking representation for A if there is n ∈ N and T ∈ M n ( A )with the property that for every irreducible ∗ -representation σ : A → B ( H σ ) notunitarily equivalent to π we have k π ( n ) ( T ) k > k σ ( n ) ( T ) k . In this case, we say that T peaks at π . For our purposes, we will need a slightweakening of this notion, which in a sense is “local”. We say that T peaks locally at π if for every irreducible ∗ -representation σ : A → B ( H σ ) not unitarily equivalentto π and every finite-dimensional subspace F ⊂ H σ we have k π ( n ) ( T ) k > k P F ( n ) σ ( n ) ( T ) | F ( n ) k . If S ⊂ A is a subset and there is n ∈ N along with T ∈ M n ( S ) such that T peakslocally at π , then π is said to be a locally peaking representation for S .As a consequence of Theorem 3.4 below, we will also exhibit an example of anatural C ∗ -algebra for which there is a locally peaking representation but there areno peaking representations (see Example 2). Hence, our local notion is genuinelymore flexible than the original one.The reader may wonder why unitary equivalence is used instead of weak equiv-alence in the previous definitions; this alternative would also encode some kind ofpeaking phenomenon. There are three reasons motivating our choice. First, whatwe are hoping to accomplish using locally peaking representations is to identifyboundary representations (see Theorem 3.1). Using weak equivalence rather thanunitary equivalence in our definition would result in a class of representations whichdoes not necessarily consist entirely of boundary representations. Indeed, this canbe inferred from [21, Example 6.6.3]; the authors are grateful to Matthew Kennedyfor pointing this out. Second, guided by Question 1, in this paper our focus is onfinite-dimensional representations, in which case unitary equivalence coincides withweak equivalence anyway, by Lemma 2.1. Therefore, the proposed alternative ver-sion of (locally) peaking projections would reduce to the original one in our case ofinterest. The third reason is one of consistency: as mentioned above, previous oc-currences of peaking representations have used unitary equivalence [9],[43],[13],[22]so we opt here to follow this trend.We now arrive at an important result, which illustrates why the notion of peakingrepresentation is relevant for Question 1. The reader should compare it with [9,Theorem 7.2] (see also [43, Theorem 5.2] for a related result). INITE-DIMENSIONALITY IN THE NC CHOQUET BOUNDARY 15
Theorem 3.1.
Let
A ⊂ B ( H ) be a unital operator algebra and let π be an irre-ducible ∗ -representation of C ∗ ( A ) . Assume that π is a locally peaking representationfor A . Then, π is a boundary representation for A .Proof. There is n ∈ N along with A ∈ M n ( A ) such that k A k ≥ k π ( n ) ( A ) k > k P F ( n ) σ ( n ) ( A ) | F ( n ) k for every irreducible ∗ -representation σ : C ∗ ( A ) → B ( H σ ) not unitarily equivalentto π and every finite-dimensional subspace F ⊂ H σ . In view of Theorem 2.10, thereis a boundary representation β : C ∗ ( A ) → B ( H β ) for A and a finite-dimensionalsubspace G ⊂ H β such that k P G ( n ) β ( n ) ( A ) | G ( n ) k = k A k . Necessarily, this implies that π is unitarily equivalent to β , and hence is a boundaryrepresentation. (cid:3) In light of this result, it is desirable to efficiently identify locally peaking repre-sentations. This will be accomplished in the following subsection with the aid ofcertain special projections.3.1.
Peaking projections.
As part of our overarching analogy with classical peakpoints, the previous developments replaced points with irreducible ∗ -representationsto arrive at the notion of peaking representations. Another non-commutative inter-pretation of peak points replaces points with projections in the bidual; these shouldbe thought of as analogues of characteristic functions.In this subsection, we show how certain peaking projections can be used toproduce locally peaking representations. This is beneficial for us, as peaking pro-jections have been thoroughly studied since the appearance of [34] (see also [10]and the references therein). In particular, there are known deep results that char-acterize these projections via concrete conditions that we will manage to verify insome cases.We recall the definition (see [34, Definition 3.5 and Theorem 5.1]). A closedprojection q ∈ A ∗∗ is a peaking projection if there is a contraction t ∈ A with theproperty that tq = q and k tp k < p ∈ A ∗∗ is a closed projection with pq = 0. In this case, we say that t peaks at q . Note then that necessarily k t k = 1.In addition, if there is a ∗ -representation π of A such that s π = q , then π ( t ) = π ∗∗ ( t s π ) = π ∗∗ ( s π ) = I. We aim to give a refinement of a deep non-commutative counterpart to the classicalGlicksberg peak point theorem [29, Theorem II.12.7] based on non-trivial workof Hay [34], Blecher–Hay–Neal [11] and Read [46]. The sharper conclusion thatwe obtain upon specializing to the separable setting does not seem to have beenpreviously recorded, although it may have been known to experts. The fact thatseparability allows for a cleaner result is a reflection of the familiar fact that thetheory of peak points for uniform algebras is much more definitive in the case wherethe underlying compact Hausdorff space is metrizable; see for instance [29, LemmaII.12.2].
Theorem 3.2.
Let
A ⊂ B ( H ) be a separable unital operator algebra and let q ∈A ⊥⊥ be a closed projection. Then, there is an element t ∈ A peaking at q such that / I ≤ t ∗ t ≤ I . Furthermore, t has the property that k b ψ ( q ) k = 1 whenever ψ is amatrix state on C ∗ ( A ) satisfying k ψ ( t ∗ t ) k = 1 .Proof. We may assume that q / ∈ A , for otherwise we may simply choose t = q . ByLemma 2.7, the algebra ( I − q ) A ∗∗ ( I − q ) ∩ A has a contractive approximate unit.By [46, Theorem 1.1], there is in fact a contractive approximate unit ( b j ) j ∈ J suchthat k I − b j k ≤ j ∈ J . Observe that b j ( I − q ) = b j or b j q = 0 forevery j ∈ J . Next, because A is norm separable, so is the subset { b j : j ∈ J } .Let Γ ⊂ { b j : j ∈ J } be a countable dense subset. Note that q / ∈ A and ( b j )converges to I − q in the weak- ∗ topology of B ( H ) ∗∗ [12, Proposition 2.5.8], so thatΓ is infinite and we write Γ = { c n : n ∈ N } . Since ( I − c n ) ∗ ( I − c n ) ≤ I , we inferRe c n ≥ n ≥
1. Define t = 12 I + ∞ X n =1 n +1 ( I − c n ) ∈ A . We conclude that k t k ≤ tq = q . Note that t ∗ t = 14 I + ∞ X n =1 n +1 ( I − Re c n ) + ∞ X n =1 n +1 ( I − c n ) ! ∗ ∞ X n =1 n +1 ( I − c n ) ! . Since c n is a contraction, we see that I − Re c n ≥ n ≥
1, whence t ∗ t ≥ / I . Moreover, if ψ is a matrix state on C ∗ ( A ) then0 ≤ I − ψ (Re c n ) ≤ I, n ≥ k ψ ( t ∗ t ) k ≤
14 + ∞ X n =1 n +1 k I − ψ (Re c n ) k + ∞ X n =1 n +1 ! ≤
14 + ∞ X n =1 n +1 + 14 = 1 . Hence k ψ ( t ∗ t ) k = 1 holds only when k − ψ (Re c n ) k = 1 for every n ≥
1. Because( b j ) converges to I − q in the weak- ∗ topology of B ( H ) ∗∗ , there is a subsequenceof ( ψ ( c n )) that converges to b ψ ( I − q ) ≥ norm , seeing as ψ is a matrix state.Consequently, a subsequence of ( ψ (Re c n )) converges to b ψ ( I − q ) in norm, and k b ψ ( q ) k ≥ lim inf n →∞ k I − ψ (Re c n ) k . We conclude that k b ψ ( q ) k = 1 whenever k ψ ( t ∗ t ) k = 1. Invoking [34, Theorem 5.1]we see that t peaks at q . (cid:3) The previous result allows us to give characterizations, in the separable setting,of peaking projections.
Corollary 3.3.
Let
A ⊂ B ( H ) be a separable unital operator algebra and let q ∈ B ( H ) ∗∗ be a closed projection. Then, there is a contraction t ∈ A that peaks at q ifand only if q ∈ A ⊥⊥ .Proof. This follows upon combining [34, Proposition 5.6] with Theorem 3.2. (cid:3)
INITE-DIMENSIONALITY IN THE NC CHOQUET BOUNDARY 17
Peaking support projections.
We now return to our analysis of peakingrepresentations. Our first task is to show that irreducible ∗ -representations witha peaking support projection must be locally peaking. This is consistent with theclassical commutative setting. Indeed, let X be a compact metric space and let A ⊂ C ( X ) be a uniform algebra. For ξ ∈ X , we let π ξ : C ( X ) → C be thecharacter of evaluation at ξ . It is well known that s π ξ s π ξ ′ = 0 whenever ξ = ξ ′ (seefor instance [15, Proposition 2.3]). In particular, we see that if some contraction a ∈ A peaks at s π ξ , then ξ is a peak point for the function a , which is equivalentto a peaking at π ξ . A similar statement holds in the general non-commutativecontext, at least locally. Theorem 3.4.
Let
A ⊂ B ( H ) be a separable unital operator algebra and let π be a finite-dimensional irreducible ∗ -representation of C ∗ ( A ) such that s π ∈ A ⊥⊥ .Then, π is a locally peaking representation for A .Proof. Let a ∈ A be the contraction obtained from an application of Theorem 3.2to the closed projection s π ∈ A ⊥⊥ . In particular, a ∈ A peaks at s π so that1 = k a k = k π ( a ) k . Let σ : C ∗ ( A ) → B ( H σ ) be an irreducible ∗ -representation and let F ⊂ H σ be afinite-dimensional subspace. Define a matrix state ψ : C ∗ ( A ) → B ( F ) as ψ ( t ) = P F σ ( t ) | F , t ∈ C ∗ ( A ) . Assume that k ψ ( a ) k = 1. By the Schwarz inequality [44, Proposition 3.3], we inferthat k ψ ( a ∗ a ) k = 1, whence k b ψ ( s π ) k = 1 by choice of a . Now, it is readily verifiedthat b ψ ( ξ ) = P F b σ ( ξ ) | F , ξ ∈ C ∗ ( A ) ∗∗ . In particular, we infer that k b σ ( s π ) k = 1. Since σ is assumed to be irreducible,so is b σ . In turn, since s π is central we must have that b σ ( s π ) = I . This impliesthat ker π ∗∗ ⊂ ker b σ , and consequently ker π ⊂ ker σ . In other words, σ is weaklycontained in π . By Lemma 2.1 we conclude that π is unitarily equivalent to σ . Thisshows that a peaks locally at π . (cid:3) We now wish to give a non-trivial example where the previous result applies.
Example 1.
Let H = ⊕ ∞ m =1 C m and let A = Q ∞ m =1 M m . For each m ∈ N , we let π m : A → M m denote the coordinate projection. Let K ⊂ A denote the ideal ofcompact operators in A , so that K = ⊕ ∞ m =1 M m . Fix N ∈ N and let χ N = (0 , , . . . , I N , , . . . ) ∈ K . Let ( K r ) be a sequence in K converging to 0 in the weak- ∗ topology of B ( H ). Let A ⊂ A be a separable unital subalgebra containing χ N + K r for every r ∈ N . Itfollows from Lemma 2.5 that ( χ N + K r ) converges to χ N in the weak- ∗ topologyof B ( H ) ∗∗ , whence χ N ∈ A ⊥⊥ .Let ρ = π N | C ∗ ( A ) : C ∗ ( A ) → M N . We claim that χ N = s ρ . First, we note that ρ is finite-dimensional so that ρ ∗∗ = b ρ . We find ρ ∗∗ ( χ N ) = lim r →∞ ρ ( χ N + K r ) = I whence s ρ ≤ χ N . Hence, s ρ ∈ χ N C ∗ ( A ) ∗∗ . Since π N | χ N A is isometric, so is( π ∗∗ N ) | χ N A ∗∗ . Because π ∗∗ N ( χ N ) = c π N ( χ N ) = π N ( χ N ) = I = b ρ ( s ρ ) = ρ ∗∗ ( s ρ ) = π ∗∗ N ( s ρ ) we must have s ρ = χ N , as claimed.Finally, if A is chosen to be rich enough so that π N (C ∗ ( A )) = M N , then ρ isirreducible, and Theorem 3.4 implies that ρ is a locally peaking representation for A . (cid:3) As another application of Theorem 3.4, we can also show how the notion of alocally peaking representation is genuinely weaker than that of a peaking represen-tation.
Example 2.
Let H denote the classical Hardy space on the open unit disc D ⊂ C .This is the Hilbert space of holomorphic functions f : D → C that can be writtenas f ( w ) = ∞ X n =0 a n w n , w ∈ D with ∞ X n =0 | a n | < ∞ . Let A ( D ) denote the disc algebra, which is the closed subalgebra of C ( D ) consistingof those functions that are holomorphic on D . In this example, we will freely usemany classical facts about these objects. The reader is referred to [35],[24],[1] forgreater detail.Given ϕ ∈ A ( D ), the corresponding multiplication operator M ϕ on B ( H ) isknown to be bounded with k M ϕ k = k ϕ k . In fact, there is a unital completelyisometric homomorphism Θ : A ( D ) → B ( H ) defined asΘ( ϕ ) = M ϕ , ϕ ∈ A ( D ) . Let T = C ∗ (Θ( A ( D ))). Then, this C ∗ -algebra contains the ideal of compact op-erators on H and the corresponding quotient T / K can be naturally identified ∗ -isomorphically with C ( T ) via a map sending M z + K to z (here, by z we denotethe identity function). Using this identification, for each ζ ∈ T we let χ ζ : T → C be the character defined as χ ζ ( t ) = ( t + K )( ζ ) , t ∈ T . Basic representation theory of C ∗ -algebras implies that up to unitary equivalence,the irreducible ∗ -representations of T are precisely the identity representation andthe characters χ ζ for ζ ∈ T .It is readily seen that T has simply no peaking representation for Θ( A ( D )) (seethe discussion following [39, Remark 3.4]). We now show on the other hand that χ is a locally peaking representation for Θ( A ( D )). To see this, define the contraction a = 12 ( I + M z ) ∈ Θ( A ( D )) . We claim that a peaks on s χ . First note that χ ( a ) = 1 so that I − a ∈ ker χ .In particular, ( I − a ) s χ = 0 so that a s χ = s χ . To show that a peaks on s χ ,by virtue of [34, Theorem 5.1] it suffices to fix a pure state ψ on T with ψ = χ and show that ψ ( a ∗ a ) <
1. In view of the above description of the irreducible ∗ -representations of T , we infer that either ψ = χ ζ for some ζ ∈ T \ { } or there isa unit vector h ∈ H such that ψ ( t ) = h th, h i , t ∈ T . INITE-DIMENSIONALITY IN THE NC CHOQUET BOUNDARY 19
Assume first that ψ = χ ζ for some ζ ∈ T \ { } . It is readily verified that ψ ( a ∗ a ) = | ζ | < . Next, let h ∈ H be a unit vector and assume that ψ ( t ) = h th, h i , t ∈ T . If ψ ( a ∗ a ) = 1, then the Cauchy-Schwarz inequality implies that a ∗ ah = h . Bydefinition of a , using that M z is an isometry we infer that12 ( M z + M ∗ z ) h = h. Since M z h = zh and M ∗ z h = ( h − h (0)) /z , the previous equation is equivalent to( w − h ( w ) = h (0) , w ∈ D . An elementary calculation reveals that the condition h ∈ H forces h = 0, which isabsurd. Hence ψ ( a ∗ a ) <
1. We conclude that a indeed peaks at s χ . By Corollary3.3 and Theorem 3.4, we conclude that χ is a locally peaking representation for A . (cid:3) Strongly peaking representations.
We now turn to a uniform version ofthe notion of peaking representation. Let A be a C ∗ -algebra and let π be anirreducible ∗ -representation of A . We let U π = { [ σ ] ∈ b A : [ σ ] = [ π ] } . Let n ∈ N and T ∈ M n ( A ). Then, T is said to peak strongly at π if k π ( n ) ( T ) k > sup [ σ ] ∈U π k σ ( n ) ( T ) k . If S ⊂ A is a subset and there is n ∈ N along with T ∈ M n ( S ) that peaks stronglyat π , then π is said to be a strongly peaking representation for S .We remark here that while strongly peaking representations are a major themeof the recent paper [22], there is seemingly very little overlap between that paperand our current work. We refer the reader to [22] for connections between theserepresentations and certain uniqueness properties of so-called “fully compressed”operator systems.If A is a uniform algebra on some compact metric space X , then the peakingrepresentations for A are precisely the characters of evaluation at points in theChoquet boundary [29, Theorem 11.3], so in particular they exist in abundance.On the other hand, the continuity of the functions in C ∗ ( A ) implies that stronglypeaking representations for C ∗ ( A ) cannot exist unless X has isolated points. Weconclude from this that strongly peaking representations may be quite rare. Whenthey do exist however, then enjoy many useful properties, as we shall see below.We start with an observation that can be extracted from the proof of [22, Corollary2.6]. Lemma 3.5.
Let A be a unital operator algebra and let π be an irreducible ∗ -representation of C ∗ e ( A ) . Then, π is a strongly peaking representation for A if andonly if it is a strongly peaking representation for C ∗ e ( A ) . Proof.
Assume that π is a strongly peaking representation for C ∗ e ( A ). Thus, thereis n ∈ N and T ∈ M n (C ∗ e ( A )) with the property that k π ( n ) ( T ) k > sup [ σ ] ∈U π k σ ( n ) ( T ) k . For each u ∈ U π , let σ u be an irreducible ∗ -representation of C ∗ e ( A ) such that[ σ u ] = u . Then, the ∗ -representation L u ∈U π σ u is not completely isometric onC ∗ e ( A ) and hence it is not injective either. By Lemma 2.11, we conclude thatthe set { σ u : u ∈ U π } is not completely norming for A , so there is m ∈ N and A ∈ M m ( A ) such that sup [ σ ] ∈U π k σ ( m ) ( A ) k < k A k . On the other hand, the set { π } ∪ { σ u : u ∈ U π } is clearly completely norming forC ∗ e ( A ), so we must have that k A k = k π ( m ) ( A ) k . We conclude that A peaks stronglyat π . The converse is trivial. (cid:3) Recall that in the setting of Question 1, we are dealing with RFD C ∗ -algebras. In-terestingly, strongly peaking representations must necessarily be finite-dimensionalin that case. In fact, we can show even more. Theorem 3.6.
Let A be an RFD C ∗ -algebra and let π be an irreducible ∗ -repre-sentation. Assume that there is n ∈ N and T ∈ M n ( A ) with the property that k π ( n ) ( T ) k > sup [ σ ] ∈W π k σ ( n ) ( T ) k where W π = { [ σ ] ∈ b A : σ π } . Then, π is finite-dimensional.Proof. Write π : A → B ( H π ) and choose ε > k π ( n ) ( T ) k ≥ (1 + ε ) k σ ( n ) ( T ) k , σ ∈ W π . By virtue of Theorem 2.3, there is a net of ∗ -representations π λ : A → B ( H π ) , λ ∈ Λsuch that π λ ( A ) is finite-dimensional for every λ ∈ Λ andlim λ π λ ( s ) ξ = π ( s ) ξ, s ∈ A , ξ ∈ H π . Choose 0 < δ < δ (1 + ε ) >
1. There is µ ∈ Λ such that k π ( n ) µ ( T ) k ≥ δ k π ( n ) ( T ) k . Since π µ ( A ) is finite-dimensional, there are finite-dimensional irreducible ∗ -represen-tations ρ , . . . , ρ r of A with the property that π µ ( s ) ρ ( s ) ⊕ ρ ( s ) ⊕ . . . ⊕ ρ r ( s ) , s ∈ A is a ∗ -isomorphism. Hence k π ( n ) µ ( T ) k = max ≤ j ≤ r k ρ ( n ) j ( T ) k . Thus, there is 1 ≤ k ≤ r such that k ρ ( n ) k ( T ) k ≥ δ k π ( n ) ( T ) k whence k ρ ( n ) k ( T ) k ≥ δ (1 + ε ) k σ ( n ) ( T ) k , σ ∈ W π . INITE-DIMENSIONALITY IN THE NC CHOQUET BOUNDARY 21
Since δ (1 + ε ) >
1, we infer that [ ρ k ] / ∈ W π whence ρ k is weakly equivalent to π .Because ρ k is finite-dimensional, so must be π . (cid:3) In the notation of the previous result, we see that W π ⊂ U π , so Theorem 3.6applies in particular to strongly peaking representations. We now obtain a conse-quence which addresses Question 1. Corollary 3.7.
Let A be a unital operator algebra such that C ∗ e ( A ) is RFD. Let π be a strongly peaking ∗ -representation for C ∗ e ( A ) . Then, π is a finite-dimensionalboundary representation for A .Proof. Simply combine Theorem 3.1, Lemma 3.5 and Theorem 3.6. (cid:3)
In view of Question 1 and of the previous result, it is worthwhile to track downstrongly peaking representations among the finite-dimensional ones. We recordnext a useful observation: a finite-dimensional irreducible ∗ -representation of a C ∗ -algebra A is necessarily strongly peaking when its support projection belongs to A itself (and not only to the larger bidual A ∗∗ ). Lemma 3.8.
Let A be a C ∗ -algebra. Let π be a finite-dimensional irreducible ∗ -representation of A with support projection s π lying in A . Then, s π peaks stronglyat π .Proof. Let σ be an irreducible ∗ -representation of A which is not unitary equivalentto π . By Lemma 2.1, we see that ker π is not contained in ker σ , whence σ ( s π ) = I as( I − s π ) A ∗∗ = ker π ∗∗ . Since σ is irreducible and s π is central, this forces σ ( s π ) = 0.We conclude that k π ( s π ) k = 1 > [ σ ] ∈U π k σ ( s π ) k so s π peaks strongly at π . (cid:3) The requirement that the support projection lies in A is equivalent to it beingsimultaneously closed and open [2, Theorem II.18] in the non-commutative topologydiscussed in Subsection 2.4.We also note that there are known examples of unital operator algebras A and ofinjective, infinite-dimensional, irreducible ∗ -representations π of C ∗ e ( A ) that are notboundary representations for A [21, Example 6.6.3]. In this case, s π = I ∈ C ∗ e ( A ),so in light of Theorem 3.1 and Lemma 3.5, we see that the statement of Lemma3.8 cannot be strengthened to cover infinite-dimensional representations.We now give an elementary application. Corollary 3.9.
Let A be a unital operator algebra such that C ∗ e ( A ) contains a non-zero finite-dimensional closed two-sided ideal. Then, there is a finite-dimensionalstrongly peaking boundary representation for A .Proof. Let J ⊂ C ∗ e ( A ) be a non-zero finite-dimensional closed two-sided ideal. Let J ′ be a minimal non-zero closed two-sided ideal in J . There is a central projection e ∈ C ∗ e ( A ) such that J = C ∗ e ( A ) e . ThusC ∗ e ( A ) J ′ = C ∗ e ( A ) e J ′ = JJ ′ ⊂ J ′ and J ′ C ∗ e ( A ) = J ′ e C ∗ e ( A ) = J ′ J ⊂ J ′ . We infer that J ′ is a minimal non-zero closed two-sided ideal of C ∗ e ( A ), which isstill finite-dimensional. Therefore, there is a central projection z ∈ C ∗ e ( A ) such that J ′ = C ∗ e ( A ) z . Consider the unital ∗ -representation π : C ∗ e ( A ) → J ′ defined as π ( t ) = tz, t ∈ C ∗ e ( A ) . By Lemma 2.6 we obtain that s π = z . Now, J ′ is a simple finite-dimensional C ∗ -algebra so there is a positive integer n and a ∗ -isomorphism σ : J ′ → M n . Then,the support projection for the irreducible finite-dimensional ∗ -representation σ ◦ π is simply z . By Lemma 3.8, we see that z peaks strongly at σ ◦ π . Therefore, σ ◦ π is a boundary representation for A by Theorem 3.1. Moreover, σ ◦ π is a stronglypeaking representation for A by virtue of Lemma 3.5. (cid:3) The next result gives a sufficient condition for a support projection to be closedand open. Given a bounded linear operator T on a Hilbert space, we denote itsspectrum by spec( T ). Theorem 3.10.
Let A be a unital C ∗ -algebra and let ∆ be a set of irreducible ∗ -representations of A which is separating for A . Assume that there is a positiveelement t ∈ A and a finite-dimensional ∗ -representation π ∈ ∆ with the property min spec( π ( t )) > sup σ ∈ ∆ ,σ π k σ ( t ) k . Then, the support projection of π lies in A and peaks strongly at π .Proof. Upon scaling if necessary, we may assume that k π ( t ) k = 1. Put r = sup σ ∈ ∆ ,σ π k σ ( t ) k and R = min { λ : λ ∈ spec( π ( t )) } . Thus, we have 0 ≤ r < R ≤
1. Choose a continuous function f : [0 , → [0 ,
1] withthe property that f = 1 on the interval [ R,
1] and f = 0 on the interval [0 , r ]. If σ is an irreducible ∗ -representation of A with σ ∼ π , then spec( π ( t )) = spec( σ ( t )).Since ∆ is assumed to be separating for A , we conclude that spec( t ) coincides withthe closure of spec( π ( t )) ∪ [ σ ∈ ∆ ,σ π spec( σ ( t )) . Thus, spec( t ) ⊂ [0 , r ] ∪ [ R, . If we let p = f ( t ), then we see that p is a self-adjoint projection in A . Furthermore,we have π ( p ) = π ( f ( t )) = f ( π ( t )) = I since f = 1 on spec( π ( t )) ⊂ [ R, σ ∈ ∆ is such that σ π , then f = 0 on spec( σ ( t )) ⊂ [0 , r ] so σ ( p ) = σ ( f ( t )) = f ( σ ( t )) = 0 . We also infer that π ( ps − sp ) = π ( s ) − π ( s ) = 0and σ ( ps − sp ) = 0 − σ ∈ ∆ , σ π and every s ∈ A . Hence, using that ∆ is separating for A , we conclude that p is a non-zero central projection and A p ⊂ T σ ∈ ∆ ,σ π ker σ .Observe next that ker M σ ∈ ∆ ,σ ∼ π σ = ker π. INITE-DIMENSIONALITY IN THE NC CHOQUET BOUNDARY 23
Using once again that ∆ is separating for A , we see that A p is ∗ -isomorphic to M σ ∈ ∆ σ ! ( A p ) ∼ = M σ ∈ ∆ ,σ ∼ π σ ( A p ) . By Lemma 2.1, we see that every σ ∈ ∆ with σ ∼ π is in fact unitarily equivalentto π , whence A p is ∗ -isomorphic to π ( A p ), and in particular is a non-zero finite-dimensional closed two-sided ideal.Since π ( p ) = I we have s π ≤ p . In particular, s π lies in A ∗∗ p . This last setcoincides with the weak- ∗ closure of A p inside of A ∗∗ . But A p is finite-dimensionaland hence weak- ∗ -closed already, so that s π ∈ A p ⊂ A . We conclude from Lemma3.8 that s π peaks strongly at π . (cid:3) We close this section with another application of our ideas. This example isrelated to [22, Theorem 3.9], which is obtained by other means.
Example 3.
Let A be a unital operator algebra with the property that C ∗ e ( A ) isRFD. We may thus assume that there is a set { r λ : λ ∈ Λ } of positive integers withthe property that A ⊂ C ∗ e ( A ) ⊂ Y λ ∈ Λ M r λ and that each natural projection π λ : C ∗ e ( A ) → M r λ is surjective. Hence, each π λ is a finite-dimensional irreducible ∗ -representation, and the set ∆ = { π λ : λ ∈ Λ } isseparating for C ∗ e ( A ). Assume also that there is a positive element t = ( t λ ) λ ∈ Λ ∈ C ∗ e ( A ) such that there is λ ∈ Λ withmin spec( t λ ) > sup λ = λ k t λ k . Then, we may apply Theorem 3.10 to conclude that the support projection of π λ belongs to C ∗ e ( A ) and peaks strongly at π λ . By Lemma 3.5, we see that π λ isa strongly peaking representation for A . In particular, we see that π λ is a finite-dimensional boundary representation for A by Theorem 3.1. (cid:3) RFD C ∗ -envelopes and C ∗ -liminality In this section, we study unital operator spaces whose C ∗ -envelopes are assumedto admit a large supply of a finite-dimensional boundary representations. Our firstresult is a characterization of the residual finite-dimensionality of the C ∗ -envelope,in the spirit of Theorem 2.3. Theorem 4.1.
Let M be a unital operator space. Consider the following state-ments. (i) The algebra C ∗ e ( M ) is RFD. (ii) Every boundary representation for M on C ∗ e ( M ) admits a finite-dimensionalapproximation. (iii) Every boundary representation for M on C ∗ e ( M ) admits a finite-dimensionalapproximation consisting of ∗ -representations with the essential unique exten-sion property with respect to M . (iv) The set of unitary equivalence classes of finite-dimensional boundary repre-sentations for M is dense in \ C ∗ e ( M ) . (v) There is a set of finite-dimensional boundary representations for M which isseparating for C ∗ e ( M ) .Then, we have that (v) ⇐⇒ (iv) ⇐⇒ (iii) = ⇒ (ii) ⇐⇒ (i) .Proof. (v) = ⇒ (iv): This is an immediate consequence of the definition of thetopology on the spectrum of C ∗ e ( M ).(iv) = ⇒ (iii): This argument closely follows the ideas of [26]. Let F denotethe set of pure states on C ∗ e ( M ) whose GNS representation is a finite-dimensionalboundary representation for M . By assumption, we may invoke Lemma 2.2 to findthat F is weak- ∗ dense in the pure states on C ∗ e ( M ).Let π : C ∗ e ( M ) → B ( H ) be a boundary representation for M . For our purposes,it is no loss of generality to assume that H is infinite-dimensional. Let ξ ∈ H be aunit vector. Let ϕ : C ∗ e ( M ) → C be the state defined as ϕ ( t ) = h π ( t ) ξ, ξ i , t ∈ C ∗ e ( M ) . Since π is irreducible, ϕ is a pure state. By the previous paragraph, there is a net( ψ α ) of states in F that converges to ϕ in the weak- ∗ topology. Let σ α : C ∗ e ( M ) → B ( H α ) be the GNS representation of ψ α . Then, σ α is a finite-dimensional boundaryrepresentation for M . Arguing as in the proof of [26, Theorem 2.4], we find anisometry V α : H α → H such thatlim α V α σ α ( t ) V ∗ α h = π ( t ) h, t ∈ C ∗ e ( M ) , h ∈ H . By Lemma 2.9, we see that the net ( V α σ α ( · ) V ∗ α ) α is a finite-dimensional approx-imation for π consisting of ∗ -representations with the essential unique extensionproperty with respect to M .(iii) = ⇒ (v): Let t ∈ C ∗ e ( M ) be a non-zero element. By Lemma 2.11, there is aboundary representation π : C ∗ e ( M ) → B ( H π ) for M with π ( t ) = 0. Let ξ ∈ H π be a unit vector such that π ( t ) ξ = 0. By assumption, there is a finite-dimensionalapproximation π λ : C ∗ e ( M ) → B ( H π ) , λ ∈ Λfor π consisting of ∗ -representations with the essential unique extension propertywith respect to M . For each λ ∈ Λ, let E λ = π λ (C ∗ e ( M )) H π and note that thisis a finite-dimensional subspace. Let ρ λ : C ∗ e ( M ) → B ( E λ ) be the unital finite-dimensional ∗ -representation defined as ρ λ ( t ) = π λ ( t ) | E λ , t ∈ C ∗ e ( M ) . Then, ρ λ has the unique extension property with respect to M . We see thatlim λ ∈ Λ ρ λ ( t ) P E λ ξ = lim λ ∈ Λ π λ ( t ) ξ = π ( t ) ξ = 0whence there is λ ∈ Λ such that ρ λ ( t ) = 0. Since ρ λ is finite-dimensional, itcan be written as the direct sum of boundary representations for M by Lemma2.8. In particular, there is a boundary representation β for M on C ∗ e ( M ) suchthat β ( t ) = 0. We conclude that there is a set of finite-dimensional boundaryrepresentations for M which is separating for C ∗ e ( M ).(iii) = ⇒ (ii): Trivial.(ii) = ⇒ (i): Let t ∈ C ∗ e ( M ) be a non-zero element. By Lemma 2.11, there is aboundary representation π : C ∗ e ( M ) → B ( H π ) for M with π ( t ) = 0. Let ξ ∈ H π INITE-DIMENSIONALITY IN THE NC CHOQUET BOUNDARY 25 be a unit vector such that π ( t ) ξ = 0. By assumption, there is a finite-dimensionalapproximation π λ : C ∗ e ( M ) → B ( H π ) , λ ∈ Λfor π . For each λ ∈ Λ, let E λ = π λ (C ∗ e ( M )) H π and note that this is a finite-dimensional subspace. Let ρ λ : C ∗ e ( M ) → B ( E λ ) be the unital finite-dimensional ∗ -representation defined as ρ λ ( t ) = π λ ( t ) | E λ , t ∈ C ∗ e ( M ) . We see that lim λ ∈ Λ ρ λ ( t ) P E λ ξ = lim λ ∈ Λ π λ ( t ) ξ = π ( t ) ξ = 0whence there is λ ∈ Λ such that ρ λ ( t ) = 0. We conclude that there is a set offinite-dimensional ∗ -representations of C ∗ e ( M ) which is separating for C ∗ e ( M ).(i) = ⇒ (ii): This is an immediate consequence of Theorem 2.3. (cid:3) Recall that Question 1 asks whether C ∗ e ( A ) being RFD forces the existence ofeven a single finite-dimensional boundary representation. This offers some perspec-tive regarding the potential validity of the implication (i) = ⇒ (v) above.We make a few related remarks. Inside the spectrum of C ∗ e ( M ), let D denote theset of unitary equivalence classes of finite-dimensional irreducible ∗ -representations,and let B denote the set of unitary equivalence classes of boundary representationsfor M . It follows from Theorems 2.3 and 2.12 that both these subsets are dense,but we do not know if these two large sets intersect non-trivially. Finer topologicalproperties may be useful in resolving this. Indeed, inspired by the statement ofthe Baire Category theorem, it is natural to wonder if the sets are of type G δ forinstance. This seems unlikely, as [23, Proposition 3.6.3] implies that D is in fact oftype F σ in \ C ∗ e ( M ).As discussed in Subsection 2.2, there are examples of unital C ∗ -algebras thatare RFD but not liminal. In particular, the set D defined above is a proper subsetof the spectrum in this case. Moreover, the set B defined above can also be aproper subset of the spectrum, even in the classical situation of a uniform algebra:there are known examples of uniform algebras on compact metric spaces where theChoquet boundary is a proper (albeit dense) subset of the Shilov boundary (see forinstance [45, page 42]).4.1. C ∗ -liminality. In this subsection, we approach our main problem from an-other angle. Herein, we study unital operator spaces with the property that all boundary representations are finite-dimensional. The hope is that a thorough un-derstanding of this extremal situation may shed some light on Question 1.A unital operator space M is said to be C ∗ -liminal if every boundary representa-tion for M on C ∗ e ( M ) is finite-dimensional. When M is a C ∗ -algebra, a boundaryrepresentation is simply an irreducible ∗ -representation, so this notion coincideswith the usual one introduced in Subsection 2.2. We remark that a certain relatednotion of what one may call “C ∗ -subhomogeneity” was studied recently in [4] in adifferent context.It follows from Theorem 2.4 that a unital C ∗ -algebra is liminal if and only if it isNFD. Along similar lines, Theorem 2.10 implies that a C ∗ -liminal unital operatoralgebra is necessarily NFD. But the converse is false in general. We illustratethis below (Example 4) with a rather pathological example of a finite-dimensional operator algebra with a unique boundary representation (up to unitary equivalence)and such that this boundary representation is infinite-dimensional. As preparation,we need a basic fact. Lemma 4.2.
Let
M ⊂ B ( H ) be a finite-dimensional unital operator space. Con-sider the set B M ⊂ M ( B ( H )) consisting of those elements of the form (cid:20) λI x µI (cid:21) where λ, µ ∈ C , x ∈ M . Then, B M is a finite-dimensional unital operator alge-bra and C ∗ e ( B M ) ∼ = M (C ∗ e ( M )) . Moreover, every boundary representation for B M on M (C ∗ e ( M )) is unitarily equivalent to one of the form β (2) for some boundaryrepresentation β for M on C ∗ e ( M ) .Proof. A routine calculation shows that B M is a finite-dimensional unital operatoralgebra and that C ∗ ( B M ) = M (C ∗ ( M )). By [17, Lemma 3.1] we find C ∗ e ( B M ) ∼ = M (C ∗ e ( M )) . Let π : M (C ∗ e ( M )) → B ( H π ) be a boundary representation for B M .Then, there is a boundary representation β : C ∗ e ( M ) → B ( H β ) for M with theproperty that π is unitarily equivalent to β (2) by the main result of [36]. (cid:3) We can now give the announced example of a finite-dimensional operator algebrawith no finite-dimensional boundary representation.
Example 4.
By [16, Theorem 6.2], there exists a Hilbert space H and an operator T ∈ B ( H ) with the property that the norm-closed unital algebra A generated by T is infinite-dimensional and has the identity representation as its only boundary rep-resentation (up to unitary equivalence). Let M = C I + C T . Then, C ∗ ( M ) = C ∗ ( A )and it is easily verified that a boundary representation for M on C ∗ ( M ) is neces-sarily a boundary representation for A as well. Thus, the identity representationof C ∗ ( M ) is the only boundary representation for M , up to unitary equivalence.In particular, C ∗ e ( M ) ∼ = C ∗ ( M ).Let B M ⊂ M ( B ( H )) be the finite-dimensional unital operator algebra as inLemma 4.2. Then, C ∗ e ( B M ) ∼ = M (C ∗ ( M )) and up to unitary equivalence, theonly boundary representation for B M on M (C ∗ ( M )) is id (2) , which is infinite-dimensional. Hence B M is not C ∗ -liminal. Note also that [17, Theorem 3.5] impliesthat B M is NFD. (cid:3) In fact, things can be even worse, and the C ∗ -envelope of a finite-dimensionalunital operator algebra can have no finite-dimensional ∗ -representations whatso-ever. Example 5.
Let H be a Hilbert space and let U, V ∈ B ( H ) be isometries such that U U ∗ + V V ∗ = I . Let M = C I + C U + C V . Then, C ∗ ( M ) is ∗ -isomorphic to theCuntz algebra O , which is infinite-dimensional and simple [19, Theorem V.4.7]. Inparticular, the Shilov ideal of M in C ∗ ( M ) is trivial, so C ∗ e ( M ) ∼ = C ∗ ( M ) = O .Let B M ⊂ M ( B ( H )) be the finite-dimensional unital operator algebra as inLemma 4.2. We find C ∗ e ( B M ) ∼ = M ( O ) and C ∗ e ( B M ) has no finite-dimensional ∗ -representation. (cid:3) As discussed in the introduction, the previous pair of examples illustrates someof the subtleties inherent to Question 1. In particular, the existence of finite-dimensional boundary representations appears to lie much deeper than the mere
INITE-DIMENSIONALITY IN THE NC CHOQUET BOUNDARY 27 finite-dimensionality of the original operator algebra. It is conceivable that Ques-tion 1 may thus have a negative answer in general. An operator algebra witnessingthis pathology would need to have an RFD C ∗ -envelope that is not liminal. Unfor-tunately, we do not have a large supply of examples of such C ∗ -algebras. A standardone is the full group C ∗ -algebra of the free group on two generators, as mentioned inSubsection 2.2. Generally speaking however, natural examples from group theorytend to admit finite-dimensional boundary representations, as we show next. Example 6.
Let G be a discrete group that is not virtually abelian. As explainedin [18], it then follows from [49],[50] that the full group C ∗ -algebra C ∗ ( G ) is not lim-inal. Fix some generating subset S for G . Let A ⊂ C ∗ ( G ) be the unital subalgebragenerated by the unitaries corresponding to the elements of S . It is readily seen thatC ∗ ( A ) = C ∗ ( G ). Moreover, by definition we see that A contains unitaries that gen-erate C ∗ ( G ) as a C ∗ -algebra, so it follows that every irreducible ∗ -representation ofC ∗ ( G ) is a boundary representation for A (see for instance [38, Lemma 5.5]). Con-sequently C ∗ e ( A ) ∼ = C ∗ ( G ). But C ∗ ( G ) always admits the character correspondingto the trivial representation of G , which is then a finite-dimensional boundary rep-resentation for A . (cid:3) Next, we exhibit another method for constructing a unital operator algebrawhose C ∗ -envelope is RFD and not liminal, while admitting many finite-dimensionalboundary representations. Example 7.
Let X be a compact metric space and let A ⊂ C ( X ) be a uniformalgebra. Assume that X is the Shilov boundary of A . Let R be a unital RFD C ∗ -algebra which is not liminal. It follows from [37] that the boundary representationsfor A ⊗ min R on C ( X ) ⊗ min R are precisely those of the form ε ξ ⊗ π where π isan irreducible ∗ -representation of R and ε ξ is the character on C ( X ) of evaluationat a point ξ in the Choquet boundary of A . In particular, because R is RFD,we conclude that there are many finite-dimensional boundary representations for A ⊗ min R on C ( X ) ⊗ min R .Next, we note that C ( X ) ⊗ min R ∼ = C ( X, R ) [44, Proposition 12.5]. From thisidentification, it is readily seen that the Shilov ideal of A ⊗ min R is trivial, so thatC ∗ e ( A ⊗ min R ) ∼ = C ( X ) ⊗ min R ∼ = C ( X, R ) . In particular, we see that the C ∗ -envelope is RFD and not liminal. (cid:3) Because boundary representations always factor through the C ∗ -envelope, a uni-tal operator space M is C ∗ -liminal whenever C ∗ e ( M ) is liminal in the classicalsense. In the next development we obtain a sort of partial converse. First, we givea characterization of C ∗ -liminality in terms of certain pure linear maps.Given a unital operator space M and a unital completely contractive map ψ : M → B ( H ψ ), we say that ψ is finite-dimensional if there exists a finite-dimensionalunital ∗ -representation σ : C ∗ e ( M ) → B ( H σ ) and an isometry V : H ψ → H σ suchthat ψ ( a ) = V ∗ σ ( a ) V, a ∈ M . Note in particular that H ψ must be finite-dimensional in this case, so only matrixstates can ever be finite-dimensional in this sense. Proposition 4.3.
Let M be a unital operator space. Then, M is C ∗ -liminal if andonly if all pure unital completely contractive maps on M are finite-dimensional. Proof.
Assume first that M is C ∗ -liminal. Let ψ : M → B ( H ψ ) be a pure unitalcompletely contractive map. Applying [20, Theorem 2.4], we see that ψ dilates toa boundary representation for M on C ∗ e ( M ), and thus ψ is finite-dimensional.Conversely, assume that all pure unital completely contractive maps on M arefinite-dimensional. Let π : C ∗ e ( M ) → B ( H π ) be a boundary representation for M .The restriction π | M is a pure unital completely contractive map [27, Proposition2.12], and thus it is finite-dimensional by assumption. In particular, H π must befinite-dimensional. (cid:3) We record a consequence of the previous result.
Corollary 4.4.
Let M be a C ∗ -liminal unital operator space and let B ⊂ M be a C ∗ -algebra with the same unit as M . Then, B is liminal.Proof. Let π be an irreducible ∗ -representation of B . By [6, Theorem 1.4.2], wesee that π is a pure unital completely positive map on B , and thus it admits apure extension to a unital completely contractive map π ′ on M [39, Corollary 2.3].By Proposition 4.3, we see that π ′ is finite-dimensional, so that π must act on afinite-dimensional space. (cid:3) For C ∗ -algebras, the statement of the previous corollary is well known and followsfor instance from Theorem 2.4.Let M be a unital operator space. We say that a unital completely contractivemap ψ : M → B ( H ψ ) is locally finite-dimensional if whenever N ⊂ M is a unitalfinite-dimensional subspace, there exist a finite-dimensional unital ∗ -representation σ : C ∗ e ( M ) → B ( H σ ) and an isometry V : H ψ → H σ such that ψ ( a ) = V ∗ σ ( a ) V, a ∈ N . Note once again that this forces H ψ to be finite-dimensional. The next resultis a minor refinement of a recent result of Hartz and Lupini [33, Theorem 1.5].Roughly speaking, it says that all matrix states of a C ∗ -liminal operator space arelocally finite-dimensional. Thus, locally we can remove the purity requirement ofProposition 4.3. Theorem 4.5.
Let M be a C ∗ -liminal unital operator space. Then, every matrixstate of M is locally finite-dimensional.Proof. Throughout the proof, we let T denote the operator system generated by M inside of C ∗ e ( M ). Let N ⊂ M be a unital finite-dimensional subspace and let S denote the operator system generated by N inside of C ∗ e ( M ). Note that S isstill finite-dimensional.We start with a preliminary observation. Let ω : S → B ( H ω ) be a pure unitalcompletely positive map. It follows from [39, Corollary 2.3] that there is a pureunital completely positive map Ω : T → B ( H ω ) extending ω . In light of [20,Theorem 2.4], we know that Ω can be dilated to a boundary representation for M on C ∗ e ( M ). By assumption, this means that Ω (and hence ω ) can be dilated to aunital finite-dimensional ∗ -representation of C ∗ e ( M ).We now turn to proving the desired statement. Let ψ : N → M r be a matrixstate and let Ψ denote its unique unital completely positive extension to S . Invoke[33, Theorem 2.9] (see also [33, Section 3]) to conclude that Ψ is a finite matrixconvex combination of matrix extreme points of the matrix state space of S . Inturn, by [28, Theorem B], we infer that the matrix extreme points among matrix INITE-DIMENSIONALITY IN THE NC CHOQUET BOUNDARY 29 states are pure matrix states on S . As shown in the previous paragraph, all purematrix states on S can be dilated to a finite-dimensional ∗ -representation of C ∗ e ( M ).Arguing now as in the proof of [33, Lemma 3.2], we see that Ψ dilates to a finite-dimensional unital ∗ -representation of C ∗ e ( M ). Thus, there is a finite-dimensionalunital ∗ -representation σ of C ∗ e ( M ) and an isometry V with the property thatΨ( s ) = V ∗ σ ( s ) V, s ∈ S . Restricting to N , this yields the desired statement for ψ . (cid:3) A few remarks concerning this theorem are in order. First, recall that the mo-tivation behind this result is the question asking whether C ∗ -liminality of a unitaloperator space implies liminality of its C ∗ -envelope. Although we do not manageto prove this, Theorem 4.5 supports this possibility. Indeed, if M is a C ∗ -algebra,then the conclusion is known to imply liminality [33, Proposition 1.4]. Further-more, the conclusion of the previous theorem is necessary for the liminality of theC ∗ -envelope, by [33, Theorem 1.5].While we do not know if the converse to Theorem 4.5 holds, we can at leastprove the following. Theorem 4.6.
Let M be a unital operator space. Assume that every matrix stateon M is locally finite-dimensional. Then, every unital ∗ -representation of C ∗ e ( M ) with the unique extension property with respect to M admits a finite-dimensionalapproximation. In particular, C ∗ e ( M ) is RFD.Proof. Let β : C ∗ e ( M ) → B ( H β ) be a unital ∗ -representation with the uniqueextension property with respect to M . Without loss of generality, we may assumethat H β is infinite-dimensional, for otherwise the desired result is trivial. Let Λdenote the collection of pairs ( N , X ) where N ⊂ M is a unital finite-dimensionalsubspace and
X ⊂ H β is a finite-dimensional subspace. Then, Λ is a directed set.Let λ = ( N , X ) ∈ Λ. The map a P X β ( a ) | X , a ∈ M is a matrix state on M . By assumption, there is a finite-dimensional unital ∗ -representation σ λ : C ∗ e ( M ) → B ( K λ ) and an isometry V λ : X → K λ with theproperty that P X β ( a ) | X = V ∗ λ σ λ ( a ) V λ , a ∈ N . Since H β is assumed to be infinite-dimensional, there is an isometry W λ : K λ → H β .Define ρ λ : C ∗ e ( M ) → B ( H β ) as ρ λ ( t ) = W λ σ λ ( t ) W ∗ λ , t ∈ C ∗ e ( M ) . This is a ∗ -representation and ρ λ (C ∗ e ( M )) H β ⊂ W λ K λ so that ρ λ (C ∗ e ( M )) H β isfinite-dimensional. Again, since H β is assumed to be infinite-dimensional, we canextend the isometry W λ V λ : X → H β to a unitary U λ on H β . Note that U λ | X = W λ V λ and P X U ∗ λ = V ∗ λ W ∗ λ so we find(1) P X β ( a ) | X = P X U ∗ λ ρ λ ( a ) U λ | X , a ∈ N . Next, for each λ ∈ Λ we define a ∗ -representation π λ : C ∗ e ( M ) → B ( H β ) as π λ ( t ) = U ∗ λ ρ λ ( t ) U λ , t ∈ C ∗ e ( M ) . We find π λ (C ∗ e ( M )) H β = U ∗ λ ρ λ (C ∗ e ( M )) H β which is again finite-dimensional. Let ψ : C ∗ e ( M ) → B ( H β ) be the limit of a subnet( π λ µ ) µ in the pointwise weak- ∗ topology; this exists by compactness [44, Theorem7.4]. In particular, ψ is a completely positive map.Let a ∈ M and ξ, η ∈ H β . Let µ such that λ µ = ( N , X ) ∈ Λ satisfies a ∈ N and ξ, η ∈ X . Then, using Equation (1) we obtain h β ( a ) ξ, η i = h P X β ( a ) | X ξ, η i = h P X π λ µ ( a ) | X ξ, η i = h π λ µ ( a ) ξ, η i whence h β ( a ) ξ, η i = lim µ h π λ µ ( a ) ξ, η i = h ψ ( a ) ξ, η i . Therefore, ψ and β agree on M . By the unique extension property of β with respectto M , we conclude that β ( t ) = ψ ( t ) = lim µ π λ µ ( t ) , t ∈ C ∗ e ( M )where the limit exists in the weak- ∗ topology of B ( H β ). By [23, Paragraph 3.5.2],we conclude that ( π λ µ ) µ is a finite-dimensional approximation for β .Finally, we may invoke Theorem 4.1 to see that C ∗ e ( M ) is RFD. (cid:3) Note that by Theorem 4.1, in order to establish that C ∗ e ( A ) is RFD above, itwould have sufficed to produce a finite-dimensional approximation for an arbitraryboundary representation β . In other words, we could have assumed that β wasirreducible. Nevertheless, even then it is not clear that a matrix state of the form a P X β ( a ) | X , a ∈ M would necessarily be pure. This explains why we need to assume that every ma-trix state is locally finite-dimensional, as opposed to simply the pure ones, andemphasizes the relevance of Theorem 4.5.Let us now summarize our findings related to C ∗ -liminality. Corollary 4.7.
Let M be a unital operator space. Consider the following state-ments. (i) The operator space M is C ∗ -liminal. (ii) Every matrix state of M is locally finite-dimensional. (iii) The algebra C ∗ e ( M ) is RFD.Then, we have (i) ⇒ (ii) ⇒ (iii) . If M is in fact a unital operator algebra thatsatisfies (ii) , then M must be NFD.Proof. (i) ⇒ (ii) ⇒ (iii): This follows by combining Theorems 4.5 and 4.6.Assume now that M is a unital operator algebra and every matrix state of M is locally finite-dimensional. Let n ∈ N and let A ∈ M n ( M ). Write A = [ a ij ]and let N ⊂ M be the unital finite-dimensional subspace generated by { a ij : 1 ≤ i, j ≤ n } . By Theorem 2.10, there is a matrix state ψ : M → B ( F ) such that k ψ ( n ) ( A ) k = k A k . By assumption, we infer the existence of a finite-dimensionalunital ∗ -representation π : C ∗ e ( M ) → B ( H π ) and of an isometry V : F → H π suchthat ψ ( b ) = V ∗ π ( b ) V, b ∈ N . INITE-DIMENSIONALITY IN THE NC CHOQUET BOUNDARY 31
In particular, we see that ψ ( n ) ( A ) = V ∗ ( n ) π ( n ) ( A ) V ( n ) whence k π ( n ) ( A ) k = k A k . We conclude that M is indeed NFD. (cid:3) By choosing M to be a unital C ∗ -algebra which is RFD but not liminal, we seethat (iii) (i) and (iii) (ii) above. Furthermore, Example 5 shows that (ii) canfail for NFD unital operator algebras. As mentioned before, we do not know if (ii) ⇒ (i). In [33], this is shown to hold in the self-adjoint context by using that (i) isequivalent to being NFD for C ∗ -algebras, by virtue of Theorem 2.4. Unfortunately,that equivalence fails in our generality. References [1] Jim Agler and John E. McCarthy,
Pick interpolation and Hilbert function spaces , Gradu-ate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002.MR1882259 (2003b:47001)[2] Charles A. Akemann,
The general Stone-Weierstrass problem , J. Functional Analysis (1969), 277–294. MR0251545[3] , Approximate units and maximal abelian C ∗ -subalgebras , Pacific J. Math. (1970),543–550. MR0264406[4] Alexandu Aleman, Michael Hartz, John McCarthy, and Stefan Richter, Multiplier tests andsubhomogeneity of multiplier algebras , preprint arXiv:2008.00981 (2020).[5] R. J. Archbold,
On residually finite-dimensional C ∗ -algebras , Proc. Amer. Math. Soc. (1995), no. 9, 2935–2937. MR1301006[6] William Arveson, Subalgebras of C ∗ -algebras , Acta Math. (1969), 141–224. MR0253059(40 An invitation to C ∗ -algebras , Springer-Verlag, New York-Heidelberg, 1976. GraduateTexts in Mathematics, No. 39. MR0512360[8] , The noncommutative Choquet boundary , J. Amer. Math. Soc. (2008), no. 4, 1065–1084. MR2425180 (2009g:46108)[9] , The noncommutative Choquet boundary II: hyperrigidity , Israel J. Math. (2011),349–385. MR2823981[10] David P. Blecher,
Noncommutative peak interpolation revisited , Bull. Lond. Math. Soc. (2013), no. 5, 1100–1106. MR3105002[11] David P. Blecher, Damon M. Hay, and Matthew Neal, Hereditary subalgebras of operatoralgebras , J. Operator Theory (2008), no. 2, 333–357. MR2411049[12] David P. Blecher and Christian Le Merdy, Operator algebras and their modules—an operatorspace approach , London Mathematical Society Monographs. New Series, vol. 30, The Claren-don Press, Oxford University Press, Oxford, 2004. Oxford Science Publications. MR2111973[13] Xinxin Chen, Adam Dor-on, Langwen Hui, Christopher Linden, and Yifan Zhang,
Doob equiv-alence and non-commutative peaking for Markov chains , preprint arXiv:1911.10423 (2019).[14] Man Duen Choi,
The full C ∗ -algebra of the free group on two generators , Pacific J. Math. (1980), no. 1, 41–48. MR590864[15] Rapha¨el Clouˆatre, Non-commutative peaking phenomena and a local version of the hyper-rigidity conjecture , Proc. Lond. Math. Soc. (3) (2018), no. 2, 221–245. MR3851322[16] Rapha¨el Clouˆatre and Michael Hartz,
Multiplier algebras of complete Nevanlinna-Pick spaces:dilations, boundary representations and hyperrigidity , J. Funct. Anal. (2018), no. 6, 1690–1738. MR3758546[17] Rapha¨el Clouˆatre and Christopher Ramsey,
Residually finite-dimensional operator algebras ,J. Funct. Anal. (2019), no. 8, 2572–2616. MR3990728[18] Kristin Courtney and Tatiana Shulman,
Elements of C ∗ -algebras attaining their norm in afinite-dimensional representation , Canad. J. Math. (2019), no. 1, 93–111. MR3928257[19] Kenneth R. Davidson, C ∗ -algebras by example , Fields Institute Monographs, vol. 6, AmericanMathematical Society, Providence, RI, 1996. MR1402012 (97i:46095)[20] Kenneth R. Davidson and Matthew Kennedy, The Choquet boundary of an operator system ,Duke Math. J. (2015), no. 15, 2989–3004. MR3430455 [21] ,
Noncommutative Choquet theory , preprint arXiv:1905.08436 (2019).[22] Kenneth R. Davidson and Benjamin Passer,
Strongly peaking representations and compres-sions of operator systems , preprint arXiv:2005.11582 (2020).[23] Jacques Dixmier, C ∗ -algebras , North-Holland Publishing Co., Amsterdam-New York-Oxford,1977. Translated from the French by Francis Jellett, North-Holland Mathematical Library,Vol. 15. MR0458185[24] Ronald G. Douglas, Banach algebra techniques in operator theory , Second, Graduate Textsin Mathematics, vol. 179, Springer-Verlag, New York, 1998. MR1634900[25] Michael A. Dritschel and Scott A. McCullough,
Boundary representations for families ofrepresentations of operator algebras and spaces , J. Operator Theory (2005), no. 1, 159–167. MR2132691 (2006a:47095)[26] Ruy Exel and Terry A. Loring, Finite-dimensional representations of free product C ∗ -algebras , Internat. J. Math. (1992), no. 4, 469–476. MR1168356[27] Douglas Farenick and Ryan Tessier, Purity of the embeddings of operator systems into their c ∗ - and injective envelopes , preprint arXiv:2006.08501 (2020).[28] Douglas R. Farenick, Extremal matrix states on operator systems , J. London Math. Soc. (2) (2000), no. 3, 885–892. MR1766112[29] Theodore W. Gamelin, Uniform algebras , Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969.MR0410387[30] Guihua Gong, Huaxin Lin, and Zhuang Niu,
Classification of finite simple amenable Z -stable C ∗ -algebras , preprint arXiv:1501.00135 (2015).[31] Don Hadwin, A lifting characterization of RFD C ∗ -algebras , Math. Scand. (2014), no. 1,85–95. MR3250050[32] Masamichi Hamana, Injective envelopes of operator systems , Publ. Res. Inst. Math. Sci. (1979), no. 3, 773–785. MR566081 (81h:46071)[33] Michael Hartz and Martino Lupini, Dilation theory in finite dimensions and matrix convexity ,preprint arXiv:1910.03549 (2019).[34] Damon M. Hay,
Closed projections and peak interpolation for operator algebras , IntegralEquations Operator Theory (2007), no. 4, 491–512. MR2313282[35] Kenneth Hoffman, Banach spaces of analytic functions , Dover Publications, Inc., New York,1988. Reprint of the 1962 original. MR1102893[36] Alan Hopenwasser,
Boundary representations on C ∗ -algebras with matrix units , Trans. Amer.Math. Soc. (1973), 483–490. MR322522[37] , Boundary representations and tensor products of C ∗ -algebras , Proc. Amer. Math.Soc. (1978), no. 1, 95–98. MR0482241[38] Ali S¸. Kavruk, Nuclearity related properties in operator systems , J. Operator Theory (2014), no. 1, 95–156. MR3173055[39] Craig Kleski, Boundary representations and pure completely positive maps , J. Operator The-ory (2014), no. 1, 45–62. MR3173052[40] Tom-Lukas Kriel, An introduction to matrix convex sets and free spectrahedra , Complex Anal.Oper. Theory (2019), no. 7, 3251–3335. MR4020034[41] Meghna Mittal and Vern I. Paulsen, Operator algebras of functions , J. Funct. Anal. (2010), no. 9, 3195–3225. MR2595740 (2011h:47150)[42] Paul S. Muhly and Baruch Solel,
An algebraic characterization of boundary representations ,Nonselfadjoint operator algebras, operator theory, and related topics, 1998, pp. 189–196.MR1639657[43] M. N. N. Namboodiri, S. Pramod, P. Shankar, and A. K. Vijayarajan,
Quasi hyperrigidityand weak peak points for non-commutative operator systems , Proc. Indian Acad. Sci. Math.Sci. (2018), no. 5, Paper No. 66, 14. MR3869539[44] Vern Paulsen,
Completely bounded maps and operator algebras , Cambridge Studies in Ad-vanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR1976867(2004c:46118)[45] Robert R. Phelps,
Lectures on Choquet’s theorem , Second Ed., Lecture Notes in Mathematics,vol. 1757, Springer-Verlag, Berlin, 2001. MR1835574[46] C. J. Read,
On the quest for positivity in operator algebras , J. Math. Anal. Appl. (2011),no. 1, 202–214. MR2796203[47] Shˆoichirˆo Sakai, C ∗ -algebras and W ∗ -algebras , Springer-Verlag, New York-Heidelberg, 1971.Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60. MR0442701 INITE-DIMENSIONALITY IN THE NC CHOQUET BOUNDARY 33 [48] M. Takesaki,
Theory of operator algebras. I , Encyclopaedia of Mathematical Sciences,vol. 124, Springer-Verlag, Berlin, 2002. Reprint of the first (1979) edition, Operator Alge-bras and Non-commutative Geometry, 5. MR1873025[49] Elmar Thoma, ¨Uber unit¨are Darstellungen abz¨ahlbarer, diskreter Gruppen , Math. Ann. (1964), 111–138. MR160118[50] ,
Eine Charakterisierung diskreter Gruppen vom Typ I , Invent. Math. (1968), 190–196. MR248288[51] Aaron Tikuisis, Stuart White, and Wilhelm Winter, Quasidiagonality of nuclear C ∗ -algebras ,Ann. of Math. (2) (2017), no. 1, 229–284. MR3583354 Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, CanadaR3T 2N2
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