Nonunital operator systems and noncommutative convexity
aa r X i v : . [ m a t h . OA ] J a n NONUNITAL OPERATOR SYSTEMS ANDNONCOMMUTATIVE CONVEXITY
MATTHEW KENNEDY, SE-JIN KIM, AND NICHOLAS MANOR
Abstract.
We establish the dual equivalence of the category of(potentially non-unital) operator systems and the category of pointedcompact nc (noncommutative) convex sets, extending a result ofDavidson and the first author. We then apply this dual equivalenceto establish a number of results about operator systems, some ofwhich are new even in the unital setting.For example, we show that the maximal and minimal C*-coversof an operator system can be realized in terms of the C*-algebraof continuous nc functions on its nc quasistate space, clarifyingrecent results of Connes and van Suijlekom. We also characterize“C*-simple” operator systems, i.e. operator systems with simpleminimal C*-cover, in terms of their nc quasistate spaces.We develop a theory of quotients of operator systems that ex-tends the theory of quotients of unital operator algebras. In addi-tion, we extend results of the first author and Shamovich relatingto nc Choquet simplices. We show that an operator system is aC*-algebra if and only if its nc quasistate space is an nc Bauersimplex with zero as an extreme point, and we show that a sec-ond countable locally compact group has Kazhdan’s property (T)if and only if for every action of the group on a C*-algebra, the setof invariant quasistates is the quasistate space of a C*-algebra.
Contents
1. Introduction 22. Preliminaries on operator systems 63. Pointed noncommutative convex sets 114. Categorical duality 17
First author supported by Canadian Natural Sciences and Engineering ResearchCouncil (NSERC) Discovery Grant number 2018-202107.Second author supported by Canadian Natural Sciences and Engineering Re-search Council (NSERC) PGS-D Scholarship number 396162013, and the Euro-pean Research Council (ERC) Grant number 817597 under the European Union’sHorizon 2020 research and innovation programme.Third author supported by the Canadian Natural Sciences and Engineering Re-search Council (NSERC) PGS-D Scholarship number 401226864.
5. Pointed noncommutative functions 236. Minimal and maximal C*-covers 247. Characterization of unital operator systems 298. Quotients of operator systems 319. Noncommutative faces 3510. C*-simplicity 3711. Characterization of C*-algebras 3912. Stable equivalence 4213. Dynamics and Kazhdan’s property (T) 43References 471.
Introduction
Werner’s notion of a (generalized, i.e. potentially nonunital) operatorsystem is an axiomatic, representation-independent characterization ofconcrete operator systems, which are self-adjoint subspace of boundedoperators acting on a Hilbert space. Werner [24] showed that everyconcrete operator system satisfies the axioms of an abstract operatorsystem, and conversely that every abstract operator system is isomor-phic to a concrete operator system, thereby generalizing an importantresult of Choi and Effros [3] for unital operator systems.Recently, Davidson and the first author [6] introduced a theory ofnoncommutative convex sets and noncommutative functions. A keystarting point for the theory is the dual equivalence between the cat-egory of compact noncommutative convex sets and the category ofclosed unital operator systems. On the one hand, this equivalence al-lows the rich theory of operator systems and C*-algebras to be appliedto problems in noncommutative convexity. On the other hand, recentresults suggest that that the perspective of noncommutative convexitycan also provide new insight on operator systems and C*-algebras (seee.g. [5, 7, 15]).In this paper we will establish a similar dual equivalence between thecategory of operator systems in the sense of Werner and a category ofobjects that we call pointed noncommutative convex sets. These arecertain pairs consisting of a compact noncommutative convex set alongwith a distinguished point in the set. We will then consider a numberof applications of this equivalence.Before stating our results, we will first briefly review some of thebasic ideas from the theory of noncommutative convexity.A compact nc (noncommutative) convex set is a graded set K = ⊔ K n , where each graded component K n is an ordinary compact convex ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 3 subset of the set M n ( E ) of n × n matrices over an operator space E , andthe graded components are related by requiring that K is closed underdirect sums and compressions. The union is taken over all n ≤ κ forsome sufficiently large infinite cardinal number κ depending on K . Thefact that κ is infinite is an essential part of the theory, being necessaryfor e.g. the existence of extreme points. In the separable setting, ittypically suffices to take κ = ℵ .The conditions on K are equivalent to requiring that K is closedunder nc convex combinations, meaning that P α ∗ i x i α i ∈ K n for everybounded family of points { x i ∈ K n i } and every family of scalar matrices { α i ∈ M n i ,n } .The prototypical example of a compact nc convex set is the nc statespace of a unital operator system S defined by K = ⊔ K n , where K n =UCP( S, M n ) is the set of unital completely positive maps from S intothe space M n of n × n matrices. The dual equivalence in [6] impliesthat S is isomorphic to the operator system A( K ) of continuous affinenc functions on K , and that, on the other hand, if K is a compact ncconvex set, then K is affinely homeomorphic to the nc state space ofthe operator system A( K ) . In particular, every compact nc convex setarises as the nc state space of an operator system.For a (generalized) operator system S , it is necessary to insteadconsider the nc quasistate space of S . This is the pair ( K, z ) consistingof the compact nc convex set K = ⊔ K n , where K n = CCP( S, M n ) isthe set of completely contractive and completely positive maps from S into M n , and z ∈ K is the zero map.More generally, a pointed compact nc convex set is a pair ( K, z ) consisting of a compact nc convex set K and a distinguished point z ∈ K . However, it turns out that not every pair ( K, z ) arises asthe nc quasistate space of an operator system. This is an importantpoint that explains many of the difficulties that arise in the non-unitalsetting. Therefore, in order to obtain the desired the dual equivalencebetween operator systems and pointed compact nc convex sets, it isnecessary to impose an additional constraint.Specifically, we say that the pair ( K, z ) is a pointed compact ncconvex set if the operator system A( K, z ) ⊆ A( K ) of continuous affinenc functions on K that vanish at z to have nc quasistate space ( K, z ) .Our results will imply that this property is equivalent to ( K, z ) arisingas the state space of a compact nc convex set.We consider pointed compact nc convex sets and functions on pointedcompact nc convex sets in Section 3 and Section 5 respectively. Thefollowing two results establishing the above-mentioned dual equivalenceare the main results in Section 4. M. KENNEDY, S.J. KIM, AND N. MANOR
Theorem A.
An operator system S with nc quasistate space ( K, z ) is isomorphic to the operator system A( K, z ) ⊆ A( K ) of continuousaffine nc functions on K that vanish at z . Hence ( K, z ) is a pointedcompact nc convex set if and only if it arises as the nc quasistate spaceof an operator system.Theorem A is the key ingredient in the dual equivalence between thecategory of generalized operator systems and the category of pointedcompact nc convex sets. Theorem B.
The category
OpSys of generalized operator systems isdually equivalent to the category
PoNCConv of pointed compact ncconvex sets.An important consequence of Theorem A is that essentially all of theresults from [6] about unital operator systems apply to (generalized)operator systems. For example, in Section 6, we establish characteriza-tions of the maximal and minimal C*-covers of an operator system interms of the C*-algebra of continuous nc functions on its nc quasistatespace. As a corollary, we recover results about the minimal C*-cover(i.e. the C*-envelope) recently obtained by Connes and van Suijlekom[4].
Theorem C.
Let ( K, z ) be a pointed compact nc convex set.(1) The C*-algebra C( K, z ) of pointed continuous nc functions on ( K, z ) is the maximal C*-cover of A( K, z ) .(2) Let I ∂K denote the boundary ideal in the C*-algebra C( K ) ofcontinuous nc functions on K relative to the unital operatorsystem A( K ) , so that the C*-algebra C( K ) /I ∂K ∼ = C( ∂K ) isthe minimal unital C*-cover of A( K ) , and let I ( ∂K,z ) = I ∂K ∩ C( K, z ) . Then the C*-algebra C( K, z ) /I ( ∂K,z ) is the minimalC*-cover of A( K, z ) .In Section 8, as another application of the dual equivalence betweenoperator systems and pointed compact nc convex sets, we develop atheory of quotients of generalized operator systems that extends thetheory of quotients of unital operator systems developed by Kavruk,Paulsen, Todorov and Tomforde [14]. Theorem D.
Let S be an operator system and let J ⊆ S be thekernel of a completely contractive and completely positive map on S .There is a unique pair ( S/J, ϕ ) consisting of an operator system S/J and a morphism ϕ : S → S/J with the property that whenever T isan operator system and ψ : S → T is a completely contractive andcompletely positive map with J ⊆ ker ψ , then ψ factors through ϕ . In ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 5 other words, there is a completely contractive and completely positivemap ω : S/J → T such that ψ = ω ◦ ϕ .We also obtain some results that are new even for unital operatorsystems. In Section 10, we establish a characterization of operatorsystems that are C*-simple, meaning that their minimal C*-cover issimple. We refer to Section 6 for the definition of the spectral topology. Theorem E.
An operator system S with nc quasistate space ( K, z ) isC*-simple if and only if the closed nc convex hull of any nonzero pointin the spectral closure of ∂K contains ∂K \ { z } .In Section 11, we establish a characterization of operator systemsthat are isomorphic to C*-algebras in terms of their nc quasistatespaces, extending a result for unital operator systems from [15]. Theorem F.
Let S be an operator system with nc quasistate space ( K, z ) . Then S is a C*-algebra if and only if K is an nc Bauer simplexand z is an extreme point. The result also holds for unital operatorsystems with nc quasistate spaces replaced by nc state spaces.In Section 12, we make another connection to the recent work ofConnes and van Suijlekom [4]. They consider operator systems S and T that are stably equivalent in the sense that the minimal tensor product S ⊗ min K is isomorphic to the minimal tensor product T ⊗ min K , where K denotes the C*-algebra of compact operators. The next result is acharacterization of stable equivalence of operator systems in terms oftheir nc quasistate spaces. Theorem G.
Let S and T be operator systems with nc quasistatespaces ( K, z ) and ( L, w ) respectively. Let K and id K denote the zeromap and the identity representation respectively of K . Then S and T are stably isomorphic if the closed nc convex hulls of ∂K ⊗ { K , id K } and ∂L ⊗ { K , id K } are pointedly affinely homeomorphic with respectto the points z ⊗ K and w ⊗ K (see Section 12).Finally, in Section 13 we establish the following characterization ofsecond countable locally compact groups with property (T), extendinga result from [15] for discrete groups acting on unital C*-algebras, aswell as a result of Glasner and Weiss from [9] for second countablelocally compact groups acting on unital commutative C*-algebras. Theorem H.
A second countable locally compact group G has Kazh-dan’s property (T) if and only if for every action of the group on aC*-algebra, the set of invariant quasistates is the quasistate space of aC*-algebra. The result also holds for unital C*-algebras with quasistatespaces replaced by state spaces. M. KENNEDY, S.J. KIM, AND N. MANOR Preliminaries on operator systems
In this section we will recall the notion of a matrix ordered oper-ator space and introduce the notion of a generalized (i.e. potentiallynonunital) operator system. For a reference on operator spaces andunital operator systems, we refer the reader to the books of Paulsen[18] and Pisier [20].Let E be a self-adjoint operator space, i.e. such that E = E ∗ . Welet E h = { x ∈ E : x = x ∗ } denote the set of self-adjoint elements in E . For n ∈ N , we will write M n ( E ) for the operator space of n × n matrices over E , and we will write M n for M n ( C ) . A matrix cone over E is a disjoint union P = ( P n ) n ∈ N of closed subsets P n ⊆ M n ( E ) h suchthat(1) P n ∩ − P n = 0 for all n ∈ N and(2) AP n A ∗ ⊆ P m for all A ∈ M m,n and m, n ∈ N . Definition 2.1. A matrix ordered operator space is a pair ( E, P ) con-sisting of a self-adjoint *-vector space E and a matrix cone P over E .For n ∈ N , an element in M n ( E ) is positive if it belongs to P n . Remark 2.2.
When referring to a matrix ordered space, we will typi-cally omit the positive cone unless we need to refer to it explicitly. Notethat if E is a matrix ordered space then for m ∈ N , the space M m ( E ) isa matrix ordered space in a canonical way. Specifically, letting P denotethe matrix cone for E , ( M m ( E ) , Q ) is a matrix ordered space, where Q = ( Q n ) n ∈ N is the matrix cone defined by identifying M m ( M n ( E )) with M mn ( E ) in the obvious way and setting Q n = P mn .Let E be a matrix ordered operator space. An element e ∈ E isan archimedean order unit for E if for every x ∈ E h , there is a scalar α > such that − αe ≤ x ≤ αe , and if x + αe ≥ for all α > , then x ≥ . It is an archimedean matrix order unit for E if for every n ∈ N , n ⊗ e is an archimedean order unit for M n ( E ) .If E is a matrix ordered operator space, then an archimedean matrixorder unit e ∈ E induces a norm k · k e on M n ( E ) for each n ∈ N ,defined by k x k e = inf (cid:26) α > (cid:18) α n ⊗ e xx ∗ α n ⊗ e (cid:19) ≥ (cid:27) for x ∈ M n ( E ) . The next definition is equivalent to the definition of an operatorsystem given by Choi and Effros [3].
ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 7
Definition 2.3. A unital operator system S is a complete matrix or-dered space with an archimedean matrix order unit S that is distin-guished in the sense that for each n , the norm on M n ( S ) coincides withthe norm k · k S from above. Remark 2.4.
Although not strictly necessary, it will be convenientfor the purposes of this paper to assume that operator systems arecomplete. If S is a unital operator system, then the distinguishedarchimedean order unit S is uniquely determined by the property thatfor s ∈ S with s ≥ , k s k ≤ if and only if s ≤ S .Let ( E, P ) and ( F, Q ) be matrix ordered spaces and let ϕ : E → F be a bounded map. We will write ϕ n : M n ( E ) → M n ( F ) for the linearmap defined by ϕ n = id n ⊗ ϕ . Definition 2.5.
Let ( E, P ) and ( F, Q ) be matrix ordered operatorspaces. A linear map ϕ : E → F is contractive if k ϕ k ≤ , and completely contractive if ϕ n is contractive for all n ∈ N . It is isometric if k ϕ ( x ) k = k x k for all x ∈ E , and completely isometric if k ϕ n ( x ) k = k x k for all n ∈ N and all x ∈ M n ( E ) . Similarly, it is positive if ϕ ( P ) ⊆ Q ,and completely positive if ϕ n is positive for each n ∈ N . The map ϕ isa complete order isomorphism if it is completely positive and invertiblewith a completely positive inverse. It is a complete order embedding ifit is completely positive and invertible on its range with a completelypositive inverse. Remark 2.6.
For unital operator systems, these definitions agree withthe usual definitions. Furthermore, because the norm on a unital op-erator system is completely determined by the matrix order, a unitalmap between unital operator systems is completely isometric if andonly if it is a complete order embedding. However, this is not true forarbitrary matrix ordered spaces (see [24]).We will write
UnOpSys for the category of unital operator systemswith unital completely positive maps (equivalently, unital complete or-der homomorphisms) as morphisms. We will refer to unital completeorder isomorphisms as isomorphisms , and to unital complete order em-beddings as embeddings .Choi and Effros [3, Theorem 4.4] showed that every unital operatorsystem is isomorphic to a concrete unital operator system, meaningthat there is a unital completely isometric map into some B ( H ) , where B ( H ) denotes the C*-algebra of bounded linear operators acting ona Hilbert space H . We will be interested in matrix ordered spacessatisfying an appropriate analogue of this property. M. KENNEDY, S.J. KIM, AND N. MANOR
Specifically, we are interested in matrix ordered spaces with a com-pletely isometric complete order embedding into some B ( H ) . It turnsout that not every matrix ordered space has this property. FollowingConnes and van Suijlekom [4], we will make use of Werner’s [24] char-acterization of matrix ordered spaces with this property in terms ofpartial unitizations (see below), although other characterizations arealso known (see e.g. [21]).The next definition is [24, Definition 4.1] (see also [4, Definition2.11]). Definition 2.7.
Let E be a matrix ordered operator space. The partialunitization of E is the matrix ordered operator space ( E ♯ , P ) , where E ♯ = E ⊕ C and the matrix cone P = ( P n ) is defined by specifyingthat for each n ∈ N , P n ⊆ M n ( E ♯ ) h = M n ( E ) h ⊕ ( M n ) h consists ofall pairs ( x, α ) ∈ M n ( E ) h ⊕ ( M n ) h satisfying α ≥ and ϕ ( α − / ǫ xα − / ǫ ) ≥ − for all ǫ > and ϕ ∈ CCP( E, M n ) , where α ǫ = α + ǫ n and CCP( E, M n ) denotes the space of completelycontractive and completely positive maps from E to M n . We will referto the map E → E ♯ : x → ( x, as the canonical inclusion map , andwe will refer to the map E ♯ → C : ( x, α ) → α as the projection ontothe scalar summand .The next result is contained in [24, Section 4] (see also [4, Proposition2.12] and [4, Lemma 2.13]). Theorem 2.8.
Let E be a matrix ordered space.(1) The partial unitization E ♯ is a unital operator system.(2) Let ι : E → E ♯ denote the canonical inclusion map and let τ : E ♯ → C denote the projection onto the scalar summand.Then ι is completely contractive and completely positive and τ is unital and positive, and the following sequence is split exact: E E ♯ C . ι τ (3) Let F be a matrix ordered space and let ϕ : E → F be a com-pletely contractive and completely positive map. Then the uni-tization ϕ ♯ : E ♯ → F ♯ defined by ϕ ♯ (( x, α )) = ( ϕ ( x ) , α ) for ( x, α ) ∈ E ♯ is unital and completely positive. Furthermore, if ϕ is a completely isometric complete order isomorphism then ϕ ♯ is a unital complete order isomorphism. Remark 2.9.
Note that E = E ♯ , even if E is already unital. For aC*-algebra A , the partial unitization A ♯ coincides with the usual C*-algebraic unitization of A , and hence is a unital C*-algebra. ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 9
It follows from the representation theorem of Choi and Effros [3] forunital operator systems that if E is a matrix ordered operator spacewith partial unitization E ♯ and the canonical inclusion map E → E ♯ is completely isometric, then there is a completely isometric completeorder isomorphism of E onto a self-adjoint subspace of bounded op-erators acting on a Hilbert space. Following [4], this motivates thefollowing definition. Definition 2.10.
We will say that a complete matrix ordered space S isan operator system if the canonical inclusion map S → S ♯ is completelyisometric, in which case we will refer to S ♯ as the unitization of S . Remark 2.11.
As in the unital case, it is not strictly necessary toassume that operator systems are complete. For an operator system S , we will identify S with its image in S ♯ under the canonical inclusionmap. In particular, if T is an operator system and ϕ : S → T iscompletely contractive and completely positive, then we will view theunitization ϕ ♯ : S ♯ → T ♯ as an extension of ϕ . Remark 2.12. If S is a unital operator system, then it follows from[24, Lemma 4.9] that the identity map on S factors through the canon-ical inclusion map S ♯ . In particular, this implies that the canonicalinclusion map is completely isometric, so S is an operator system inthe sense of Definition 2.10. Remark 2.13.
Let S and T be operator systems and let ϕ : S → T bea completely contractive completely positive map. If ϕ is a completelyisometric complete order isomorphism, then Theorem 2.8 implies thatthe unitization ϕ ♯ : S ♯ → T ♯ is a complete order isomorphism. However,if ϕ is merely a completely isometric complete order embedding, thenit is not necessarily true that the unitization ϕ ♯ is a complete orderembedding (see Example 2.14). We will need to take this into accountwhen we define embeddings between operator systems below.In the following example, we construct operator systems S and T and a completely isometric complete order embedding ϕ : S → T suchthat the unitization S ♯ → T ♯ is not a complete order embedding. Thefundamental issue is that completely contractive completely positivemaps on the image of S in T do not necessarily extend to completelycontractive completely positive maps on T (see [21, Section 6]). Example 2.14.
Define a, b ∈ M by a = (cid:20) − (cid:21) , b = (cid:20) − / (cid:21) . Let S = span { a } and B = span { , b } ∼ = C . Then S is a non-unitaloperator system and B is a unital C*-algebra. Define ϕ : S → B by ϕ ( αa ) = αb for α ∈ C . We claim that ϕ is a completely isometriccomplete order embedding, but that the unitization ϕ ♯ : S ♯ → B ♯ isnot completely isometric.Note that M n ( S ) = span { α ⊗ a : α ∈ M n } . Since k ϕ ( α ⊗ a ) k = k α ⊗ b k = k α k = k α ⊗ a k ,ϕ is completely isometric. Also, α ⊗ a ≥ if and only if α ⊗ b ≥ ifand only if α = 0 , so α is a complete order embedding.It is not difficult to see that for λ ∈ [ − , the map ϕ λ : S → C defined by ϕ λ ( αa ) = λα for α ∈ C is a quasistate, i.e. is completelycontractive and completely positive. Furthermore, if ψ : S → C isa quasistate, then ψ = ϕ λ for some λ ∈ [ − , . Hence the set ofquasistates on S can be identified with [ − , .We will see in Section 4.4 that this implies that the state space ofthe unitization S ♯ is [ − , . Since [ − , is a simplex, it follows froma classical result of Bauer that S ♯ = span { , a } ∼ = C (see e.g. [15]).Note that B ♯ ∼ = C . We can identify B = C with the first twocoordinates of C . Then ϕ ♯ ( α + βa ) = α + βb . In particular, ϕ ♯ ( + a ) = + b . Since + a ≥ but + b , it followsthat ϕ ♯ is not a complete order embedding.We will write OpSys for the category of operator systems with com-pletely contractive and completely positive maps as morphisms. Wewill refer to completely isometric complete order isomorphisms as iso-morphisms . Motivated by Remark 2.13, for operator systems S and T , we will refer to a completely isometric complete order embedding ϕ : S → T as an embedding if the unitization ϕ ♯ : S ♯ → T ♯ is anembedding in the category of unital operator systems.Werner was able to isolate the precise obstruction to a matrix orderedoperator space being an operator system in the sense of Definition 2.10.The next result is [24, Lemma 4.8]. Theorem 2.15.
Let E be a matrix ordered operator space with partialunitization E ♯ . For each n , let ν n : M n ( E ) → R ≥ denote the mapdefined by ν n ( x ) = sup ϕ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ϕ n ( x ) ϕ n ( x ) ∗ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) , for x ∈ M n ( E ) . where the supremum is taken over all maps ϕ ∈ CCP( E, C ) . Then ν n is a norm on M n ( E ) . The inclusion E → E ♯ is completely isometricif and only if for each n , the norm on M n ( E ) coincides with ν n . ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 11 Pointed noncommutative convex sets
A key result from [6] is the dual equivalence between the categoryof unital operator systems and the category of compact nc convex sets.In this section we will review the definition of a compact nc convex setand introduce the definition of a pointed compact nc convex set. InSection 4, we will show that the category of operator systems is dualto the category of pointed compact nc convex sets.3.1.
Noncommutative convex sets.
Let E be an operator space.For nonzero (potentially infinite) cardinals m and n , let M m,n ( E ) de-note the operator space of m × n matrices over X with the propertythat the set of finite submatrices are uniformly bounded. For brevity,we will write M n ( E ) for M n,n ( E ) , M m,n for M m,n ( C ) and M n for M n ( C ) . Restricting to matrices with uniformly bounded finite subma-trices ensures that matrices over E can be multiplied on the left andright by scalar matrices of the appropriate size. We identify M n withthe C*-algebra of bounded operators acting on a Hilbert space H n ofdimension n .If E is a dual operator space with distinguished predual E ∗ , thenthere is a natural operator space isomorphism M n ( E ) ∼ = CB( E ∗ , M n ) ,where CB( E ∗ , M n ) denotes the space of completely bounded maps from E ∗ to M n . We equip M n ( E ) with the corresponding point-weak*topology.Let M ( E ) = ⊔ n M n ( E ) , where the union is taken over all nonzerocardinal numbers n . Once again, for brevity, we will write M for M ( C ) . Although M ( E ) is a proper class and not a set, we will onlybe interested in subsets, so this will not present any set-theoretic dif-ficulties. More generally, we will consider disjoint unions over nonzerocardinal numbers n of subsets of M n ( E ) . For a subset X ⊆ M ( E ) and a cardinal number n , we will write X n for the graded component X n = X ∩ M n ( E ) . Definition 3.1.
Let E be an operator space. An nc convex set over E is a graded subset K = ⊔ n K n with K n ⊆ M n ( E ) that is closed underdirect sums and compressions, meaning that(1) P α i x i α ∗ i ∈ K n for every bounded family of points x i ∈ K n i andevery family of isometries α i ∈ M n,n i satisfying P α i α ∗ i = 1 n .(2) β ∗ xβ ∈ K n for every x ∈ K n and every isometry β ∈ M m,n .If E is a dual operator space, so that each M n ( E ) is equipped withthe weak* topology discussed above, then we will say that K is closed if each K n is closed. Similarly, we will say that K is compact if each K n is compact. The most important examples of compact nc convex sets are non-commutative state spaces of operator systems. The next definition is[6, Example 2.2.6].
Definition 3.2.
Let S be a unital operator system. The nc statespace of S is the set K = ⊔ n K n defined by K n = UCP( S, M n ) . Here, UCP( S, M n ) denotes the space of unital completely positive maps from S to M n . Elements in K are referred to as nc states on S . Remark 3.3.
Note that the set K is nc convex and compact sinceeach UCP( S, M n ) is compact.The following characterization of compact nc convex sets as sets thatare closed under nc convex combinations is often useful. In particular,it makes the analogy between nc convex sets and ordinary convex setsmore explicit. The next result is [6, Proposition 2.2.8]. Proposition 3.4.
Let E be a dual operator space and let K = ⊔ n K n forclosed subsets K n ⊆ M n ( E ) . Then K is nc convex if and only if it isclosed under nc convex combinations , meaning that P α ∗ i x i α i ∈ K n forevery bounded family of points x i ∈ K n i and every family α i ∈ M n i ,n satisfying P α ∗ i α i = 1 n . One of the most important justifications for the utility of noncom-mutative convexity is the fact that there is a robust notion of extremepoint for which a noncommutative analogue of the Krein-Milman the-orem [6, Theorem 6.4.2] holds, meaning that every compact nc convexset is generated by its extreme points.
Definition 3.5.
Let K be a compact nc convex set. A point x ∈ K n is extreme if whenever x is written as a finite nc convex combination x = P α ∗ i x i α i for { x i ∈ K n i } and nonzero { α i ∈ M n i ,n } satisfying P α ∗ i α i = 1 n , then each α i is a positive scalar multiple of an isometry β i ∈ M n i ,n satisfying β ∗ i x i β i = x and each x i decomposes with respectto the range of α i as a direct sum x i = y i ⊕ z i for y i , z i ∈ K with y i unitarily equivalent to x . The set of all extreme points is ∂K = ⊔ ( ∂K ) n .The morphism between nc convex sets are the continuous affine non-commutative maps. The next definition is [6, Definition 2.5.1]. Definition 3.6.
Let K and L be compact nc convex sets. A map θ : K → L is an nc map if it is graded, respects direct sums and isunitarily equivariant, meaning that(1) θ ( K n ) ⊆ L n for all n , ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 13 (2) θ ( P α i x i α ∗ i ) = P α i θ ( x i ) α ∗ i for every bounded family { x i ∈ K n i } and every family of isometries { α i ∈ M n i ,n } satisfying P α ∗ i α i = 1 n ,(3) θ ( α ∗ xα ) = α ∗ θ ( x ) α for every x ∈ K m and every unitary α ∈M n .An nc map θ is affine if, in addition, it is equivariant with respect toisometries, meaning that(3’) θ ( α ∗ xα ) = α ∗ θ ( x ) α for every x ∈ K m and every isometry α ∈M m,n .An affine nc map θ is continuous if the restriction f | K n is continuouswith respect to the point-strong topology on K n and L n for each n . Itis bounded if k θ k ∞ < ∞ , where k θ k ∞ denotes the the uniform norm k θ k ∞ = sup x ∈ K k θ ( x ) k . Finally, θ is an (affine) homeomorphism and K and L are (affinely) homeomorphic if θ is continuous and has acontinuous (affine) nc inverse. Remark 3.7.
We will consider an appropriate notion of continuity formore general nc maps in Section 5.We will write
NCConv for the category of compact nc convex setswith continuous affine nc maps as morphisms. We will refer to affine nchomeomorphisms as isomorphisms , and to injective continuous affinenc maps as embeddings .The next definition is [6, Definition 3.2.1].
Definition 3.8.
Let K be a compact nc convex set. We will write A( K ) for the unital operator system of all continuous affine nc functions from K to M . Remark 3.9.
The fact that A( K ) is a unital operator system is dis-cussed in [6, Section 3.2].For a point x ∈ K n , the corresponding evaluation map A( K ) →M n : a → a ( x ) is an nc state on A( K ) . Moreover, by [6, Theorem3.2.2], every nc state on A( K ) is given by evaluation at some point in K (we will say more about this in Section 4). It will be convenient toidentify points in K with the corresponding nc state on A( K ) .Following [6, Section 3.2], for each n we will identify the unital op-erator system M n (A( K )) with the space of continuous affine nc mapsfrom K to M n ( M ) in the obvious way.3.2. Pointed noncommutative convex sets.
In this section we in-troduce the notion of a pointed compact nc convex set, of which the most important examples will be nc quasistate spaces of operator sys-tems. Before introducing the definition of a pointed compact nc convexset, we require the definition of a pointed continuous affine nc function.
Definition 3.10.
Let ( K, z ) be a pair consisting of a compact nc con-vex set K and a point z ∈ K . We will say that a continuous affinenc function a ∈ A( K ) is pointed if f ( z ) = 0 . We let A( K, z ) ⊆ A( K ) denote the space of pointed continuous affine nc functions on K . Remark 3.11.
The space A( K, z ) is a matrix ordered space with ma-trix cone P = ⊔ P n inherited from A( K ) . Specifically, for n ∈ N , thepositive cone on P n consists of the positive functions in M n (A( K, z )) .Since A( K, z ) is a closed self-adjoint subspace of the unital operatorsystem A( K ) , it follows that A( K, z ) is an operator system.The most important examples of pointed compact nc convex setswill be nc quasistate spaces of operator systems. The idea to utilize ncquasistate spaces in this setting was inspired by the importance of thequasistate space of a non-unital C*-algebra. Definition 3.12.
Let S be an operator system. The nc quasistatespace of S is the pair ( K, z ) , where K = ⊔ n K n is defined by K n =CCP( S, M n ) and z ∈ K is the zero map. Here, CCP( S, M n ) denotesthe space of completely contractive and completely positive maps from S to M n . We will refer to elements of K as nc quasistates on S . Remark 3.13.
Note that the set K is nc convex and compact sinceeach CCP( S, M n ) is compact.We are now ready to introduce the definition of a pointed compactnc convex set. Definition 3.14.
Let ( K, z ) be a pair consisting of a compact nc con-vex set K and a point z ∈ K . We will say that ( K, z ) is a pointedcompact nc convex set if every nc quasistate on the operator system A( K, z ) belongs to K , i.e. is evaluation at a point in K . Remark 3.15.
Since K is the nc state space of the unital operatorsystem A( K ) , ( K, z ) is a pointed compact nc convex set if and only ifevery nc quasistate on A( K, z ) extends to an nc state on A( K ) .By definition, a pointed compact nc convex set is the nc quasistatespace of an operator system. In Section 4, we will show that the ncquasistate space of every operator system is a pointed compact nc con-vex set. The proof of this fact is non-trivial. However, we are now ableto give some examples. ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 15
Example 3.16.
Define K = ⊔ K n by K n = { α ∈ ( M n ) h : − n ≤ α ≤ n } , for n ∈ N . Then K is a compact nc convex set (see [6, Example 2.2.4]). Let z = 0 .We will show that the pair ( K, z ) is a pointed compact nc convex set.The unital operator system A( K ) is given by A( K ) = span { A( K ) , a } ,where a ∈ A( K, z ) is the coordinate function a ( α ) = α for α ∈ K .Hence A( K, z ) = span { a } . In fact, A( K, z ) and A( K ) are isomorphicto the operator systems S and S ♯ from Example 2.14.If θ : A( K, z ) → M n is an nc quasistate, then there is a self-adjoint β ∈ M n with − n ≤ β ≤ n such that θ ( αa ) = αβ for α ∈ C .Conversely, it is easy to check that every self-adjoint β ∈ M n with − n ≤ beta ≤ n gives rise to an nc quasistate on A( K, z ) of thisform. Hence the nc quasistate space of A( K, z ) is K . Therefore, ( K, z ) is a pointed compact nc convex set.Note that A( K, z ) ♯ = A( K ) . In Corollary 4.7, we will show that thisproperty characterizes pointed pointed compact nc convex sets.The next example shows that not every pair ( K, z ) consisting of acompact nc convex set and a point z ∈ K is a pointed compact ncconvex set. Example 3.17.
Define K = ⊔ K n by K n = { α ∈ ( M n ) h : − n ≤ α ≤ n } , for n ∈ N . Then as in Example 3.16, K is a compact nc convex set. Let z = 0 .We will show that the pair ( K, z ) is not a pointed compact nc convexset.The unital operator system A( K ) is given by A( K ) = span { A( K ) , b } ,where b ∈ A( K, z ) is the coordinate function b ( α ) = α for α ∈ K .Hence A( K, z ) = span { a } . In fact, A( K ) is isomorphic to the C*-algebra B from Example 2.14.Define θ : A( K, z ) → C by θ ( αb ) = − α . Since A( K, z ) does notcontain any positive elements, the matrix cone of A( K, z ) is zero, so itis easy to check that θ is an nc quasistate. However, θ does not extendto an nc state on A( K ) since A( K ) + b ≥ , while + θ ( b ) = − .Hence θ does not belong to K and ( K, z ) is not a pointed compact ncconvex set.We will now establish a geometric characterization of pointed com-pact nc convex sets. Proposition 3.18.
Let ( K, z ) be a pair consisting of a compact ncconvex set K and a point z ∈ K . Then ( K, z ) is pointed if and only if whenever self-adjoint a ∈ M m (A( K, z )) satisfies a ( x ) ≤ m ⊗ n for all x ∈ K n , then θ m ( a ) ≤ m ⊗ p for all nc quasistates θ : A( K, z ) → M p .Proof. If ( K, z ) is pointed, then every nc quasistate on A( K, z ) belongsto K , so the condition trivially holds. Conversely, suppose that ( K, z ) is not pointed. Then there is an nc quasistate θ : A( K, z ) → M n suchthat θ / ∈ K . Identifying K with its image in M (A( K, z ) ∗ ) and viewing θ as a point in M n (A( K, z ) ∗ ) , the nc separation theorem [6, Theorem2.4.1] implies there is a self-adjoint element a ∈ M n (A( K, z )) such that θ ( a ) n ⊗ n but a ( x ) ≤ n ⊗ p for all x ∈ K p . (cid:3) Example 3.17 is a single instance of a general class of examples.
Corollary 3.19.
Let ( K, z ) be a pair consisting of a compact nc convexset and a point z ∈ K such that the matrix cone for A( K, z ) is zero.Then ( K, z ) is pointed if and only if whenever x ∈ K n satisfies αz ( n ) +(1 − α ) x ∈ K n for < α < , then αz ( n ) − (1 − α ) x ∈ K n . Here z ( n ) ∈ K n denotes the direct sum of n copies of z .Proof. Suppose that ( K, z ) is pointed and x ∈ K n satisfies αz ( n ) + (1 − α ) x ∈ K n for < α < . Then since the positive cone of A( K, z ) is zero,the map θ : A( K, z ) → M n defined by θ ( a ) = αa ( z ( n ) ) − (1 − α ) a ( x ) = − (1 − α ) a ( x ) for a ∈ A( K, z ) is an nc quasistate. Hence θ is givenby evaluation at a point in K which must be αz ( n ) − (1 − α ) x . Hence αz ( n ) − (1 − α ) x ∈ K .Conversely, suppose that whenever x ∈ K n satisfies αz ( n ) + (1 − α ) x ∈ K n for < α < , then αz ( n ) − (1 − α ) x ∈ K n . If self-adoint a ∈ M m (A( K, z )) satisfies a ( x ) ≤ m ⊗ n for all x ∈ K n , then for < α < , (1 − α ) a ( x ) = a ( αz ( n ) + (1 − α ) x ) ≤ m ⊗ n and − (1 − α ) a ( x ) = a ( αz ( n ) − (1 − α ) x ) ≤ m ⊗ n . Then taking α → implies − m ⊗ n ≤ a ( x ) ≤ m ⊗ n . Hence k a k ∞ ≤ . It follows that if θ : A( K, z ) → M n is an nc quasis-tate on A( K ) , then θ ( a ) ≤ m ⊗ n . Therefore, by Proposition 3.18, ( K, z ) is pointed. (cid:3) Example 3.20.
Let K denote the nc state space of M and let z = Tr ,where Tr ∈ K denotes the normalized trace. Then identifying M with A( K ) , A( K, z ) = (cid:26)(cid:20) α βγ − α (cid:21) : α, β, γ ∈ C (cid:27) . ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 17
The matrix cone of A( K, z ) is clearly zero. Define θ : M → C by θ (cid:18)(cid:20) α βγ δ (cid:21)(cid:19) = α for (cid:20) α βγ δ (cid:21) ∈ M . Then Tr + θ ∈ K . But Tr − θ / ∈ K . Hence by Corollary 3.19,the pair ( K, z ) is not pointed.We now define the category of pointed compact nc convex sets. Definition 3.21.
Let ( K, z ) and ( L, w ) be pointed nc convex sets. Wewill say that an affine nc map θ : K → L is pointed if θ ( z ) = w . Wewill say that ( K, z ) and ( L, w ) are pointedly affinely homeomorphic ifthere is a pointed affine homeomorphism from ( K, z ) to ( L, w ) .We will write PoNCConv for the category of pointed compact ncconvex sets with pointed continuous affine nc maps as morphisms. Wewill refer to pointed affine nc homeomorphisms as isomorphisms , andto pointed injective continuous affine nc maps as embeddings .4.
Categorical duality
In this section we will prove the dual equivalence between the cate-gory
OpSys of operator systems and the category
PoNCConv of pointedcompact nc convex sets. We begin by reviewing the details of the dualequivalence between the category
UnOpSys of unital operator systemsand the category
NCConv of compact nc convex sets from [6].4.1.
Categorical duality for unital operator systems.
The dualequivalence between the category of unital operator systems and thecategory of compact nc convex sets was developed in [6, Section 3]. Itis closely related to a similar dual equivalence established by Websterand Winkler [23].The next result combines [6, Theorem 3.2.2] and [6, Theorem 3.2.3].
Theorem 4.1.
Let K be a compact nc convex set. The nc state spaceof the unital operator system A( K ) is isomorphic to K . For a unitaloperator system S with nc state space K , the map S → A( K ) : s → ˆ s defined by ˆ s ( x ) = x ( s ) for s ∈ S, x ∈ K is an isomorphism. The dual equivalence between the category
UnOpSys of unital opera-tor systems and the category
NCConv of compact nc convex sets followsfrom Theorem 4.1. The contravariant functor
UnOpSys → NCConv isdefined in the following way: (1) A unital operator system S is mapped to its nc state space.(2) For unital operator systems S and T with nc state spaces K and L respectively, a morphism ϕ : S → T is mapped to themorphism ϕ d : L → K defined by ϕ d ( y )( a ) = ϕ ( a )( y ) , for y ∈ L and a ∈ A( K ) . The inverse functor
NCConv → UnOpSys is defined in the followingway:(1) A compact nc convex set K is mapped to the unital operatorsystem A( K ) .(2) If K and L are compact nc convex sets and ψ : L → K isa morphism, then the corresponding morphism ψ d : A( K ) → A( L ) is defined by ψ d ( a )( y ) = a ( ψ ( y )) , for a ∈ A( K ) and y ∈ L. The next result summarizes this discussion. It is [6, Theorem 3.2.5].
Theorem 4.2.
The contravariant functors
UnOpSys → NCConv and
NCConv → UnOpSys defined above are inverses. Hence the categories
UnOpSys and
NCConv are dually equivalent.
We will make use of the following result in the next section.
Proposition 4.3.
Let K and L be compact nc convex sets. Let ϕ :A( K ) → A( L ) be a unital completely positive map and let ϕ d : L → K denote the continuous affine map obtained by applying Theorem 4.2 to ϕ . Then ϕ is completely isometric if and only if ϕ d is surjective.Proof. if ϕ d is surjective, then for a ∈ M n (A( K )) , k ϕ ( a ) k ∞ = sup y ∈ L k ϕ ( a )( y ) k ∞ = sup y ∈ L k a ( ϕ d ( y )) k ∞ = sup x ∈ K k a ( x ) k = k a k ∞ . Hence ϕ is completely isometric.Conversely, suppose that ϕ is completely isometric. Let S = ϕ (A( K )) .Then S is a unital operator system. Let M denote the nc state spaceof S , so that S is isomorphic to A( K ) . It follows from Arveson’s ex-tension theorem that the restriction map r : L → M is surjective. Let ψ : M → K denote the continuous affine nc map obtained by restrict-ing the range of ϕ to S and applying Theorem 4.2. Then ϕ d = ψ ◦ r .Theorem 4.2 implies that ψ is an affine homeomorphism. Since r issurjective, it follows that ϕ d is surjective. (cid:3) ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 19
Categorical duality for operator systems.
Let ( K, z ) be apair consisting of a compact nc convex set K and a point z ∈ K .Observe that for a point x ∈ K , viewed as a unital completely positivemap on A( K ) , the restriction x | A( K,z ) is an nc quasistate. For brevity,it will be convenient to simultaneously view points in K as nc stateson A( K ) and nc quasistates on A( K, z ) . We will take care to ensurethat this does not cause any confusion. Proposition 4.4.
Let S be an operator system with nc quasistate space ( K, z ) and let L denote the nc state space of the unitization S ♯ . Theunitization map K → L : x → x ♯ is an affine homeomorphism withinverse given by the restriction map L → K : y → y | S . Hence S ♯ isisomorphic to A( K ) .Proof. For x ∈ K , the unitization x ♯ is the unique unital completelypositive extension of x to S ♯ . Hence x ♯ ∈ L . On the other hand, for y ∈ L , the restriction y | S is completely contractive and completelypositive, so y | S ∈ K . Then by uniqueness, ( y | S ) ♯ = y . It follows thatthe unitization map is a bijection with inverse given by the restrictionmap.It is clear that the restriction map from L to K is continuous andaffine. From above, the restriction to each L n is a continuous bijectiononto K n . Since L n is compact, it follows that this restriction is ahomeomorphism. Hence the restriction map is a homeomorphism.The fact that S ♯ is isomorphic to A( K ) now follows from Theo-rem 4.1. (cid:3) Theorem 4.5.
Let S be an operator system with nc quasistate space ( K, z ) . Then S is isomorphic to A( K, z ) .Proof. For s ∈ S , the function ˆ s : K → M defined by ˆ s ( x ) = x ( s ) for x ∈ K is a pointed continuous affine nc function. Define ϕ : S → A( K, z ) by ϕ ( s ) = ˆ s for s ∈ S . We will show that ϕ is a completelyisometric complete order isomorphism. It is clear that ϕ is completelypositive. The fact that it is completely isometric follows from Theorem2.15.To see that ϕ is surjective, we will first consider another descriptionof ϕ . Let A( S ∗ ) denote the space of continuous affine functions on S ∗ ,where S ∗ is equipped with the weak* topology. Also, let A( S ∗ , z ) ⊆ A( S ∗ ) denote the subspace of functions that are pointed at z , i.e. zeroat z .For s ∈ S , let ˜ s : S ∗ → C denote the function defined by ˜ s ( x ) = x ( s ) for x ∈ S ∗ . Then ˜ s is continuous and linear, and ˜ s ( z ) = 0 ,so ˜ s ∈ A( S ∗ , z ) . On the other hand, since functions in A( S ∗ , z ) are continuous linear functions on S ∗ , every function in A( S ∗ , z ) is of thisform.Identify K with a compact convex subset of S ∗ and let A( K ) and A( K , z ) denote the spaces of continuous affine functions and pointedcontinuous affine functions on K respectively. Then for s ∈ S , ˜ s | K ∈ A( K , z ) ⊆ A( K ) and ˜ s | K = ˆ s | K . Moreover, Kadison’s representa-tion theorem [12] (see [19] for a complex version) implies that A( K ) isorder isomorphic to the unital operator system A( K ) . Since elementsin A( K, z ) are completely determined by their restriction to K , it fol-lows that ϕ is obtained by composing the map S → A( K , z ) : s → ˜ s | K with this order isomorphism.By [1, Corollary I.1.5], the range of the restriction map A( S ∗ ) → A( K ) is uniformly dense in A( K ) . Observe that this implies that therange of the restriction map A( S ∗ , z ) → A( K , z ) is uniformly densein A( K , z ) . Indeed, for a ∈ A( S ∗ ) , a − a ( z )1 A( S ∗ ) ∈ A( S ∗ , z ) , so anynet in A( S ∗ ) with an image under the restriction map that convergesuniformly to a function in A( K , z ) can be replaced by a net in A( S ∗ , z ) with the same property. It follows from above that ϕ has dense range.Since ϕ is completely isometric, and in particular is isometric, it followsthat ϕ is surjective.Finally, to see that ϕ is a complete order isomorphism, let P = ⊔ P n and Q = ⊔ Q n denote the matrix cones of S and A( K, z ) respectively.If ϕ is not a complete order isomorphism, then there is s ∈ M n ( S ) such that s / ∈ P n but ϕ ( s ) ∈ Q n . Suppose that this is the case. Wewill apply a separation argument to obtain a contradiction.Identify S with its image under the canonical embedding into itsbidual S ∗∗ and define M ⊆ M ( S ∗∗ ) by M = P = ⊔ P m , where theclosure is taken with respect to the weak* topology. Since P is ncconvex, M is nc convex. Hence M is a weak* closed nc convex set.Furthermore, since P n is convex and uniformly closed, it is weaklyclosed, implying s / ∈ M . Therefore, by the nc separation theorem[6, Theorem 2.4.1] there is a self-adjoint normal completely boundedlinear map ψ : S ∗∗ → M n such that ψ ( s )
6≥ − n ⊗ n but ψ ( t ) ≥− n ⊗ p for all t ∈ M p .Since ψ is normal, it can be identified with the unique normal ex-tension of a map ψ : S → M n satisfying ψ ( s )
6≥ − n ⊗ n but ψ ( t ) ≥ − n ⊗ p for all t ∈ P p . Then in particular, ψ ( s ) . However,since P is closed under multiplication by positive scalars, for t ∈ P p and α > , ψ ( t ) ≥ − α − n ⊗ p . Taking α → ∞ implies ψ ( t ) ≥ . Hence ψ ≥ . Multiplying ψ by a sufficiently small positive scalar, we obtaina quasistate x ∈ K such that x ( s ) . But then ˆ s ( x ) = x ( s ) , so ϕ ( s ) = ˆ s , contradicting the assumption that ϕ ( s ) ∈ Q n . (cid:3) ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 21
Corollary 4.6.
Let S be an operator system with nc quasistate space ( K, z ) . Then ( K, z ) is a pointed compact nc convex set.Proof. By Theorem 4.5, we can identify S with the operator system A( K, z ) , and by definition, every nc quasistate on A( K, z ) belongs to K . (cid:3) Corollary 4.7.
Let ( K, z ) be a pair consisting of a compact nc convexset K and a point z ∈ K . The following are equivalent:(1) The pair ( K, z ) is a pointed compact nc convex set.(2) The nc quasistate space of the operator system A( K, z ) is ( K, z ) .(3) The operator system A( K, z ) satisfies A( K, z ) ♯ = A( K ) .Proof. (1) ⇒ (2) If ( K, z ) is a pointed compact nc convex set thenby definition every nc quasistate on A( K, z ) belongs to K . Since ev-ery point in K is an nc quasistate on A( K, z ) , it follows that the ncquasistate space of A( K, z ) is ( K, z ) .(2) ⇒ (3) If the nc quasistate space of A( K, z ) is ( K, z ) , then Propo-sition 4.4 implies that the nc state space of A( K, z ) ♯ is K . It followsfrom Theorem 4.2 that A( K, z ) ♯ = A( K ) .(3) ⇒ (1) If A( K, z ) ♯ = A( K ) , then since every nc quasistate on A( K, z ) extends to an nc state on A( K, z ) ♯ , and since K is nc statespace of A( K ) , it follows that every nc quasistate of A( K, z ) belongsto K . Hence ( K, z ) is a pointed compact nc convex set. (cid:3) The next result follows immediately from Theorem 4.5 and Corollary4.7. It is an analogue of the representation theorem [6, Theorem 3.2.3].
Theorem 4.8.
Let S be an operator system with nc quasistate space ( K, z ) . The map S ♯ → A( K ) : s → ˆ s defined by ˆ s ( x ) = x ♯ ( s ) for x ∈ K, is a unital complete order isomorphism that restricts to a completelyisometric complete order isomorphism from S to A( K, z ) . Hence S isisomorphic to A( K, z ) . Theorem 4.5 and Corollary 4.7 imply the dual equivalence of thecategory
OpSys of operator systems and the category
PoNCConv ofpointed compact nc convex sets. The contravariant functor
OpSys → PoNCConv is defined in the following way:(1) An operator system S is mapped to its nc quasistate space.(2) For operator systems S and T with nc quasistate spaces ( K, z ) and ( L, w ) respectively, a morphism ϕ : S → T is mapped tothe morphism ϕ d : L → K defined by ϕ d ( y )( a ) = ϕ ( a )( y ) , for y ∈ L and a ∈ A( K, z ) . The inverse functor
PoNCConv → OpSys is defined in the followingway:(1) A pointed compact nc convex set ( K, z ) is mapped to the op-erator system A( K, z ) .(2) If ( K, z ) and ( L, w ) are compact nc convex sets and ψ : L → K is a morphism, then the corresponding morphism ψ d : A( K, z ) → A( L, w ) is defined by ψ d ( a )( y ) = a ( ψ ( y )) , for a ∈ A( K, z ) and y ∈ L. The next result summarizes this discussion.
Theorem 4.9.
The contravariant functors
OpSys → PoNCConv and
PoNCConv → OpSys defined above are inverses. Hence the categories
OpSys and
PoNCConv are dually equivalent.
The next result characterizing isomorphic operator systems is ananalogue of [6, Corollary 3.2.6]. It follows immediately from Theorem4.9.
Corollary 4.10.
Let ( K, z ) and ( L, w ) be compact pointed nc convexsets. Then A( K, z ) and A( L, w ) are isomorphic if and only if ( K, z ) and ( L, w ) are pointedly affinely homeomorphic. Hence two operatorsystems are isomorphic if and only if their nc quasistate spaces arepointedly affinely homeomorphic. We saw in Example 2.14 that if S and T are operator systems and ϕ : S → T is a completely contractive complete order embedding, thenit is not necessarily true that the unitization ϕ ♯ : S ♯ → T ♯ is completelyisometric. In other words, ϕ is not necessarily an embedding. However,we can now state necessary and sufficient conditions for ϕ to be anembedding.The following result follows immediately from Theorem 4.2, Theorem4.9 and the discussion preceding the statements of these results. Lemma 4.11.
Let ( K, z ) and ( L, w ) be pointed compact nc convexsets and let ϕ : A( K, z ) → A( L, w ) be a completely contractive andcompletely positive map. Let ϕ d : L → K denote the correspondingcontinuous affine map defined as in Theorem 4.9. Then ϕ d coincideswith the continuous affine map obtained by applying Theorem 4.2 tothe unitization ϕ ♯ : A( K ) → A( L ) . Corollary 4.12.
Let ( K, z ) and ( L, w ) be pointed compact nc convexsets. Let ϕ : A( K, z ) → A( L, w ) be a completely contractive and com-pletely positive map and let ϕ d : L → K denote the pointed continuousaffine map given by applying Theorem 4.9 to ϕ . Then ϕ is an embed-ding if and only if ϕ d is surjective. ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 23
Proof.
By Lemma 4.11, the map ϕ d coincides with the map obtainedby applying Theorem 4.2 to the unitization ϕ ♯ : A( K ) → A( L ) . ByProposition 4.3, ϕ ♯ is completely isometric if and only if ϕ d is an em-bedding. (cid:3) Pointed noncommutative functions
Noncommutative functions.
In order to define a more generalnotion of continuous nc function, it is necessary to introduce the point-strong topology on a compact nc convex set K . This is the weakesttopology on each K n making the maps K n → C : x → ξ ∗ a ( x ) η contin-uous for all a ∈ A( K ) and all vectors ξ, η ∈ H n .The following definition is essentially [6, Definition 4.2.1]. Definition 5.1.
Let K be a compact nc convex set. An nc function on K is an nc map f : K → M in the sense of Definition 3.6. An ncfunction f is continuous if it is continuous with respect to the point-strong topology on K from above. We will write B( K ) and C( K ) forthe unital C*-algebras of bounded and continuous nc functions on K respectively. Remark 5.2.
It is clear that A( K ) ⊆ C( K ) ⊆ B( K ) . The producton B( K ) is the pointwise product, meaning that for f, g ∈ B( K ) and x ∈ K , ( f g )( x ) = f ( x ) g ( x ) . The adjoint is defined by f ∗ ( x ) = f ( x ) ∗ for f ∈ B( K ) and x ∈ K . By [6, Theorem 4.4.3], C( K ) = C * (A( K )) .We will say more about the C*-algebra C( K ) in Section 6.For x ∈ K n , we will write δ x : B( K ) → M n for the point evaluation*-homomorphism defined by δ x ( f ) = f ( x ) for f ∈ B( K ) . This is anoncommutative analogue of an evaluation functional, since for f ∈ C( K ) , δ x ( f ) = f ( x ) .Elements in the enveloping von Neumann algebra C( K ) ∗∗ can nat-urally be identified with bounded nc functions on K . Specifically, for x ∈ K n , the *-homomorphism δ x : C( K ) → M n has a unique extensionto a normal *-homomorphism δ ∗∗ x : C( K ) ∗∗ → M n . For f ∈ C( K ) ∗∗ ,the function ˜ f : K → M defined by ˜ f ( x ) = δ ∗∗ x ( f ) for x ∈ K is abounded nc function and hence belongs to B( K ) . In fact, much morecan be said.The following result is contained in [6, Theorem 4.4.3] and [6, Corol-lary 4.4.4]. Theorem 5.3.
Let K be a compact nc convex set. The map σ :C( K ) ∗∗ → B( K ) defined as above is a normal *-isomorphism that re-stricts to a normal unital complete order isomorphism from A( K ) ∗∗ onto the unital operator system A b ( K ) of bounded affine nc functions. Pointed noncommutative functions.Definition 5.4.
Let ( K, z ) be a pointed compact nc convex set. Wewill say that an nc function f : K → M is pointed if f ( z ) = 0 . Welet B( K, z ) denote the space of pointed bounded nc functions on K .Similarly, we let C( K, z ) = C( K ) ∩ B( K, z ) denote the space of pointedcontinuous nc functions on K . Remark 5.5.
It is clear that B( K, z ) is a closed two-sided ideal of B( K ) and that C( K, z ) is a closed two-sided ideal of C( K ) . In particular, B( K, z ) and C( K, z ) are C*-algebras. Furthermore, it follows from theidentification C( K ) ∗∗ = B( K ) that the representation δ z is normal on B( K ) . Hence B( K, z ) is a weak*-closed ideal of B( K ) . Proposition 5.6.
Let ( K, z ) be a pointed compact nc convex set. Then C( K, z ) ♯ = C( K ) and C( K, z ) = C * (A( K, z )) .Proof. By Corollary 4.7, A( K, z ) ♯ = A( K ) . Hence A( K ) = A( K, z ) + C A( K ) . Since C( K ) = C * (A( K )) , it follows that C( K ) = C( K, z ) + C .Hence C( K, z ) = C * (A( K, z )) .To see that C( K, z ) ♯ = C( K ) , it suffices to show that for any *-homomorphism π : C( K, z ) → M n , there is a unital *-homomorphism ˜ π : C( K ) → M n extending π . The restriction π | A( K,z ) is an nc qua-sistate, so by the assumption that ( K, z ) is pointed, it is given byevaluation at a point x ∈ K n . Then the unital *-homomorphism δ x : C( K ) → M n extends π | A( K,z ) . Since A( K, z ) generates C( K, z ) , itfollows that δ x | C( K,z ) = π . (cid:3) The next result follows from restricting the *-isomorphism in thestatement of Theorem 5.3.
Theorem 5.7.
Let ( K, z ) be a compact pointed nc convex set. Thenthe map C( K, z ) ∗∗ → B( K, z ) : f → ˜ f defined by ˜ f ( x ) = δ ∗∗ x ( f ) for f ∈ C( K, z ) ∗∗ and x ∈ K, is a normal *-isomorphism of von Neumann algebras that restrictsto a normal completely isometric complete order isomorphism from A( K, z ) ∗∗ onto the operator system A b ( K, z ) of pointed bounded affinenc functions. Minimal and maximal C*-covers
The deepest results in [6] arise from the interplay between unital op-erator systems of continuous affine nc functions on compact nc convexsets and unital C*-covers of nc functions on the sets. Connes and vanSuijlekom [4] introduced an analogous notion of C*-cover for operator
ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 25 systems. In this section we will review the notion of a unital C*-coverof a unital operator system before considering the more general notionof a C*-cover of an operator system.6.1.
Minimal and maximal unital C*-covers.
Let S be a unitaloperator system.(1) A pair ( A, ι ) consisting of a unital C*-algebra A and an embed-ding ι : S → A is a unital C*-cover of S if A = C * ( ι ( S )) .(2) If ( A ′ , ι ′ ) is another unital C*-cover of S , then we will say that ( A, ι ) and ( A, ι ′ ) are equivalent if there is a unital *-isomorphism π : A → A ′ such that π ◦ ι = ι ′ .(3) We will say that a unital C*-cover ( A, ι ) of S is maximal iffor any unital C*-cover ( B, ϕ ) of S , there is a surjective unital*-homomorphism σ : A → B such that ϕ = σ ◦ ι . S A = C ∗ ( ι ( S )) B = C ∗ ( ϕ ( S )) ιϕ σ (4) We will say that a unital C*-cover ( A, ι ) of S is minimal iffor any unital C*-cover ( B, ϕ ) of S , there is a surjective unital*-homomorphism π : B → A such that π ◦ ϕ = ι . S A = C ∗ ( ι ( S )) B = C ∗ ( ϕ ( S )) ιϕ π The existence and uniqueness of the maximal unital C*-cover of aunital operator system was established by Kirchberg and Wassermann[16]. The following result is non-trivial. It is implied by [6, Theorem4.4.3].
Theorem 6.1.
Let K be a compact nc convex set. The maximal unitalC*-cover for the unital operator system A( K ) is the C*-algebra C( K ) of continuous nc functions on K . The existence and uniqueness of the minimal unital C*-cover of aunital operator system was established by Hamana [10]. The results in[6] and [15] imply a description in terms of the nc state space of theoperator system, which we will now describe.Let K be a compact nc convex set. It follows from Theorem 6.1 thatthere is a surjective *-homomorphism π from C( K ) onto the minimalunital C*-cover of A( K ) . A result of Dritschel and McCullough [8] implies that ker π is the boundary ideal in C( K ) relative to A( K ) , i.e.the unique largest ideal in C( K ) with the property that the restrictionof the corresponding quotient *-homomorphism to A( K ) is completelyisometric.Let I ∂K = ker π and let C( ∂K ) = C( K ) /I ∂K . We will refer to C( ∂K ) as the minimal unital C*-cover of A( K ) . In order to explain this choiceof notation and give a description of C( ∂K ) in terms of K , we requirethe spectral topology from [15, Section 9]. Definition 6.2.
Let K be a compact nc convex set. We will say thata point x ∈ K is reducible if x is unitarily equivalent to a direct sum x ≃ y ⊕ z for points y, z ∈ K . We will say that x is irreducible if it isnot reducible, and we will write Irr( K ) for the set of irreducible pointsin K . Remark 6.3.
Note that a point x ∈ K is irreducible if and only if thecorresponding *-homomorphism δ x is. In particular, ∂K ⊆ Irr( K ) .Let K be a compact nc convex set. Let Spec(C( K )) denote the C*-algebraic spectrum of C( K ) , i.e. the set of unitary equivalence classesof irreducible representations of C( K ) equipped with the hull-kerneltopology. For a point x ∈ Irr( K ) , we have already observed that the*-homomorphism δ x is irreducible. Hence letting [ δ x ] denote the uni-tary equivalence class of δ x , [ δ x ] ∈ Spec(C( K )) . Note that the map Irr( K ) → Spec(C( K )) : x → [ δ x ] is surjective. Definition 6.4.
The spectral topology on Irr( K ) is the pullback of thehull-kernel topology on Spec(C( K )) . Specifically, the open subsets of Irr( K ) are the preimages of open subsets of Spec(C( K )) under the map Irr( K ) → Spec(C( K )) : x → [ δ x ] .The results in [15, Section 9] imply that I ∂K = { f ∈ C( K ) : f ( x ) = 0 for all x ∈ ∂K } , where ∂K denotes the closure of ∂K with respect to the spectral topol-ogy on Irr( K ) .6.2. Minimal and maximal C*-covers.
Connes and van Suijlekom[4] introduced an analogue for operator systems of a unital C*-coverof a unital operator system from Section 6, which they refer to as aC ♯ -cover. We will instead refer to C*-covers. Definition 6.5.
Let S be an operator system.(1) We will say that a pair ( A, ι ) consisting of a C*-algebra A andan embedding ι : S → A is a C*-cover of S if A = C * ( ι ( S )) . ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 27 (2) If ( A ′ , ι ′ ) is another C*-cover of S , then we will say that ( A, ι ) and ( A, ι ′ ) are equivalent if there is a *-isomorphism π : A → A ′ such that π ◦ ι = ι ′ .(3) We will say that a C*-cover ( A, ι ) of S is maximal if for anyC*-cover ( B, ϕ ) of S there is a surjective *-homomorphism σ : A → B such that ϕ = σ ◦ ι . S A = C ∗ ( ι ( S )) B = C ∗ ( ϕ ( S )) ιϕ σ (4) We will say that a C*-cover ( A, ι ) of S is minimal if for anyC*-cover ( B, ϕ ) of S , there is a surjective *-homomorphism π : B → A such that π ◦ ϕ = ι . S A = C ∗ ( ι ( S )) B = C ∗ ( ϕ ( S )) ιϕ π Remark 6.6.
Let ( K, z ) be a pointed compact nc convex set. If ( A, ι ) is a C*-cover for A( K, z ) , then since ϕ is an embedding, the unitization ϕ ♯ : A( K ) → A ♯ is an embedding. Hence ( A ♯ , ι ♯ ) is a unital C*-coverof A( K ) .The existence and uniqueness of the minimal C*-cover of an op-erator system was established in [4, Theorem 2.2.5] under the nameC ♯ -envelope. In this section we will prove the existence and unique-ness of the maximal C*-cover, and we will describe the maximal andminimal C*-covers of an operator system in terms of the maximal andminimal unital C*-covers of its unitization. Proposition 6.7.
Let S be an operator system. If the maximal andminimal C*-covers of S exist, then they are unique up to equivalence.Proof. Let ( A, ι ) and ( A ′ , ι ′ ) be maximal C*-covers for S . Then bydefinition there are surjective homomorphisms σ : A → A ′ and σ ′ : A ′ → A such that ι ′ = σ ◦ ι and ι = σ ′ ◦ ι ′ . Hence σ − = σ ′ , so σ is a*-isomorphism and hence ( A, ι ) and ( A, ι ′ ) are equivalent. The prooffor the minimal C*-cover is similar. (cid:3) Theorem 6.8.
Let ( K, z ) be a compact pointed nc convex set.(1) The C*-algebra C( K, z ) is a maximal C*-cover for A( K, z ) withrespect to the canonical inclusion. (2) Let I ∂K denote the boundary ideal in the C*-algebra C( K ) ofcontinuous nc functions on K relative to A( K ) , so that the C*-algebra C( K ) /I ∂K ∼ = C( ∂K ) is the minimal unital C*-cover of A( K ) , and let I ( ∂K,z ) = I ∂K ∩ C( K, z ) . Then the C*-algebra C( K, z ) /I ( ∂K,z ) is the minimal C*-cover of A( K, z ) with respectto the quotient *-homomorphism.Proof. (1) Let ( B, ϕ ) be a C*-cover for A( K, z ) . We can assume that B ⊆ M n for some n , so that ϕ = x and B = δ x (C( K, z )) for some x ∈ K n . It follows that C( K, z ) is a maximal C*-cover for A( K, z ) with respect to the canonical inclusion.(2) Since C( K ) /I ∂K is a unital C*-cover for A( K ) , C( K, z ) /I ( ∂K,z ) is a C*-cover for A( K, z ) . To see that it is minimal, it suffices to showthat if ( B, ϕ ) is any C*-cover for A( K, z ) , then ker σ ⊆ I ( ∂K,z ) .By (1), there is a surjective unital *-homomorphism σ : C( K, z ) → B such that σ | A( K,z ) = ϕ . The unitization σ ♯ : C( K ) → B ♯ is a unital*-homomorphism satisfying σ ♯ | A( K ) = ϕ ♯ . Since ϕ is an embedding, ϕ ♯ is completely isometric, so ker σ ♯ ⊆ I ∂K . Hence ker σ ⊆ I ( ∂K,z ) . (cid:3) Definition 6.9.
Let ( K, z ) be a pointed compact nc convex set. Let I ( ∂K,z ) denote the ideal in C( K, z ) from Theorem 6.8 and let C( ∂K, z ) =C( K, z ) /I ( ∂K,z ) . We will refer to C( ∂K, z ) as the minimal C*-cover of A( K, z ) , and we will refer to the corresponding quotient *-homomorphismas the canonical embedding of A( K, z ) into C( ∂K, z ) . Remark 6.10.
The ideal I ( ∂K,z ) = ker π is a pointed analogue of theboundary ideal from Section 6.1. It is the largest ideal in C( K, z ) such that the corresponding quotient *-homomorphism restricts to anembedding of A( K, z ) . Example 6.11.
Define a, b ∈ M by a = (cid:20) − (cid:21) , b = (cid:20) − / (cid:21) . Let S = span { a } and T = span { b } . Then S and T are nonunitaloperator systems, and it is not difficult to verify that S and T areisomorphic to the operator systems considered in Example 3.16 andExample 3.17 respectively.Let ( K, z ) denote the nc quasistate space of S . Note that this isthe same ( K, z ) from Example 3.16. Since K = [ − , is a simplex,the results in [15] imply that ∂K = ∂K = {− , } . Hence identifying S with A( K, z ) , the minimal C*-cover of S ♯ = A( K ) is C( ∂K ) =C( {− , } ) ∼ = C . ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 29
Let ι : A( K ) → C( ∂K ) denote the canonical embedding. Then ι (A( K, z )) ∼ = { ( − α, α ) : α ∈ C } ∼ = C . Hence C( ∂K , z ) ∼ = C . Note that C( ∂K, z ) is unital even though A( K, z ) is nonunital.Define θ : S → T defined by θ ( αa ) = αb for α ∈ C . Then arguing asin Example 3.17, θ is an isomorphism. Hence the minimal C*-cover of T is also isomorphic to C . Example 6.12.
Let A be a C*-algebra. Then A is clearly a C*-coverof itself with respect to the identity map. By definition, there is a sur-jective *-homomorphism π : A → C *min ( A ) that is completely isometricon A . Therefore, π is a *-isomorphism, implying A = C *min ( A ) .Let K be a compact nc convex set. A useful fact implied by [5,Theorem 3.4] and [6, Proposition 5.2.4] is that the direct sum of thepoints in ∂K extends to a faithful representation of the minimal unitalC*-cover C( ∂K ) . Specifically, define y ∈ K by y = ⊕ x ∈ ∂k x . Thenthe *-homomorphism δ y satisfies ker δ y = ker I ∂K . Hence δ y (C( K )) ∼ =C( ∂K ) . The following result is an analogue of this fact for the minimalC*-cover of an operator system. Proposition 6.13.
Let ( K, z ) be a pointed compact nc convex set. De-fine y ∈ K by y = ⊕ x ∈ ∂K \{ z } x . Then the *-homomorphism δ y satisfies ker δ y = I ( ∂K,z ) , where I ( ∂K,z ) is the ideal from Theorem 6.8. Hence δ y (C( K, z )) ∼ = C( ∂K , z ) .Proof. By [6, Proposition 5.2.4], every point in ∂K , considered as annc state on A( K ) , factors through C( ∂K ) . For x ∈ ∂K \ { z } , it followsfrom the nc separation theorem [6, Corollary 2.4.2] that δ x is nonzero on A( K, z ) . Hence every nc state on A( K, z ) in ∂K \ { z } factors through C( ∂K, z ) . (cid:3) We will say more about the minimal C*-cover in Section 7.7.
Characterization of unital operator systems
In this section we will apply the results from Section 6 to establisha characterization of operator systems that are unital in terms of theirnc quasistate space. We note that a closely related problem, of char-acterizing operator spaces that are unital operator systems, has beenconsidered by Blecher and Neal [2].
Theorem 7.1.
Let ( K, z ) be a pointed compact nc convex set. Thefollowing are equivalent for a pointed continuous affine nc function e ∈ A( K, z ) : (1) The function e is a distinguished archimedean matrix order unitfor A( K, z ) .(2) The image of e under the canonical embedding of A( K, z ) into C( ∂K , z ) is the identity.(3) For every n and every x ∈ ( ∂K \ { z } ) n , e ( x ) = 1 n .Proof. (1) ⇒ (2) Suppose that e is a distinguished archimedean matrixorder unit for A( K, z ) . Then A( K, z ) is a unital operator system, soit follows from [3, Theorem 4.4] that there is y ∈ K n such that y isa unital complete isometry on A( K, z ) with e ( y ) = 1 n . Then ker δ y is contained in the boundary ideal I ∂K,z from Remark 6.10. It followsthat the canonical embedding of A( K, z ) into C( ∂K , z ) factors through y , and hence maps e to the identity.(2) ⇒ (3) Suppose that the image of e under the canonical embeddingof A( K, z ) into C( ∂K , z ) is the identity. Proposition 6.13 implies thatthe restriction to A( K, z ) of every nc quasistate in ∂K \ { z } factorsthrough C( ∂K, z ) . It follows that for x ∈ ( ∂K \ { z } ) n , e ( x ) = 1 .(3) ⇒ (1) Suppose that for every n and every x ∈ ( ∂K \ { z } ) n , e ( x ) = 1 n . Then it follows from Proposition 6.13 that the image of e under the canonical embedding of A( K, z ) into C( ∂K , z ) is the identity.It follows that e is a distinguished archimedean matrix order unit for C( ∂K, z ) , and hence also for A( K, z ) . (cid:3) Corollary 7.2.
Let S be an operator system with nc quasistate space ( K, z ) . The following are equivalent:(1) The operator system S is unital.(2) There is e ∈ S such that for every n and every x ∈ ( ∂K \ { z } ) n , e ( x ) = 1 n The next result is [4, Theorem 2.25 (ii)].
Corollary 7.3.
Let S be a unital operator system. Then the minimalunital C*-cover of S and the minimal C*-cover of S coincide.Proof. Let ( A, ι ) and ( B, κ ) denote the minimal unital C*-cover of S and the minimal C*-cover of S respectively. It follows from Theorem7.1 and Proposition 6.13 that B is unital. Hence by the universalproperty of A , there is a surjective *-homomorphism π : A → B suchthat π ◦ ι = κ . On the other hand, by the universal property of B ,there is a surjective *-homomorphism σ : B → A such that σ ◦ κ = ι .Hence A and B are isomorphic. (cid:3) ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 31 Quotients of operator systems
In this section we will utilize the dual equivalence between the cate-gory of operator systems and the category of pointed compact nc con-vex sets to develop a theory of quotients for operator systems. We willshow that the theory developed here extends the theory of quotientsfor unital operator systems developed by Kavruk, Paulsen, Todorovand Tomforde [14]. We note that the the theory of quotients for uni-tal operator systems can be developed in a similar way using the dualequivalence between the category of unital operator systems and thecategory of compact nc convex sets from [6, Section 3].
Definition 8.1.
Let S be an operator system and let ( K, z ) denote thenc quasistate space of S . We will say that a subset J ⊆ S is a kernel if there is an nc quasistate x ∈ K such that J = ker x . Remark 8.2.
For x ∈ K n , the closure of the image x ( S ) ⊆ M n is anoperator system. Hence J is a kernel if and only if there is an operatorsystem T and a completely contractive and completely positive map ϕ : S → T with ker ϕ = J .Let S be an operator system and let ( K, z ) denote the nc quasistatespace of S . For a subset Q ⊆ S , the annihilator of Q is Q ⊥ = { x ∈ K : a ( x ) = 0 for all a ∈ Q } . Note that Q ⊥ is a closed nc convex set.Similarly, for a subset X ⊆ K , the annihilator of X is X ⊥ = { a ∈ S : a ( x ) = 0 for all x ∈ X } .The next result is a noncommutative analogue of [1, II.5.3]. Lemma 8.3.
Let K be a compact nc convex set and let X ⊆ K bea subset. Then X ⊥⊥ = Y ∩ K , where Y ⊆ M (A( K ) ∗ ) denotes theclosed nc convex hull generated by ⊔ span X n , where span X n is takenin M n (A( K ) ∗ ) .Proof. It is clear that Y ∩ K ⊆ X ⊥⊥ . For the other inclusion, supposefor the sake of contradiction there is z ∈ ( X ⊥⊥ ) n \ ( Y ∩ K ) . Then z / ∈ Y . Hence by the nc separation theorem [6, 2.4.1], there is a self-adjoint element a ∈ A( K ) satisfying a ( z ) n ⊗ n but a ( y ) ≤ n ⊗ p for all y ∈ Y p . Since each Y p is a subspace, this forces a ( y ) = 0 forall y ∈ Y p . Hence viewing a as an n × n matrix a = ( a ij ) over A( K ) , a ij ( y ) = 0 for all y ∈ Y . In particular, a ij ( x ) = 0 for all x ∈ X . Hence a ij ∈ X ⊥ for all i, j . Since z ∈ X ⊥⊥ , it follows that a ij ( z ) = 0 for all i, j . Therefore, a ( z ) = 0 , giving a contradiction. (cid:3) Proposition 8.4.
Let ( K, z ) be a pointed compact nc convex set. Asubset J ⊆ A( K, z ) is a kernel if and only if J = J ⊥⊥ . If J is a kernel and M = J ⊥ , then the completely contractive completely positiverestriction map A( K, z ) → A( M, z ) has kernel J . Moreover, z ∈ M and the pair ( M, z ) is a pointed compact nc convex set.Proof. Suppose that J = J ⊥⊥ . Let M = J ⊥ . Then J = M ⊥ . Let r :A( K, z ) → A( M, z ) denote the restriction map. Then r is completelycontractive and completely positive and ker r = M ⊥ = J . Hence J isa kernel.Conversely, suppose that J is a kernel. It is clear that J ⊆ J ⊥⊥ . Forthe other inclusion, choose x ∈ K such that J = ker x . Let T denotethe closure of the image A( K, z )( x ) ⊆ M n . Then T is an operatorsystem. Letting ( L, w ) denote the nc quasistate space of T , we canidentify T with A( L, w ) . Let ψ : L → K denote the continuous affinemap obtained by applying Theorem 4.9 to x . Then for a ∈ J and y ∈ L , a ( x )( y ) = a ( ψ ( y )) . Hence ψ ( L ) ⊆ J ⊥ , so for a ∈ J ⊥⊥ and y ∈ L , a ( ψ ( y )) = a ( x )( y ) , i.e. a ( x ) = 0 . Hence J ⊥⊥ ⊆ J , so J = J ⊥⊥ .If J is a kernel and M = J ⊥ , then clearly z ∈ M . To see that ( M, z ) is a pointed compact nc convex set, let θ : A( M, z ) → M n be an ncquasistate. Let r : A( K, z ) → A( M, z ) denote the restriction map fromabove. Then the composition θ ◦ r is an nc quasistate on A( K, z ) . Since ( K, z ) is a pointed compact nc convex set, by definition there is x ∈ K such that θ ◦ r = x . Since x factors through r , x ∈ J ⊥ = M . (cid:3) Definition 8.5.
Let S be an operator system and let ( K, z ) denotethe nc quasistate space of S . For a kernel J ⊆ S , we let S/J denotethe operator system A( M, z ) , where M = J ⊥ . We will refer to S/J as the quotient of S by J , and we will refer to the restriction map S → A( M, z ) obtained by identifying S with A( K, z ) as the canonicalquotient map . Remark 8.6.
Note that we have applied Theorem 4.8 to identify S with A( K, z ) . It is clear that the canonical quotient map A( K, z ) → A( M, z ) is completely contractive and completely positive.The next result characterizes operator system quotients in terms ofa natural universal property. It is an analogue of [14, Proposition 3.6]. Theorem 8.7.
Let S be an operator system and let J ⊆ S be a ker-nel. The quotient S/J is the unique operator system up to isomorphismsatisfying the following universal property: there is a completely con-tractive and completely positive map ϕ : S → S/J , and whenever T is an operator system and ψ : S → T is a completely contractive andcompletely positive map with J ⊆ ker ψ , then ψ factors through ϕ . In ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 33 other words, there is a completely contractive and completely positivemap ω : S/J → T such that ψ = ω ◦ ϕ . J S S/JT ϕ ωψ
Proof.
To see that
S/J satisfies this universal property, first note thatthe canonical quotient map ϕ : S → S/J is completely contractive andcompletely positive. Let T be an operator system and let ψ : S → T bea completely contractive and completely positive map with J ⊆ ker ψ .Letting ( K, z ) and ( L, w ) denote the nc quasistate spaces of S and T respectively, we can assume that S = A( K, z ) and T = A( L, w ) . Let ψ d : L → K denote the continuous affine nc map obtained by applyingTheorem 4.8 to ψ .Let M = J ⊥ . For a ∈ J and y ∈ L , the fact that J ⊆ ker ψ impliesthat ψ ( a )( y ) = a ( ψ d ( y )) . Hence ψ d ( L ) ⊆ J ⊥ = M . Restrictingthe codomain of ψ d to M and applying Theorem 4.8 to ψ d , we obtaina completely contractive and completely positive map ω : A( M, z ) → A( L, w ) such that ω ◦ ϕ = ψ .To see that S/J is the unique operator system with this universalproperty, suppose that R is another operator system that satisfies theproperty from the statement of the theorem, then there are surjectivecompletely contractive and completely positive maps S/J → R and R → S/J such that the composition is the identity map on
S/J . Itfollows that each of the individual maps must be a completely isometriccomplete order isomorphism. Hence R is isomorphic to S/J . (cid:3) In order to relate our theory of quotients of operator systems to thetheory of quotients of unital operator systems from [14], we require thefollowing result.
Lemma 8.8.
Let ( K, z ) be a pointed compact nc convex set such that A( K, z ) is a unital operator system and let e ∈ A( K, z ) denote thedistinguished archimedean matrix order unit. Let J ⊆ A( K, z ) be akernel and let M = J ⊥ . Then for x ∈ ∂M \ { z } , e ( x ) = 1 .Proof. Let K and K denote the closed nc convex hulls of { z } and ∂K \ { z } respectively. By Theorem 7.1, e ( x ) = 1 for x ∈ ∂K \ { z } .Hence by the continuity of e , e ( x ) = 1 for all x ∈ K . Since e ( z ) = 0 ,in particular this implies that K K . Since every point in K is adirect sum of copies of z , it follows that z ∈ ∂K .From above, for x ∈ ∂K , either e ( x ) = 1 or e ( x ) = 0 . It follows from[5, Theorem 3.4] and [6, Proposition 5.2.4] that the image of e under the canonical embedding of A( K ) into its minimal unital C*-cover C( ∂K ) is a projection in the center of C( ∂K ) .Choose x ∈ K m and let y ∈ K n be a maximal dilation of x . Thenthere is an isometry α ∈ M n,m such that x = α ∗ yα . By [6, Proposition5.2.4], the *-homomorphism δ y factors through C( ∂K ) . Hence fromabove, y decomposes as a direct sum y = y ⊕ y for y ∈ K n and y ∈ K n . This implies that x can be written as an nc convex combination x = α ∗ y α + α ∗ y α for α ∈ M n ,m and α ∈ M n ,m satisfying α ∗ α + α ∗ α = 1 m .Suppose x ∈ ( ∂M ) m and write x can be written as an nc convexcombination x = α ∗ y α + α ∗ y α as above. Then for a ∈ J , a ( x ) = α ∗ a ( y ) α + α ∗ a ( y ) α . Since y is a direct sum of copies of z , a ( y ) = 0 . Hence α ∗ a ( y ) α = 0 .From above, we can decompose y with respect to the range of α as y = (cid:20) u ∗∗ ∗ (cid:21) for u ∈ M k , and there is β ∈ M k,m such that β ∗ u β = α ∗ y α and α ∗ α + β ∗ β = 1 m .Now since y , u ∈ M and x ∈ ( ∂M ) m , it follows that either α = 0 or β = 0 . Hence either x ∈ K or x ∈ K . In the former case, x = z ,while in the latter case, e ( x ) = 1 . (cid:3) Proposition 8.9.
Let S be a unital operator system. Then for everykernel J ⊆ S , the quotient operator system S/J is unital.Proof.
Letting ( K, z ) denote the nc quasistate space of S , we can as-sume that S = A( K, z ) . Let M = J ⊥ , so that S/J = A(
M, z ) , andlet ϕ : A( K, z ) → A( M, z ) denote the canonical quotient map. Let e ∈ A( K, z ) denote the distinguished archimedean matrix order unit.Then for x ∈ ∂M \ { z } , Corollary 8.8 implies that e ( x ) = 1 . Hence byTheorem 7.1, ϕ ( e ) is an archimedean matrix order unit. (cid:3) Remark 8.10. If S is a unital operator system and J is the kernel ofa unital completely positive map, then the quotient S/J from Defini-tion 8.5 coincides with the definition of quotient in [14]. Indeed, thequotient T of S by J that they consider in their paper is the uniqueunital operator system satisfying a universal property analogous to theproperty in Theorem 8.7 for unital completely positive maps into unitaloperator systems. By Proposition 8.9, S/J is a unital operator system,it follows from Theorem 8.7 that
S/J = T . ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 35
Lemma 8.11.
Let ( K, z ) be a pointed compact nc convex set and let J ⊆ A( K, z ) be a kernel. Let M = J ⊥ . Then the closed two-sided ideal I of C( K, z ) generated by J is I = { f ∈ C( K, z ) : f | M = 0 } . Henceletting π : C( K, z ) → C( M, z ) denote the restriction *-homomorphism, I = ker π .Proof. Let I ′ = ker π . Then I ′ = { f ∈ C( K, z ) : f | M = 0 } . ByProposition 8.4, the restriction π | A( K,z ) satisfies ker π | A( K,z ) = J , so itis clear that I ⊆ I ′ .For the other inclusion, first note that J = I ′ ∩ A( K, z ) . Hence bythe definition of I and the fact from above that I ⊆ I ′ , J ⊆ I ∩ A( K, z ) ⊆ I ′ ∩ A( K, z ) = J, implying J = I ∩ A( K, z ) . Let ρ : C( K, z ) → C( K, z ) /I denote thequotient *-homomorphism. Since the restriction ρ | A( K,z ) has kernel J ,Theorem 8.7 implies that ρ | A( K,z ) factors through A( M, z ) . It followsthat there is a completely contractive and completely positive map ω : A( M, z ) → C( K, z ) /I such that ω ◦ π | A( K,z ) = ρ | A( K,z ) .By the universal property of C( M, z ) , ω extends to a *-homomorphism σ : C( M, z ) → C( K, z ) /I . Hence I ′ ⊆ I , and we conclude that I ′ = I . (cid:3) Proposition 8.12.
Let ( K, z ) be a pointed compact nc convex set.Let J ⊆ A( K, z ) be a subset and let I denote the closed two-sidedideal of C( K, z ) generated by J . Then J is a kernel if and only if I ∩ A( K, z ) = J .Proof. If J is a kernel, then letting M = J ⊥ , Proposition 8.4 andLemma 8.11 imply that I ∩ A( K, z ) = { a ∈ A( K, z ) : a | M = 0 } = M ⊥ = J . Conversely, if I ∩ A( K, z ) = J , then letting π : C( K, z ) → C( K, z ) /I denote the quotient *-homomorphism, J = ker π | A( K,z ) . Hence J is a kernel. (cid:3) Noncommutative faces
In this section we introduce a notion of noncommutative face fornoncommutative convex sets.
Definition 9.1.
Let K be an nc convex set. We will say that an ncconvex subset F ⊆ K is an nc face if whenever x ∈ F n is written as afinite nc convex combination x = P α ∗ i x i α i for { x i ∈ K n i } and nonzero { α i ∈ M n i ,n } satisfying P α ∗ i α i = 1 n , then each x i decomposes withrespect to the range of α i as x i = y i ⊕ z i for y i ∈ F and z i ∈ K . Remark 9.2.
Note that for each n , F n is a face in K n . An important property of closed faces in classical compact convexsets is that extreme points of a face are also extreme points of theambient set. The next result implies that nc faces satisfy an analogueof this property with extreme points replaced by pure points. The nextdefinition is [6, Definition 6.1.2].
Definition 9.3.
Let K be a compact nc convex set. A point x ∈ K is pure if whenever x ∈ K n is written as a finite nc convex combination x = P α ∗ i x i α i for { x i ∈ K n i } and nonzero { α i ∈ M n i ,n } satisfying P α ∗ i α i = 1 n , then each α i is a positive scalar multiple of an isometry β i ∈ M n i ,n satisfying β ∗ i x i β i ∈ K n . Remark 9.4.
Every extreme point is pure, however the converse isfalse (see [6, Example 6.1.7]).
Proposition 9.5.
Let K be a compact nc convex set and let F ⊆ K bea closed face. A pure point x ∈ F is a pure point in K . In particular,an extreme point in x ∈ ∂F is a pure point in K .Proof. Let x ∈ F n be a pure point. Suppose that x is written asa finite nc convex combination x = P α ∗ i x i α i for { x i ∈ K n i } andnonzero { α i ∈ M n i ,n } satisfying P α ∗ i α i = 1 n . Since F is a face, each x i decomposes with respect to the range of α i as x i = y i ⊕ z i for y i ∈ F and z i ∈ K .Then x = P α ∗ i y i α i . Since x is pure in F and y i ∈ F for each i , it follows that each α i is a positive scalar multiple of an isometry β i ∈ M n i ,n satisfying β ∗ i y i β i ∈ F n . Then β ∗ i x i β i = β ∗ i y i β i ∈ F n . Hence x is pure in F . (cid:3) We will utilize the following definition to give some examples of ncfaces.
Definition 9.6.
Let S be an operator system and let J ⊆ S be a kernelin the sense of Definition 8.1. We will say that J is positively generated if it is spanned by its positive elements. Proposition 9.7.
Let ( K, z ) be a compact nc convex set and let J ⊆ A( K, z ) be a kernel. If J is positively generated, then J ⊥ is an nc face.Proof. Let M = J ⊥ and suppose that x ∈ F n is written as a finitenc convex combination x = P α ∗ i x i α i for { x i ∈ K n i } and nonzero { α i ∈ M n i ,n } satisfying P α ∗ i α i = 1 n .Then for a ∈ J , a ( x ) = P α ∗ i a ( x i ) α i . If a is positive, thensince each α ∗ i a ( x i ) α i is positive, it follows that α ∗ i a ( x i ) α i = 0 . Hencedecomposing x i with respect to the range of α i as x i = u i ⊕ v i for u i , v i ∈ K , it follows that a ( u i ) = 0 . Since J is positively generated, it ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 37 is spanned by its positive elements. Therefore, u i ∈ M . Hence M is annc face. (cid:3) Example 9.8.
Let A be a C*-algebra with nc quasistate space ( K, z ) and let x ∈ K be a *-homomorphism. Then the kernel J = ker x is anideal in A , and so is positively generated. Hence by Proposition 9.7, M = J ⊥ is an nc face. The fact that M is a face in K is a well knownclassical fact (see e.g. [1]).The next example shows that extreme points of nc faces in a compactnc convex set are not necessarily extreme points of the ambient set. Example 9.9.
For n ∈ N with n ≥ , let ( K, z ) denote the nc qua-sistate space of M n . Let x ∈ K be a pure state, i.e. an (ordinary)extreme point of the compact convex set K . Since x is pure, up tounitary equivalence it is the compression of M n onto a diagonal coor-dinate.Let F ⊆ K denote the closed nc convex hull of { x } . Observe thatevery point y ∈ F n is a direct sum of copies of x . We claim that F is annc face. To see this, first suppose that x is written as a finite nc convexcombination x = P α ∗ i x i α i for { x i ∈ K n i } and nonzero { α i ∈ M n i , } satisfying P α ∗ i α i = 1 , then the fact that x is an extreme point in K implies that each α ∗ i x i α i is a positive scalar multiple of x . Hence each x i decomposes with respect to the range of α i as x i = β i x ⊕ z i for ascalar β i > . Then x = P α ∗ i x i α i = P α ∗ i α i β i x , implying each β i = 1 for each i . Hence each x i decomposes with respect to the range of α i as x i = x ⊕ z i .It follows from above that if y is written as a finite nc convex com-bination y = P α ∗ i x i α i for { x i ∈ K n i } and nonzero { α i ∈ M n i ,n } satisfying P α ∗ i α i = 1 , then each x i decomposes with respect to therange of α i as u i ⊕ v i , where u i is the direct sum of copies of x and v i ∈ K . In particular, u i ∈ F , so F is an nc face.Now by [6, Example 6.1.8], ∂K consists of the irreducible representa-tions of M n , so x / ∈ ∂K . On the other hand, by the nc Krein-Milmantheorem [6, Theorem 6.4.2], F has at least one extreme point. Fromabove, we must have ∂F = { x } .10. C*-simplicity
In this section we will establish a characterization of operator systemswith the property that their minimal C*-cover (i.e. their C*-envelope)is simple. The characterization will be in terms of the nc quasistatespace of an operator system.
Definition 10.1.
We will say that an operator system S is C*-simple if its minimal C*-cover C *min ( S ) is simple.We will require the spectral topology on the irreducible points in acompact nc convex set from Section 6.1), which was introduced in [15,Section 9]. Recall that for a compact nc convex set K , the spectraltoplogy on the set Irr( K ) of irreducible points in K is defined in termsof the hull-kernel topology on the spectrum of the C*-algebra C( K ) .By Proposition 6.13, letting y = ⊕ x ∈ ∂K \{ z } x , the kernel of the *-homomorphism δ y on C( K, z ) is the boundary ideal I ( ∂K,z ) from The-orem 6.8. In particular, the quotient C( K, z ) /I ( ∂K,z ) is isomorphic tothe minimal C*-cover C( ∂K , z ) of A( K, z ) . The proof of the followingresult now follows exactly as in the proof of [15, Proposition 9.4]. Proposition 10.2.
Let ( K, z ) be a pointed compact nc convex set.A point x ∈ Irr( K ) belongs to the closure of ∂K \ { z } with respectto the spectral topology if and only if the corresponding representation δ x : C( K, z ) → M n factors through the minimal C*-cover C( ∂K , z ) of A( K, z ) . Theorem 10.3.
Let ( K, z ) be a pointed compact nc convex set. Theoperator system A( K, z ) is C*-simple if and only if the closed nc convexhull of any nonzero point in the spectral closure of ∂K contains ∂K \{ z } .Proof. Suppose that A( K, z ) is C*-simple, so that its minimal C*-cover C( ∂K, z ) is simple. Choose nonzero x ∈ K m in the spectral closure of ∂K \{ z } and let M ⊆ K denote the closed nc convex hull of x . Supposefor the sake of contradiction there is y ∈ ( ∂K ) n \ { z } such that y / ∈ M .By Proposition 10.2, the corresponding representation δ x : C( K, z ) →M n factors through the minimal C*-cover C( ∂K, z ) of A( K, z ) . Since C( ∂K, z ) is simple, it follows that the kernel of δ x is the boundaryideal I ( ∂K,z ) from Theorem 6.8, so the range of δ x is isomorphic to theminimal C*-cover C( K, z ) /I ( ∂K,z ) ∼ = C( ∂K, z ) . In particular, x is anembedding. Similarly, y is an embedding.By the nc separation theorem [6, Corollary 2.4.2], there is self-adjoint a ∈ M n (A( K, z )) and self-adjoint γ ∈ M n such that a ( y ) γ ⊗ n but a ( u ) ≤ γ ⊗ p for u ∈ M p . In particular, a ( x ) ≤ γ ⊗ m but a ( y ) γ ⊗ n .However, from above x and y are embeddings, meaning that they arecomplete order embeddings on A( K ) , giving a contradiction.Conversely, suppose that the closed nc convex hull of any nonzeropoint in the spectral closure of ∂K contains ∂K \ { z } . Let I be aproper ideal in C( ∂K, z ) and choose nonzero irreducible y ∈ K n such ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 39 that the *-homomorphism δ y on C( K, z ) factors through C( ∂K, z ) /I .Then by Proposition 10.2, y is in the spectral closure of ∂K . Hence byassumption the closed nc convex hull of y contains ∂K \ { z } .By [6, Theorem 6.4.3], every point in ∂K is a limit of compressions of y . Hence, replacing y with a sufficiently large amplification, there areisometries α i ∈ M p,n such that lim α ∗ i yα i = ⊕ x ∈ ∂K \{ z } x . By passing toa subnet we can assume that there is an nc state µ on C( K ) such thatthe *-homomorphism δ y satisfies lim α ∗ i δ y α i = µ in the nc state space of C( K ) . Then since µ | A( K ) = ⊕ x ∈ ∂K \{ z } x , and since extreme points in K have unique extensions to nc states on C( K ) , µ is the *-homomorphism µ = ⊕ x ∈ ∂K \{ z } δ x (see [6, Theorem 6.1.9]).By Proposition 6.13, the image of C( K, z ) under this *-homomorphismis isomorphic to C( ∂K, z ) . It follows that the canonical *-homomorphismfrom C( K, z ) onto C( ∂K, z ) factors through δ y . Hence I = 0 . Since I was arbitrary, we conclude that C( ∂K , z ) is simple. (cid:3) The following corollary applies when the set ∂K of extreme points of K is closed in the spectral topology. This is equivalent to the statementthat every nonzero irreducible representation of C( ∂K, z ) restricts toan extreme point of K . Corollary 10.4.
Let ( K, z ) be a pointed nc convex set such that ∂K is closed in the spectral topology. Then A( K, z ) is C*-simple if andonly if for every nonzero compact nc convex subset M ⊆ K , either M ∩ ∂K = ∅ or M ∩ ∂K = ∂K .Proof. Suppose that A( K, z ) is C*-simple. If M ∩ ∂K = ∅ then The-orem 10.3 implies that ∂K ⊆ M . Conversely, suppose that for everynonzero compact nc convex subset M ⊆ K , either M ∩ ∂K = ∅ or M ∩ ∂K = ∂K . By assumption, ∂K is spectrally closed, and for anypoint x ∈ ∂K , the closed nc convex hull M generated by x triviallysatisfies M ∩ ∂K = ∅ . Hence by assumption ∂K ⊆ M , so Theorem10.3 implies that A( K, z ) is C*-simple. (cid:3) Characterization of C*-algebras
A classical result of Bauer characterizes function systems that areunital commutative C*-algebras in terms of their state space. Specif-ically, he showed that if C is a compact convex set, then the unitalfunction system A( C ) of continuous affine functions on C is a unitalcommutative C*-algebra if and only if C is a Bauer simplex (see e.g.[1, Theorem II.4.3]). The first author and Shamovich [15, Theorem 10.5] introduced a def-inition of noncommutative simplex that generalizes the classical defi-nition and established a generalization of Bauer’s result for unital op-erator systems. Specifically, they showed that if K is a compact ncconvex set, then the unital operator system A( K ) of continuous affinenc functions on K is a unital C*-algebra if and only if K is an nc Bauersimplex.In this section we will extend this result by showing that an operatorsystem is a C*-algebra if and only if its nc quasistate space is a Bauersimplex with zero as an extreme point. Before introducing the notionof a Bauer simplex, we need to recall some preliminary definitions.Let K be a compact nc convex set. For a point x ∈ K , viewed asan nc state on the unital operator system A( K ) , the *-homomorphism δ x is an extension of x . We will be interested in other nc states on C( K ) that extend x . Specifically, we will be interested in nc statesthat are maximal in a certain precise sense. The following definition is[6, Definition 4.5.1]. Definition 11.1.
Let K be a compact nc convex set and let µ :C( K ) → M n be an nc state on C( K ) . The barycenter of µ is therestriction µ | A( K ) ∈ K n . The nc state µ is said to be a representingmap for its barycenter. We will say that a point x ∈ K has a uniquerepresenting map if the *-homomorphism δ x is the unique nc state on C( K ) with barycenter x .We will also require the notion of a convex nc function. The followingdefinition is [6, Definition 3.12]. Definition 11.2.
Let K be a compact nc convex set. For a boundedself-adjoint nc function f ∈ M n (B( K )) h , the epigraph of f is the set Epi( f ) ⊆ ⊔ K m × M n ( M m ) defined by Epi( f ) m = { ( x, α ) ∈ K m × M n ( M m ) : x ∈ K m and α ≥ f ( x ) } . The function f is convex if Epi( f ) is an nc convex set.Davidson and the first author introduced a notion of nc Choquetorder on the set of representing maps of a point in a compact nc convexset that plays a key role in noncommutative Choquet theory. Thefollowing definition is [6, Definition 8.2.1]. Definition 11.3.
Let K be a compact nc convex set and let µ, ν :C( K ) → M n be nc states. We say that µ is dominated by ν in the ncChoquet order and write µ ≺ c ν if µ ( f ) ≤ ν ( f ) for every n and everycontinuous convex nc function f ∈ M n (C( K )) . We will say that µ is a ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 41 maximal representing map for its barycenter if it is maximal in the ncChoquet order.
Remark 11.4.
Several equivalent characterizations of the nc Choquetorder were established in [6]. These are among the deepest results inthat paper.We are finally ready to state the definition of an nc simplex. Thefollowing definitions are [15, Definition 4.1] and [15, Definition 10.1]respectively.
Definition 11.5. (1) A compact nc convex set K is an nc simplex if every point in K has a unique maximal representing map on C( K ) .(2) An nc simplex K is an nc Bauer simplex if the extreme bound-ary ∂K is a closed subset of the set Irr( K ) of irreducible pointsin K with respect to the spectral topology. Remark 11.6.
It was shown in [15] that these definitions generalizethe classical definitions. Specifically, if C is a classical simplex thenthere is a unique nc simplex K with K = C . Furthermore, if C is aBauer simplex then K is an nc Bauer simplex.It was shown in [15, Theorem 10.5] that if K is a compact nc convexset, then the unital operator system A( K ) is a C*-algebra if and onlyif K is an nc Bauer simplex. The next example shows that the obviousgeneralization of this statement for operator systems does not hold. Example 11.7.
Let ( K, z ) be the pointed compact nc convex set fromExample 3.16, so K = ⊔ K n is defined by K n = { α ∈ ( M n ) h : − n ≤ α ≤ n } , for n ∈ N , and z = 0 . Since K = [ − , is a Bauer simplex, it follows from theabove discussion that K is the unique compact nc convex set with thisproperty and K is an nc Bauer simplex. However, A( K, z ) is not aC*-algebra. Note that z / ∈ ∂K and hence z / ∈ ∂K . Lemma 11.8.
Let S be an operator system with nc quasistate space ( K, z ) . Then S is a C*-algebra if and only if its unitization S ♯ is aC*-algebra and z ∈ ∂K .Proof. If S is a C*-algebra, say A , then its unitization A ♯ is the C*-algebraic unitization A ♯ of A , and hence is also a C*-algebra. Further-more, A is an ideal in A ♯ and z is an irreducible *-representation of A ♯ satisfying ker z = A . Hence by [6, Example 6.1.8], z ∈ ∂K . Conversely, suppose that S ♯ is a C*-algebra, say B , and z ∈ ∂K .Then by [6, Example 6.1.8], z is an irreducible representation of B .Since A( K, z ) = ker z , A( K, z ) is an ideal in B , and in particular is aC*-algebra. (cid:3) The next result extends [15, Theorem 10.5].
Theorem 11.9.
Let S be an operator system with nc quasistate space ( K, z ) . Then S is a C*-algebra if and only if K is an nc Bauer simplexand z ∈ ∂K . The result also holds for unital operator systems with ncquasistate spaces replaced by nc state spaces.Proof. By Lemma 11.8, S is isomorphic to a C*-algebra if and only if S ♯ is isomorphic to a C*-algebra and z ∈ ∂K . By [15, Theorem 10.5],the former property is equivalent to K being a Bauer simplex. (cid:3) Stable equivalence
Connes and van Suijlekom [4, Section 2.6] considered stable equiva-lence for operator systems. Operator systems S and T are said to be stably equivalent if the operator systems S ⊗ min K and T ⊗ min K are iso-morphic. Here, K = K ( H ℵ ) denotes the C*-algebra of compact opera-tors on H ℵ and the minimal tensor products S ⊗ min K and T ⊗ min K aredefined as in [13], i.e. S ⊗ min K is the closed operator system generatedby the algebraic tensor product of S and K in C *min ( S ) ⊗ min C *min ( K ) =C *min ( S ) ⊗ min K and similarly for T ⊗ min K .In this section we will describe the nc quasistate space of the sta-bilization of an operator system. This will yield a characterization ofstable equivalence in terms of nc quasistate spaces.Let S be an operator system and let ( K, z ) denote the nc quasistatespace of S . Let ( L, w ) denote the nc quasistate space of K . We obtaina completely contractive map x ⊗ y on S ⊗ min K from the theory oftensor products of operator spaces (see e.g. [20]). However, it is notimmediately obvious that x ⊗ y is an nc quasistate. The next resultimplies that it is, and moreover, that every nc quasistate on S ⊗ min K arises in this way. Theorem 12.1.
Let S be an operator system with nc quasistate space ( K, z ) and let ( L, w ) denote the nc quasistate space of K . The ncquasistate space of S ⊗ min T is ( K ⊗ L, z ⊗ w ) , where K ⊗ L denotesthe closed nc convex hull of { x ⊗ u : x ∈ K and u ∈ L } and x ⊗ u isdefined as in the above discussion. Furthermore, letting ( M, t ) denotethe nc quasistate space of S ⊗ min K , ∂M ⊆ ∂K ⊗ ∂L . ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 43
Proof.
We can identify S with A( K, z ) and identify A( K, z ) with its im-age under the canonical embedding into its minimal C*-cover C( ∂K , z ) .By [4, Proposition 2.37], the minimal C*-cover of A( K, z ) ⊗K is C( ∂M , t ) =C( ∂K, z ) ⊗ K . Every point x ∈ K m extends to an nc quasistate ˜ x : C( ∂K , z ) → M m (see Section 6). Then for u ∈ L n , we obtainan nc quasistate ˜ x ⊗ y : C( ∂K , z ) ⊗ K → M m ⊗ M n . The restriction ˜ x ⊗ y | A( K,z ) ⊗K = x ⊗ y is therefore an nc quasistate on A( K, z ) ⊗ min K .Hence K ⊗ L ⊆ M . It is clear that z ⊗ w is the zero map.For the reverse inclusion, let r ∈ ∂M be an extreme point. Then byProposition 6.13, the *-homomorphism δ r on C( M, t ) factors through C( ∂K, z ) ⊗K . Since r is extreme, [6, Theorem 6.1.9] implies that δ r is ir-reducible. Hence there is an irreducible representation π : C( ∂K , z ) →M m such that δ r is unitarily equivalent to π ⊗ u , where u ∈ L is eitherthe identity representation of K or u = w . Letting x = π | A( K,z ) ∈ K m , r | A( K,z ) ⊗ min K = x ⊗ u . In fact, it is easy to verify that since r ∈ ∂L , x ∈ ∂K . It follows from the nc Krein-Milman theorem [6, Theorem6.4.2] that M ⊆ K ⊗ L . (cid:3) Corollary 12.2.
Let S and S be operator systems with nc quasistatespaces ( K , z ) and ( K , z ) respectively. Let K and id K denote thezero map and the identity representation respectively of K . Then S and T are stably isomorphic if and only if the closed nc convex hullsof the sets ∂K ⊗ { K , id K } and ∂L ⊗ { K , id K } are pointedly affinelyhomeomorphic with respect to the points z ⊗ K and z ⊗ K .Proof. Let ( L, K ) denote the nc quasistate space of K . Then it followsfrom Theorem 12.1 and Corollary 4.10 that S and T are stably isomor-phic if and only if ( K ⊗ L, z ⊗ K ) and ( K ⊗ L, z ⊗ K ) are pointedlyaffinely homeomorphic.By Proposition 4.4, the unitization K ♯ is a unital C*-algebra with ncstate space L . Since every irreducible *-representation of K ♯ is unitarilyequivalent to K or id K , [6, Example 6.1.8] implies that L is the closednc convex hull of { K , id K } . The result now follows from Theorem 12.1and the nc Krein-Milman theorem [6, Theorem 6.4.2]. (cid:3) Dynamics and Kazhdan’s property (T)
The fact that simplices arise as fixed point sets of affine actions ofgroups on spaces of probability measures has a number of importantapplications in classical dynamics. Glasner and Weiss showed that asecond countable locally compact group has Kazhdan’s property (T) ifand only if the simplices that arise from this result are always Bauersimplices [9].
The first author and Shamovich extended these results to actions ofdiscrete groups on nc state spaces of unital C*-algebras. Specifically, itwas shown that nc simplices arise as fixed point sets of affine actions ofdiscrete groups on nc state spaces of unital C*-algebras [15, Theorem12.12]. It was further shown that a discrete group has property (T)if and only if the nc simplices that arise from this result are alwaysnc Bauer simplices [15, Theorem 14.2]. Consequently, a discrete grouphas property (T) if and only if whenever it acts on a unital C*-algebra,the set of invariant states is the state space of a unital C*-algebra[15, Corollary 14.3].In this section we will extend these results to actions of locally com-pact groups on (potentially nonunital) C*-algebras. In fact, we willsee that the hard work was already accomplished in earlier sections ofthis paper. After introducing appropriate definitions and applying thedual equivalence between the category of operator systems and the cat-egory of pointed compact nc convex sets, the proofs in [15] will applyessentially verbatim.The next definition is a slight generalization of [15, Definition 12.1]and [15, Definition 12.2].
Definition 13.1. (1) An nc dynamical system is a triple ( S, G, σ ) consisting of anoperator system S , a locally compact group G and a grouphomomorphism σ : G → Aut( S ) with the property that theorbit map G → S : g → σ g ( s ) is continuous for all s ∈ S .(2) A affine nc dynamical system is a triple ( K, G, κ ) consisting ofa compact nc convex set K , a locally compact group G and agroup homomorphism κ : G → Aut( K ) with the property thatfor each n , the orbit map G → K n : g → κ g ( x ) is continuousfor all x ∈ K n . Remark 13.2.
Unless we need to refer to σ , we will write ( S, G ) for ( S, G, σ ) and gs for σ g ( s ) . Similarly, unless we need to refer to κ , wewill write ( K, G ) for ( K, G, κ ) and gx for κ g ( x ) . If S is a C*-algebra,say A , then we will refer to ( A, G ) as a C*-dynamical system .We will utilize the fact that if ( K, z ) is a pointed compact nc convexset and (A( K, z ) , G ) is an nc dynamical system, then the dual equiv-alence from Theorem 4.9 gives rise to an affine nc dynamical system ( K, G ) , determined by a ( κ g ( x )) = σ g − ( a )( x ) , for a ∈ A( K ) , g ∈ G and x ∈ K. It seems worth pointing out that an nc dynamical system over anoperator system lifts to an nc dynamical system on its unitization.
ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 45
Lemma 13.3.
Let ( S, G, σ ) be an nc dynamical system. Define σ ♯ : G → Aut( S ) by ( σ ♯ ) g = ( σ g ) ♯ . Then ( S ♯ , G, σ ♯ ) is an nc dynamicalsystem.Proof. For g ∈ G , ( σ g ) ♯ ( s, α ) = ( σ g ( s ) , α ) for s ∈ S ♯ . It follows imme-diately that σ ♯ : G → Aut( S ♯ ) is a group homomorphism and that thecorresponding orbit maps are continuous. (cid:3) Let G be a locally compact group. Recall that a continuous unitaryrepresentation of G on a Hilbert space H is a group homomorphism ρ : G → U ( H ) such that the orbit map G → H : g → ρ ( g ) ξ iscontinuous for every ξ ∈ H . Here U ( H ) denotes the set of unitaryoperators on H .The next result follows immediately from [15, Theorem 12.12], sincewe can view the action of a non-discrete locally compact group as anaction by its discretization. Theorem 13.4.
Let ( K, G ) be an affine nc dynamical system such that K is an nc simplex. Then the fixed point set K G = { x ∈ K : gx = x for all g ∈ G } is an nc simplex. Corollary 13.5.
Let ( A, G ) be a C*-algebra and let ( K, z ) denote thenc quasistate space of A . Then the fixed point set K G is an nc simplex.Proof. By Theorem 11.9, K is an nc Bauer simplex, so the result followsimmediately from Theorem 13.4. (cid:3) Definition 13.6.
Let G be a second countable locally compact group.(1) A continuous unitary representation ρ : G → U ( H ) is said tohave almost invariant vectors if there is a net of unit vectors { ξ i ∈ H } such that for every compact subset C ⊆ G , lim i sup g ∈ C k ρ ( g ) ξ i − ξ i k = 0 . (2) The group G is said to have Kazhdan’s property (T) if everyunitary representation of G with almost invariant vectors has anonzero invariant vector.The next result is a generalization for (potentially nonunital) C*-algebras and second countable locally compact groups of [15, Theorem14.2]. Theorem 13.7.
Let A be a C*-algebra with nc quasistate space ( K, z ) and let G be a second countable locally compact group with Kazhdan’sproperty (T) such that ( A, G ) is a C*-dynamical system. The set K G of invariant nc quasistates on A is an nc Bauer simplex. If A is unital,then the result also holds for the nc state space of A instead of its ncquasistate space.Proof. The proof of [15, Theorem 14.2] works essentially verbatim here.If G is non-discrete, then it is necessary to verify that the unitary repre-sentation constructed in the proof of the dilation theorem for invariantnc states [15, Lemma 12.6] is continuous. However, this is an easyconsequence of the continuity of the orbit maps. (cid:3) The following corollary extends a result of Glasner and Weiss forcommutative C*-algebras (see [9, Theorem 1’] and [9, Theorem 2’]).
Corollary 13.8.
Let G be a second countable locally compact group.Then G has Kazhdan’s property (T) if and only if whenever A is a C*-algebra with nc quasistate space ( K, z ) and ( A, G ) is a C*-dynamicalsystem, then the set K G of invariant quasistates is pointedly affinelyhomeomorphic to the quasistate space of a C*-algebra. If A is unital,then the result also holds with the quasistate space of A replaced by itsstate space.Proof. If G has Kazhdan’s property (T), then Theorem 13.7 impliesthat K G is an nc Bauer simplex. By Lemma 11.8, z ∈ ∂K . Hence byTheorem 11.9, ( K G , z ) is pointedly affinely homeomorphic to the ncquasistate space of a C*-algebra. In particular, the set K G of invariantquasistates of A is pointedly affinely homeomorphic to the quasistatespace of a C*-algebra.Conversely, if G does not have Kazhdan’s property (T), then it fol-lows from [9, Theorem 2’] that there is a compact Hausdorff space X and a commutative C*-dynamical system (C( X ) , G ) such that thespace Prob( X ) G of invariant probability measures on X is a Poulsensimplex. Equivalently, the set ∂ (Prob( X ) G ) of extreme points of Prob( X ) G is not closed.We need to translate this to a statement about the quasistate space Q of C( X ) . Since Q is a compact convex set, the set Q G of invariantquasistates is a simplex (see e.g. [15, Corollary 12.13]). Note that Prob( X ) ⊆ Q . In fact, Q is the closed convex hull of Prob( X ) ∪ { z } ,where z denotes the zero map on C( X ) . For nonzero µ ∈ ∂ ( Q G ) , since µ ( X ) − µ ∈ Q G , it follows that µ ( X ) = 1 . Hence µ ∈ ∂ (Prob( X ) G ) .On the other hand, it is clear that ∂ (Prob( X ) G ) ⊆ ∂ ( Q G ) . Hence ∂ ( Q G ) ⊆ ∂ (Prob( X ) G ) ∪ { z } . Since ∂ (Prob( X ) G ) is not closed and z isisolated from Prob( X ) , it follows that ∂ ( Q G ) is not closed. Therefore, Q G is not a Bauer simplex. ONUNITAL OPERATOR SYSTEMS AND NC CONVEXITY 47
The result now follows from the fact that if the quasistate space of aC*-algebra (equivalently, the state space of its unitization) is a simplex,then the C*-algebra is commutative and its quasistate space is a Bauersimplex (see e.g. Theorem 11.9). (cid:3)
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