Unperforated pairs of operator spaces and hyperrigidity of operator systems
aa r X i v : . [ m a t h . OA ] F e b UNPERFORATED PAIRS OF OPERATOR SPACES ANDHYPERRIGIDITY OF OPERATOR SYSTEMS
RAPHA¨EL CLOU ˆATRE
Abstract.
We study restriction and extension properties for states on C ∗ -algebras with an eye towards hyperrigidity of operator systems. We use theseideas to provide supporting evidence for Arveson’s hyperrigidity conjecture.Prompted by various characterizations of hyperrigidity in terms of states, weexamine unperforated pairs of self-adjoint subspaces in a C ∗ -algebra. Theconfiguration of the subspaces forming an unperforated pair is in some sensecompatible with the order structure of the ambient C ∗ -algebra. We prove thatcommuting pairs are unperforated, and obtain consequences for hyperrigidity.Finally, by exploiting recent advances in the tensor theory of operator systems,we show how the weak expectation property can serve as a flexible relaxationof the notion of unperforated pairs. Introduction
The study of uniform algebras (i.e. closed unital subalgebras of commutativeC ∗ -algebras), combines concrete function theoretic ideas with abstract algebraictools [20]. It is a classical topic that has proven to be useful in operator theory.Indeed, a prototypical instance of a uniform algebra is the disc algebra of continuousfunctions on the closed unit disc which are holomorphic on the interior. Through abasic norm inequality of von Neumann, one can bring the analytic properties of thedisc algebra to bear on the theory of contractions on Hilbert space. The seminalwork of Sz.-Nagy and Foias on operator theory aptly illustrates the depth of thisinterplay [36], and to this day the link is still being exploited.In light of this highly successful symbiosis between operator theory and functiontheory, it is natural to look for further analogies. One may wish to transplantthe sophisticated machinery available for uniform algebras in the setting of generaloperator algebras. This ambitious vision was pioneered by Arveson, who instigatedan influential line of inquiry in his landmark paper [3]. Therein, he introduced thenotion of boundary representations for an operator system S , and proposed thatthese should be the non-commutative analogue of the so-called Choquet boundaryof a uniform algebra. Furthermore, he noticed that these boundary representationscould be used to construct a non-commutative analogue of the Shilov boundaryas well. Although Arveson himself was not able to fully realize this program atthe time, via the hard work of many hands [21],[17],[6],[14] the C ∗ -envelope of anoperator system was constructed by analogy with the classical situation. Nowadays,this circle of ideas is regarded as the appropriate non-commutative version of the Mathematics Subject Classification.
Key words and phrases.
Operator systems, completely positive maps, unique extension prop-erty, hyperrigidity.The author was partially supported by an NSERC Discovery Grant.
Shilov boundary of a uniform algebra and it has emerged as a ubiquitous invariantin non-commutative functional analysis [22],[9],[19],[13].Arveson also recognized that the non-commutative Choquet boundary was a richsource of intriguing questions. For instance, in [7] he proposed a tantalizing con-nection with approximation theory by recasting a classical phenomenon in operatoralgebraic terms. The classical setting is a result of Korovkin [28], which goes asfollows. For each n ∈ N , let Φ n : C [0 , → C [0 ,
1] be a positive linear map andassume that lim n →∞ k Φ n ( f ) − f k = 0for every f ∈ { , x, x } . Then, it must be the case thatlim n →∞ k Φ n ( f ) − f k = 0for every f ∈ C [0 , n ) n on the C ∗ -algebra C [0 ,
1] is uniquely determined by the operator system S spanned by 1 , x, x . This striking phenomenon was elucidated by several researchers(see for instance [1] for a recent survey), but the perspective most relevant for ourpurpose here was offered by ˇSaˇskin [37], who observed that the key property of S is that its Choquet boundary coincides with [0 , S ⊂ B ( H ) with the property that for any sequence of unital completelypositive linear maps Φ n : C ∗ ( S ) → C ∗ ( S ) , n ∈ N such that lim n →∞ k Φ n ( s ) − s k = 0 , s ∈ S we must have lim n →∞ k Φ n ( a ) − a k = 0 , a ∈ C ∗ ( S ) . In fact, Arveson introduced even more non-commutativity in this picture, and de-fined the operator system S to be hyperrigid if for any injective ∗ -representation π : C ∗ ( S ) → B ( H π )and for any sequence of unital completely positive linear mapsΦ n : B ( H π ) → B ( H π ) , n ∈ N such that lim n →∞ k Φ n ( π ( s )) − π ( s ) k = 0 , s ∈ S we must have lim n →∞ k Φ n ( π ( a )) − π ( a ) k = 0 , a ∈ C ∗ ( S ) . Note that even in the case where C ∗ ( S ) is commutative, a priori this phenomenon isstronger than the one observed by Korovkin, as we allow the maps Φ n to take valuesoutside of C ∗ ( S ). Nevertheless, in accordance with ˇSaˇskin’s insightful observation,Arveson conjectured [7] that hyperrigidity is equivalent to the non-commutativeChoquet boundary of S being as large as possible, in the sense that every irreducible ∗ -representation of C ∗ ( S ) should be a boundary representation for S . This is nowknown as Arveson’s hyperrigidity conjecture and it has garnered significant interestin recent years [26],[27],[32],[12]. Arveson himself showed in [7] that the conjecture
NPERFORATED PAIRS AND HYPERRIGIDITY 3 is valid whenever C ∗ ( S ) has countable spectrum. Recently, it was verified in [15]in the case where C ∗ ( S ) is commutative.The hyperrigidity conjecture is the main motivation behind our work here. Tech-nically speaking however, the paper is centred around extensions and restrictions ofstates on C ∗ -algebras, and these issues occupy us for the majority of the article. Wefeel this approach to hyperrigidity is very natural, but as far as we know it has notbeen carefully investigated beyond the early connection realized in [6, Theorem 8.2].In the final section of the paper, we introduce what we call “unperforated pairs”of subspaces in a C ∗ -algebra. As we show, they constitute a device that can beleveraged to gain information about states, and ultimately to detect hyperrigidity.They also highlight a novel angle of approach to the hyperrigidity conjecture.We now describe the organization of the paper more precisely. In Section 2 wegather the necessary background material. In particular, we recall that hyperrigid-ity of an operator system S is equivalent to the following unique extension property:for every unital ∗ -representation π : A → B ( H ) and every unital completely positivelinear map Π : A → B ( H ) which agrees with π on S , we have π = Π. In Section3, we explore the link between hyperrigidity and two properties of states, namelythe unique extension property and the pure restriction property. The first mainresult of that section establishes these properties as a tool to detect hyperrigidity.We summarize our findings (Theorem 3.2, Corollary 3.3 and Theorem 3.10) in thefollowing. Theorem 1.1.
Let S be an operator system and let A = C ∗ ( S ) . Assume thatevery irreducible ∗ -representation of A is a boundary representation for S . Let π : A → B ( H ) be a unital ∗ -representation and let Π : A → B ( H ) be a unitalcompletely positive extension of π | S . The following statements are equivalent. (i) We have π = Π . (ii) We have Π( A ) ⊂ π ( A ) . (iii) Every pure state on C ∗ (Π( A )) restricts to a pure state on π ( A ) . (iv) There is a family of states on C ∗ (Π( A )) which separate (Π − π )( A ) and restrict to pure states on π ( A ) . (v) Every pure state on C ∗ (Π( A )) has the unique extension property with respectto π ( A ) . (vi) There is a family of pure states on C ∗ (Π( A )) which separate (Π − π )( A ) and have the unique extension property with respect to π ( A ) . The other main result of Section 3 provides evidence supporting Arveson’s con-jecture (Theorem 3.6).
Theorem 1.2.
Let S be an operator system and let A = C ∗ ( S ) . Assume thatevery irreducible ∗ -representation of A is a boundary representation for S . Let π : A → B ( H ) be a unital ∗ -representation and let Π : A → B ( H ) be a unitalcompletely positive extension of π | S . Then, the subspace ( π − Π)( A ) contains nostrictly positive element. In Section 4, we delve deeper into the unique extension property for states. Basedon a general construction (Theorem 4.3), we exhibit natural examples where theunique extension property is satisfied by an abundance of states, which is relevantin view of Theorem 1.1.Finally, in Section 5 we introduce the notion of an unperforated pair. A pair( S , T ) of self-adjoint subspaces in a unital C ∗ -algebra is said to be unperforated if RAPHA¨EL CLOUˆATRE whenever a ∈ S and b ∈ T are self-adjoint elements with a ≤ b , we may find anotherself-adjoint element b ′ ∈ T such that k b ′ k ≤ k a k and a ≤ b ′ ≤ b . This providesa mechanism to construct families of states with pure restrictions (Theorem 5.2).The precise relation to hyperrigidity is illustrated in the following (Corollaries 5.3and 5.5). Theorem 1.3.
Let S be a separable operator system and let A = C ∗ ( S ) . Assumethat every irreducible ∗ -representation of A is a boundary representation for S .Let π : A → B ( H ) be a unital ∗ -representation and let Π : A → B ( H ) be aunital completely positive extension of π | S . Then, the pair ((Π − π )( A ) , π ( A )) isunperforated if and only if Π = π . In particular, this is satisfied if (Π − π )( A ) commutes with π ( A ) . Unperforated pairs appear to be elusive in the absence of some form of commu-tativity. Accordingly, we aim to find a meaningful relaxation of that notion. Basedon recent advances in the tensor theory of operator systems and the so-called tightRiesz interpolation property, we propose that the weak expectation property isan appropriate relaxation. Our position is substantiated by the following result(Theorem 5.7).
Theorem 1.4.
Let A be a unital C ∗ -algebra and let B ⊂ A be a unital separable C ∗ -subalgebra with the weak expectation property. Let a ∈ A be a self-adjoint elementand let ε > . Then, there is a sequence ( β n ) n of self-adjoint elements in B withthe following properties. (1) We have k β n k ≤ (1 + ε ) k a k for every n ∈ N and lim sup n →∞ k β n k ≤ k a k . (2) We have lim sup n →∞ ψ ( β n ) ≤ inf { ψ ( b ) : b ∈ B , b ≥ a } and sup { ψ ( c ) : c ∈ B , c ≤ a } ≤ lim inf n →∞ ψ ( β n ) for every state ψ on B . As an application of the previous result, we refine Theorem 3.10 in the presenceof the weak expectation property (Corollary 5.10).2.
Preliminaries
Operator systems and completely positive maps.
Let B ( H ) denote theC ∗ -algebra of bounded linear operators on some Hilbert space H . An operatorsystem S is a unital self-adjoint subspace of B ( H ). Due to work of Choi andEffros [11], operator systems can be defined in a completely abstract fashion, butthe previous “concrete” definition will suffice for our present purposes. Likewise,we will always assume that C ∗ -algebras are concretely represented on a Hilbertspace. For each positive integer n , we denote by M n ( S ) the complex vector spaceof n × n matrices with entries in S , and regard it as a unital self-adjoint subspaceof B ( H ( n ) ). A linear map ϕ : S → B ( H ϕ )induces a linear map ϕ ( n ) : M n ( S ) → B ( H ( n ) ϕ ) NPERFORATED PAIRS AND HYPERRIGIDITY 5 defined as ϕ ( n ) ([ s ij ] i,j ) = [ ϕ ( s ij )] i,j for each [ s ij ] i,j ∈ M n ( S ) . The map ϕ is said to be completely positive if ϕ ( n ) ispositive for every positive integer n .For most of the paper, we will be dealing with unital completely positive mapswith one-dimensional range. Such a map ψ : S → C is called a state . The set ofstates on S is denote by S ( S ). It is a weak- ∗ closed convex subset of the closedunit ball of the dual space S ∗ , and so in particular it is compact in the weak- ∗ topology.The structure of unital completely positive maps on C ∗ -algebras is elucidatedby the Stinespring construction, a generalization of the classical Gelfand-Naimark-Segal (GNS) construction associated to a state. More precisely, given a unital C ∗ -algebra A and a unital completely positive map ϕ : A → B ( H ), there is a Hilbertspace H ϕ , an isometry V ϕ : H → H ϕ and a unital ∗ -representation σ ϕ : A → B ( H ϕ )satisfying ϕ ( a ) = V ∗ ϕ σ ϕ ( a ) V ϕ , a ∈ A and H ϕ = [ σ ϕ ( A ) V ϕ H ] . Here and throughout, given a subset
V ⊂ H we denote by[ V ] the smallest closed subspace of H containing V . The triple ( σ ϕ , H ϕ , V ϕ ) is calledthe Stinespring representation of ϕ , and it is unique up to unitary equivalence. Thefollowing fact is standard. Lemma 2.1.
Let A be a unital C ∗ -algebra and let B ⊂ A be a unital C ∗ -subalgebra.Let ψ : A → B ( H ) be a unital completely positive map and let ϕ = ψ | B . Then,there is an isometry W : H ϕ → H ψ such that W V ϕ = V ψ and σ ψ ( b ) W = W σ ϕ ( b ) , b ∈ B . Proof.
We first note that if b , . . . , b n ∈ B and ξ , . . . , ξ n ∈ H then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X j =1 σ ψ ( b j ) V ψ ξ j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = n X j,k =1 h V ∗ ψ σ ψ ( b ∗ k b j ) V ψ ξ j , ξ k i = n X j,k =1 h ψ ( b ∗ k b j ) ξ j , ξ k i = n X j,k =1 h ϕ ( b ∗ k b j ) ξ j , ξ k i = n X j,k =1 h V ∗ ϕ σ ϕ ( b ∗ k b j ) V ϕ ξ j , ξ k i = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X j =1 σ ϕ ( b j ) V ϕ ξ j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Using that H ϕ = [ σ ϕ ( B ) V ϕ H ], a routine argument shows that there is an isometry W : H ϕ → H ψ such that W n X j =1 σ ϕ ( b j ) V ϕ ξ j = n X j =1 σ ψ ( b j ) V ψ ξ j for every b , . . . , b n ∈ B and ξ , . . . , ξ n ∈ H . It follows readily that W V ϕ = V ψ and W σ ϕ ( b ) = σ ψ ( b ) W, b ∈ B . (cid:3) RAPHA¨EL CLOUˆATRE
Purity, extreme points and Choquet integral representation.
Let S be an operator system. A completely positive map ψ : S → B ( H ) is said to be pure if whenever ϕ : S → B ( H ) is a completely positive map with the property that ψ − ϕ is also completely positive, we must have that ϕ = tψ for some 0 ≤ t ≤ ∗ -algebra areprecisely those for which the associated Stinespring representations are irreducible[3, Corollary 1.4.3].We let S p ( S ) denote the collection of pure states. It is a standard fact that a stateis pure if and only if it is an extreme point of S ( S ) (see for instance [16, Proposition2.5.5], the proof of which is easily adapted to the setting of an operator system). Asubtlety arises for unital completely positive maps with higher dimensional ranges:it follows from [38, Example 2.3] and [18] that a matrix state ψ : S → B ( C n ) ispure if and only if it is a so-called matrix extreme point.The following tool will be important for us. It follows from [8, Theorem 4.2] (seealso [34, Chapter 3]). Recall that if S is separable, then the weak- ∗ topology on S ( S ) is compact and metrizable. Theorem 2.2.
Let S be a separable operator system and let ψ be a state on S .Then, there is a regular Borel probability measure on S ( S ) concentrated on S p ( S ) and with the property that ψ ( s ) = Z S p ( S ) ω ( s ) dµ ( ω ) , s ∈ S . Unique extension property, boundary representations and hyper-rigidity.
One important property of completely positive maps on operator systemsis that they satisfy a generalization of the Hahn-Banach extension theorem. Indeed,let S ⊂ B ( H ) be an operator system and let ϕ : S → B ( H ϕ ) be a completely pos-itive map. Then, by Arveson’s extension theorem [3], there is another completelypositive map ψ : B ( H ) → B ( H ϕ ) with the property that ψ | S = ϕ . In particular,a completely positive map on S always admits at least one completely positiveextension to any operator system T ⊂ B ( H ) containing S . We denote the set ofsuch extensions by E ( ϕ, T ). This notation will be used consistently throughout thepaper.In general, the set of extensions may contain more than one element, and thispossibility is one of the main themes of the paper. The following fact quantifies thefreedom in choosing an extension, and it follows from a verbatim adaptation of theproof of [7, Proposition 6.2]. Lemma 2.3.
Let S ⊂ T be operator systems and let ϕ be a state on S . Then, max ψ ∈E ( ϕ, T ) ψ ( t ) = inf { ϕ ( s ) : s ∈ S , s ≥ t } and min ψ ∈E ( ϕ, T ) ψ ( t ) = sup { ϕ ( s ) : s ∈ S , s ≤ t } whenever t ∈ T is self-adjoint. Let S ⊂ T be operator systems. We say that a completely positive map ψ : T → B ( H ψ ) has the unique extension property with respect to S if the restriction ψ | S admits only one completely positive extension to T , namely ψ itself. An irreducible ∗ -representation π : C ∗ ( S ) → B ( H π ) is said to be a boundary representation for S if it has the unique extension property with respect to S . NPERFORATED PAIRS AND HYPERRIGIDITY 7
We advise the reader to exercise some care: in other works (such as [6]) the useof the terminology “unique extension property” is reserved for ∗ -representationson C ∗ ( S ). Our definition is more lenient as we do not restrict our attention to ∗ -representations and no further relation is assumed between S and T beyond merecontainment. We will recall this discrepancy in terminology whenever there is anyrisk of confusion.These notions can be used to reformulate the property of hyperrigidity consideredin the introduction. The following is [7, Theorem 2.1]; therein some special attentionis paid to separability conditions, but a quick look at the proof reveals that the nextresult holds with no cardinality assumptions. Theorem 2.4.
Let S be an operator system. Then, S is hyperrigid if and onlyif every unital ∗ -representation of C ∗ ( S ) has the unique extension property withrespect to S . The driving force behind our work is the following conjecture of Arveson [7],which claims that it is sufficient to focus on irreducible ∗ -representations to detecthyperrigidity. Arveson’s hyperrigidity conjecture.
An operator system S is hyperrigid if everyirreducible ∗ -representation of C ∗ ( S ) is a boundary representation for S .To be precise, we should point out that Arveson was more cautious and re-stricted the operator system in his conjecture to be separable. We explain whythis conjecture is especially sensible in that case. We may think of an arbitrary ∗ -representation as some kind of integral of a family of irreducible ∗ -representationsagainst some measure. Since the irreducible ∗ -representations are all assumed tohave the unique extension property with respect to S , the question then becomeswhether this property is preserved by the integration procedure. This rough sketchcan be made precise, and in fact this was the philosophy used by Arveson in [6].One of the main contributions therein [6, Theorem 6.1] establishes that if the resultof the integration procedure has the unique extension property with respect to S ,then the integrand must have it almost everywhere. Arveson’s hyperrigidity con-jecture essentially asserts the converse. Note that in the “atomic” situation wherethe integral is in fact a direct sum, this converse does indeed hold [7, Proposition4.4].Finally, we note that we choose not to make separability of our operator systems ablanket assumption, although such conditions will occasionally make an appearancefor technical reasons throughout.3. Characterizing hyperrigidity via states
In this section, we make partial progress towards verifying the hyperrigidity con-jecture and provide several different characterizations of hyperrigidity using states.Before proceeding, we make an observation that will be used numerous timesthroughout. Let S be an operator system and let A = C ∗ ( S ). Let π : A → B ( H )be a unital ∗ -representation and let Π : A → B ( H ) be a unital completely positiveextension of π | S . Then, we have(1) π ( A ) = C ∗ ( π ( S )) = C ∗ (Π( S )) ⊂ C ∗ (Π( A )) . The basic tool of this section is the following.
RAPHA¨EL CLOUˆATRE
Lemma 3.1.
Let S be an operator system and let A = C ∗ ( S ) . Assume thatevery irreducible ∗ -representation of A is a boundary representation for S . Let π : A → B ( H ) be a unital ∗ -representation and let Π : A → B ( H ) be a unitalcompletely positive extension of π | S . Then, we have that ψ ◦ Π = ψ ◦ π whenever ψ is a unital completely positive map on C ∗ (Π( A )) with the property that ψ | π ( A ) ispure.Proof. Recall that π ( A ) ⊂ C ∗ (Π( A )) by (1). Let ϕ = ψ | π ( A ) which is pure byassumption. Let ( σ ψ , H ψ , V ψ ) and ( σ ϕ , H ϕ , V ϕ ) denote the Stinespring representa-tions for ψ and ϕ respectively. By Lemma 2.1, we see that there is an isometry W : H ϕ → H ψ with the property that W V ϕ = V ψ and W ∗ σ ψ ( π ( a )) W = σ ϕ ( π ( a ))for every a ∈ A . Since π and Π agree on S , we see that the map a W ∗ σ ψ (Π( a )) W, a ∈ A is a unital completely positive extension of σ ϕ ◦ π | S . Because ϕ is pure, we inferthat σ ϕ is irreducible. In particular, σ ϕ ◦ π is an irreducible ∗ -representation of A ,and thus is a boundary representation for S . We conclude that W ∗ σ ψ (Π( a )) W = σ ϕ ( π ( a ))for every a ∈ A . Hence, using that W V ϕ = V ψ we obtain ψ (Π( a )) = V ∗ ψ σ ψ (Π( a )) V ψ = V ∗ ϕ W ∗ σ ψ (Π( a )) W V ϕ = V ∗ ϕ σ ϕ ( π ( a )) V ϕ = ϕ ( π ( a )) = ψ ( π ( a ))for every a ∈ A , and therefore ψ ◦ Π = ψ ◦ π . (cid:3) Our next task is to reformulate Lemma 3.1 in a language that is convenientlyapplicable to our purposes in the paper. Let A be a unital C ∗ -algebra and let S ⊂ A be a self-adjoint subspace. Let F be a collection of states on A . We say that thestates in F separate S if for every non-zero self-adjoint element s ∈ S we have thatsup ψ ∈F | ψ ( s ) | > . Theorem 3.2.
Let S be an operator system and let A = C ∗ ( S ) . Assume thatevery irreducible ∗ -representation of A is a boundary representation for S . Let π : A → B ( H ) be a unital ∗ -representation and let Π : A → B ( H ) be a unitalcompletely positive extension of π | S . The following statements are equivalent. (i) We have π = Π . (ii) Every pure state on C ∗ (Π( A )) restricts to a pure state on π ( A ) . (iii) There is a family of states on C ∗ (Π( A )) which separate (Π − π )( A ) and restrict to pure states on π ( A ) .Proof. If π = Π, then C ∗ (Π( A )) = π ( A ) so that (i) implies (ii). It is trivial that(ii) implies (iii) since (Π − π )( A ) ⊂ C ∗ (Π( A )) by (1). Finally, assume that there isa family F of states on C ∗ (Π( A )) which separate (Π − π )( A ) and restrict to purestates on π ( A ). To establish Π = π , it suffices to show thatsup ψ ∈F | ψ (Π( a ) − π ( a )) | = 0 NPERFORATED PAIRS AND HYPERRIGIDITY 9 for every self-adjoint element a ∈ A . This follows from an application of Lemma3.1. We conclude that (iii) implies (i). (cid:3) In view of the previous statement, we note in passing that it is generally nottrue that if every state on a unital C ∗ -algebra A restricts to be pure on a unitalC ∗ -subalgebra B , then B = A . Indeed, simply consider the trivial case of B = C I .We extract an easy consequence related to hyperrigidity. Corollary 3.3.
Let S be an operator system and let A = C ∗ ( S ) . Assume thatevery irreducible ∗ -representation of A is a boundary representation for S . Let π : A → B ( H ) be a unital ∗ -representation and let Π : A → B ( H ) be a unitalcompletely positive extension of π | S . Then, π = Π if and only if Π( A ) ⊂ π ( A ) .Proof. Assume that Π( A ) ⊂ π ( A ). Then, we have π ( S ) = Π( S ) ⊂ Π( A ) ⊂ π ( A )which implies that π ( A ) = C ∗ ( π ( S )) = C ∗ (Π( A )) . Thus, the pure states on π ( A ) coincide with those on C ∗ (Π( A )), and Theorem 3.2implies that π = Π. The converse is trivial. (cid:3) In light of Theorem 3.2, it behooves us to understand the states on a unital C ∗ -algebra A which restrict to be pure on a unital C ∗ -subalgebra B . Fixing a state ψ on A and allowing B to vary (while still being non-trivial), it is sometimes possibleto arrange for the restriction ψ | B to be pure as well; see [23] and references therein.Typically however, one does not expect purity of the restriction, as easy examplesshow. Example 3.4.
Let M be the complex 2 × { e , e } be thecanonical orthonormal basis of C . Choose non-zero complex numbers γ , γ suchthat | γ | + | γ | = 1 and put ξ = γ e + γ e . Define a state ω on M as ω ( a ) = h aξ, ξ i , a ∈ M . The GNS representation of ω is seen to be unitarily equivalent to the identityrepresentation on M , which is irreducible. Thus, ω is pure. Note however that therestriction of ω to the commutative C ∗ -subalgebra C ⊕ C ⊂ M is not multiplicativesince both γ and γ are non-zero, and therefore the restriction is not pure. (cid:3) Nevertheless, the insight provided by Theorem 3.2 will guide us throughout thepaper, and it already contains non-trivial information regarding the hyperrigidityconjecture as we proceed to show next. First, we need a technical tool.
Lemma 3.5.
Let S be an operator system and let A = C ∗ ( S ) . Assume thatevery irreducible ∗ -representation of A is a boundary representation for S . Let π : A → B ( H ) be a unital ∗ -representation and let Π : A → B ( H ) be a unitalcompletely positive extension of π | S . Fix a state ϕ on π ( A ) and an element a ∈ A such that π ( a ) − Π( a ) is self-adjoint. Then, we have that sup { ϕ ( π ( c )) : c ∈ A , π ( c ) ≤ π ( a ) − Π( a ) } ≤ . Proof.
By (1), we have π ( A ) ⊂ C ∗ (Π( A )). Put x = π ( a ) − Π( a ) ∈ C ∗ (Π( A )). Weinfer from Lemma 3.1 that ψ ( x ) = 0 for every state ψ on C ∗ (Π( A )) such that ψ | π ( A ) is pure. In particular, if c ∈ A satisfies π ( c ) ≤ x and ω is a pure state on π ( A ),then we see that ω ( π ( c )) = ψ ( π ( c )) ≤ ψ ( x ) = 0for every state ψ on C ∗ (Π( A )) such that ψ | π ( A ) = ω . By the Krein-Milman theorem,the state ϕ lies in the weak- ∗ closure of the convex hull of S p ( π ( A )), and thussup { ϕ ( π ( c )) : c ∈ A , π ( c ) ≤ x } ≤ . (cid:3) Let S be an operator system and let A = C ∗ ( S ). We assume that every ir-reducible ∗ -representation of A is a boundary representation for S . Further, let π : A → B ( H ) be a unital ∗ -representation and let Π : A → B ( H ) be a unitalcompletely positive map which agrees with π on S . If the hyperrigidity conjec-ture holds, then we would have π = Π. In other words, the self-adjoint subspace( π − Π)( A ) would be trivial. The next development, which is one of the main resultof this section, establishes that this subspace cannot contain any strictly positiveelement, thus supporting Arveson’s conjecture. Theorem 3.6.
Let S be an operator system and let A = C ∗ ( S ) . Assume thatevery irreducible ∗ -representation of A is a boundary representation for S . Let π : A → B ( H ) be a unital ∗ -representation and let Π : A → B ( H ) be a unitalcompletely positive extension of π | S . Then, the subspace ( π − Π)( A ) contains nostrictly positive element.Proof. Let a ∈ A and assume that the element x = π ( a ) − Π( a ) is strictly positive,so that x ≥ δI for some δ >
0. We infer thatsup { ϕ ( π ( c )) : c ∈ A , π ( c ) ≤ x } ≥ ϕ ( π ( δI )) = δ for every state ϕ on A , which contradicts Lemma 3.5. (cid:3) Until now, the underlying theme of this section has been the purity of restrictionsof states to C ∗ -subalgebras. The dual process of extending states from a C ∗ -algebrato a larger one is also relevant for hyperrigidity, and we explore this idea next. Westart by clarifying the relation between the unique extension property for statesand the corresponding property for ∗ -representations. Theorem 3.7.
Let A be a unital C ∗ -algebra and let S ⊂ A be an operator system.The following statements hold. (1) Assume that every pure state on A has the unique extension property withrespect to S . Then, every irreducible ∗ -representation of A has the uniqueextension property with respect to S . (2) Assume that every state on A has the unique extension property with respectto S . Then, every unital ∗ -representation of A has the unique extensionproperty with respect to S . In particular, in the case where A = C ∗ ( S ) weconclude that S is hyperrigid.Proof. The two statements are established via near identical arguments, so weintertwine their proofs. Let π : A → B ( H ) be a unital ∗ -representation (irreduciblein the case of (1)) and let Π : A → B ( H ) be a unital completely positive map which NPERFORATED PAIRS AND HYPERRIGIDITY 11 agrees with π on S . We must show that π = Π on A , for which it is sufficient toestablish that h π ( a ) ζ, ζ i = h Π( a ) ζ, ζ i for every unit vector ζ ∈ H and every a ∈ A . Indeed, in that case, for each a ∈ A the numerical radius of π ( a ) − Π( a ) ∈ B ( H ) is 0, and thus π ( a ) = Π( a ).Fix henceforth a unit vector ζ ∈ H and consider the state χ on A defined as χ ( a ) = h π ( a ) ζ, ζ i , a ∈ A . If π is irreducible, it is routine to verify that the GNS representation of χ is unitarilyequivalent to π , whence χ must be pure in this case. Consider now another state ψ on A defined as ψ ( a ) = h Π( a ) ζ, ζ i , a ∈ A . We see that ψ and χ agree on S , whence they agree on A by assumption. In otherwords, h π ( a ) ζ, ζ i = h Π( a ) ζ, ζ i , a ∈ A and the proof is complete. (cid:3) In the classical case where C ∗ ( S ) is commutative, the pure states coincide withthe irreducible ∗ -representations, whence the converse of part (1) of Theorem 3.7holds. For general operator systems S however, it can happen that S is hyperrigidwhile there are some pure states on C ∗ ( S ) which do not have the unique extensionproperty with respect to S . We provide an elementary example. Example 3.8.
Let { e , e } denote the canonical orthonormal basis of C . Considerthe associated standard matrix units E , E ∈ M . Let S ⊂ M be the operatorsystem generated by I, E , E . Then, M = C ∗ ( S ). For 1 ≤ k ≤
2, we let χ k bethe vector state on M defined as χ k ( a ) = h ae k , e k i , a ∈ M . We see that the GNS representations of χ and χ are unitarily equivalent to theidentity representation on M , which is irreducible. Thus, χ and χ are both pure.Moreover, every element s ∈ S has the form s = c I + c E + c E for some c , c , c ∈ C . We note that χ k ( c I + c E + c E ) = c , ≤ k ≤ χ | S = χ | S while χ = χ . Thus, χ does not have the unique extensionproperty with respect to S .On the other hand, it is well-known that up to unitary equivalence the onlyunital ∗ -representations of M are multiples of the identity representation. Theidentity representation is a boundary representation for S by Arveson’s boundarytheorem [4, Theorem 2.1.1]. Since the unique extension property is preserved underdirect sums [7, Proposition 4.4], we conclude that every unital ∗ -representation of M has the unique extension property with respect to S . (cid:3) Our next task is to relate the unique extension property to the pure restrictionproperty. For this purpose, we recall some well-known facts which follow easilyfrom standard convexity arguments (see Subsection 2.3 about the notation usedhere).
Lemma 3.9.
Let S ⊂ T be operator systems. (1) Let χ be a pure state on T which has the unique extension property withrespect to S . Then, the restriction χ | S is pure. (2) Let ω be a pure state on S . Then, E ( ω, T ) is a weak- ∗ closed convex subsetof S ( T ) whose extreme points are pure states on T . In particular, ω admitsa unique state extension to T if and only if it admits a unique pure stateextension to T . In view of this interplay between the unique extension property for states andthe pure restriction property, we give another characterization of hyperrigidity.
Theorem 3.10.
Let S be an operator system and let A = C ∗ ( S ) . Assume thatevery irreducible ∗ -representation of A is a boundary representation for S . Let π : A → B ( H ) be a unital ∗ -representation and let Π : A → B ( H ) be a unitalcompletely positive extension of π | S . The following statements are equivalent. (i) We have π = Π . (ii) Every pure state on C ∗ (Π( A )) has the unique extension property with respectto π ( A ) . (iii) There is a family of pure states on C ∗ (Π( A )) which separate (Π − π )( A ) and have the unique extension property with respect to π ( A ) .Proof. If π = Π, then C ∗ (Π( A )) = π ( A ) so that (i) implies (ii). It is trivial that (ii)implies (iii) since (Π − π )( A ) ⊂ C ∗ (Π( A )) by (1). Assume that there is a family ofpure states on C ∗ (Π( A )) which separate (Π − π )( A ) and have the unique extensionproperty with respect to π ( A ). By Lemma 3.9, this family consists of states whichrestrict to be pure on π ( A ). Thus, π = Π by virtue of Theorem 3.2, and (iii) implies(i). (cid:3) The unique extension property for states
In the previous section, we gave several different characterizations of hyperrigid-ity in terms of states. In particular, Theorem 3.10 provides motivation to examinethe unique extension property for states in greater detail. This is the task we un-dertake in this section. First, we remark that these uniqueness considerations forpure states on a maximal abelian self-adjoint subalgebra of B ( H ) were at the heartof the famous Kadison-Singer problem [24] which was solved in [30]. The case ofgeneral subalgebras has also been studied extensively; see [2] and references therein.We now turn to examining the unique extension property for states which arenot pure. In the separable setting, those are exactly the states which are givenas the integration of a collection of pure states with respect to some probabilitymeasure (see Theorem 2.2). More generally, we have the following. Proposition 4.1.
Let A be a unital C ∗ -algebra and let S ⊂ A be an operatorsystem. Let ψ be a state on A such that ψ = Z S ( A ) χdµ ( χ ) for some Borel probability measure µ on S ( A ) . Assume that ψ has the uniqueextension property with respect to S . If χ is a state on A satisfying µ ( { χ } ) > ,then χ has the unique extension property with respect to S . NPERFORATED PAIRS AND HYPERRIGIDITY 13
Proof.
Fix a state χ on A with the property that µ ( { χ } ) >
0. Choose a state ε on A which agrees with χ on S . Next, define a state ϕ on A as ϕ ( a ) = µ ( { χ } ) ε ( a ) + Z S ( A ) \{ χ } χ ( a ) dµ ( χ ) , a ∈ A . Then, we see that ϕ and ψ agree on S , and thus by assumption we must have that ϕ = ψ . Therefore, for each a ∈ A we find that µ ( { χ } ) ε ( a ) = ϕ ( a ) − Z S ( A ) \{ χ } χ ( a ) dµ ( χ )= ψ ( a ) − Z S ( A ) \{ χ } χ ( a ) dµ ( χ )= µ ( { χ } ) χ ( a ) . Since µ ( { χ } ) >
0, we conclude that ε = χ and the proof is complete. (cid:3) In particular, the previous proposition implies that if a state with the uniqueextension property is given as some finite convex combination of states, then all ofthose have the unique extension property as well. The question asking whether thecondition on χ being an “atom” of µ can be removed from the statement appearsto be difficult. A related fact is known to hold at least in the case where S isseparable; see [6, Theorem 6.1]. In the setting of that paper however, states arereplaced by ∗ -representations and it is systematically assumed that A = C ∗ ( S ).Under these conditions, the unique extension property is known to be equivalentto a dilation theoretic maximality property [6, Proposition 2.4]. This importantcharacterization was first discovered in [31] and exploited with great success in [17].It plays a crucial role in Arveson’s proof of [6, Theorems 5.6 and 6.1], and the lackof an analogue in our context is a major obstacle to adapting his ideas.In the other direction, we exhibit an example which shows that integrating acollection of pure states with the unique extension property against some probabilitymeasure does not necessarily preserve the unique extension property. Example 4.2.
Let B ⊂ C denote the open unit ball and let S be its topologicalboundary, the sphere. Let A ( B ) denote the ball algebra , that is the algebra ofcontinuous functions on B which are holomorphic on B . Endow this algebra withthe supremum norm over B . By means of the maximum modulus principle, wemay regard A ( B ) as a unital closed subalgebra of C ( S ). Let S = A ( B ) + A ( B ) ∗ ⊂ C ( S )be the operator system generated by A ( B ) inside of C ( S ). For every λ ∈ B ,denote by ε λ the state on S uniquely determined by ε λ ( f ) = f ( λ ) , f ∈ A ( B ) . It is a classical fact [20] that S is the Choquet boundary of A ( B ). Hence, for each ζ ∈ S the state ε ζ on S has a unique extension to a state χ ζ on C ( S ). This state χ ζ is in fact the character on C ( S ) of evaluation at ζ , and in particular it is pure.Consider now the unique rotation invariant regular Borel probability measure σ on S , and let ψ σ denote the state on C ( S ) of integration against σ . By virtueof Cauchy’s formula [35, Equation 3.2.4], we have that ψ σ ( f ) = f (0) for every f ∈ A ( B ). On the other hand, let µ denote Lebesgue measure on the circle { ( ζ , ζ ) ∈ S : | ζ | = 1 , ζ = 0 } and let ψ µ denote the state on C ( S ) of integration against µ . Then, ψ µ = ψ σ .The one-variable version of Cauchy’s formula shows that ψ µ ( f ) = f (0) for every f ∈ A ( B ). In particular, we conclude that ψ σ does not have the unique extensionproperty with respect to S . Finally, note that ψ σ = Z S χ ζ dσ ( ζ )and we saw in the previous paragraph that each χ ζ is pure and has the uniqueextension property with respect to S . (cid:3) From the point of view of hyperrigidity, we see that Theorem 3.10 offers someflexibility, in the sense that it only requires that there be sufficiently many stateswith the unique extension property. Accordingly, we next aim to identify a class ofnatural examples where the unique extension property is satisfied by a separatingfamily of states. We start with a general result.Recall that if J is a closed two-sided ideal of a C ∗ -algebra A , then J admits a contractive approximate identity . In other words, there is a net ( e λ ) λ ∈ Λ of positiveelements e λ ∈ J such that k e λ k ≤ λ ∈ Λ and with the property thatlim λ k be λ − b k = lim λ k e λ b − b k = 0for every b ∈ J . Theorem 4.3.
Let A be a unital C ∗ -algebra and let J ⊂ A be a closed two-sidedideal with contractive approximate identity ( e λ ) λ ∈ Λ . Let χ be a state on A such that lim λ χ ( ae λ ) = χ ( a ) for every a ∈ A . Then, χ has the unique extension property with respect to J + C I .Proof. Let ψ be a state on A which agrees with χ on J + C I . Let ( σ ψ , H ψ , ξ ψ )be the associated GNS representation, where ξ ψ ∈ H ψ is a unit cyclic vector. Put H J = [ σ ψ ( J ) H ψ ], which is an invariant subspace for σ ψ ( A ). We can decompose H ψ as H ψ = H J ⊕ H ′ J and accordingly we have σ ψ ( a ) = σ ψ ( a ) | H J ⊕ σ ψ ( a ) | H ′ J , a ∈ A . If we let π ′ J : A → B ( H ′ J ) be the unital ∗ -representation defined as π ′ J ( a ) = σ ψ ( a ) | H ′ J , a ∈ A then it is readily verified that π ′ J ( J ) = { } . Hence, if we decompose ξ ψ = ξ J ⊕ ξ ′ J ∈ H J ⊕ H ′ J then we observe that χ ( b ) = ψ ( b ) = h σ ψ ( b ) ξ ψ , ξ ψ i = h σ ψ ( b ) ξ J , ξ J i for every b ∈ J . Note however that1 = χ ( I ) = lim λ χ ( e λ )whence k χ | J k = 1. We conclude that k ξ J k = 1 and ξ ′ J = 0, whence ξ ψ = ξ J ∈ H J .A standard verification then yieldslim λ σ ψ ( e λ ) ξ ψ = ξ ψ NPERFORATED PAIRS AND HYPERRIGIDITY 15 in the norm topology of H ψ . On the other hand, we have that ae λ ∈ J for each a ∈ A and for each λ ∈ Λ, and thus χ ( a ) = lim λ χ ( ae λ ) = lim λ ψ ( ae λ ) = lim λ h σ ψ ( ae λ ) ξ ψ , ξ ψ i = lim λ h σ ψ ( a ) σ ψ ( e λ ) ξ ψ , ξ ψ i = h σ ψ ( a ) ξ ψ , ξ ψ i = ψ ( a ) . This completes the proof. (cid:3)
We can now identify natural examples where many states have the unique ex-tension property. Recall that a subset
D ⊂ B ( H ) is said to be non-degenerate if DH = H . Corollary 4.4.
Let A ⊂ B ( H ) be a unital C ∗ -algebra and let J ⊂ A be a closedtwo-sided ideal which is non-degenerate. Let X ∈ B ( H ) be a positive trace classoperator with tr X = 1 , and let τ X be the state on A defined as τ X ( a ) = tr( aX ) , a ∈ A . Then, τ X has the unique extension property with respect to J + C I .Proof. Let ( e λ ) λ be a contractive approximate identity for J . By assumption, weknow that H = J H . A standard calculation then shows that ( e λ ) λ ∈ Λ converges tothe identity operator in the strong operator topology of B ( H ). Since τ X is weak- ∗ continuous, we conclude thatlim λ τ X ( ae λ ) = τ X ( a ) , a ∈ A . By virtue of Theorem 4.3, we conclude that τ X has the unique extension propertywith respect to J + C I . (cid:3) Note that Theorem 3.10 implies in particular that C ∗ (Π( A )) = π ( A ) if thereis a family of pure states on C ∗ (Π( A )) which separate (Π − π )( A ) and have theunique extension property with respect to π ( A ), assuming that every irreducible ∗ -representation of A is a boundary representation for S . We point out here that itis not generally the case that two unital C ∗ -algebras B ⊂ A coincide whenever thereis a family of pure states on A which separate A and have the unique extensionproperty with respect to B . The next example illustrates this phenomenon, alongwith the various properties of states considered thus far. Example 4.5.
Let H be an infinite dimensional Hilbert space. Let B be theC ∗ -algebra generated by the identity I and the ideal of compact operators K ( H ).Clearly, B = B ( H ). Recall that any non-degenerate ∗ -representation of K ( H )is unitarily equivalent to some multiple of the identity representation. Standardfacts about the representation theory of C ∗ -algebras (see the discussion preceding[5, Theorem I.3.4]) then imply that any unital ∗ -representation of B ( H ) is unitarilyequivalent to id ( γ ) ⊕ π Q where γ is some cardinal number and π Q is a ∗ -representation of B ( H ) whichannihilates K ( H ). In light of the GNS construction, this shows that a pure stateon B ( H ) is either a vector state or it annihilates K ( H ). For a general state ψ on B ( H ), we have the decomposition ψ = τ + ψ Q where τ is a positive weak- ∗ continuous linear functional on B ( H ), and ψ Q is apositive linear functional on B ( H ) which annihilates K ( H ). Furthermore, there isa positive trace class operator X ψ ∈ B ( H ) such that τ ( a ) = tr( aX ψ ) , a ∈ B ( H ) . We now carefully analyze the states on B ( H ) using this description.First, note that if ψ = τ + ψ Q where both τ and ψ Q are non-zero, then therestriction ρ = ψ | B is not pure. For then ρ − τ | B and ρ − ψ Q | B are positive linearfunctionals. However, τ | B and ψ Q | B cannot be linearly dependent as they are bothnon-zero, and ψ Q annihilates K ( H ) while τ does not.Second, assume that ψ = τ + ψ Q where ψ Q is non-zero. We claim that ψ does nothave the unique extension property with respect to B . Indeed, since B ( H ) / K ( H ) isnot merely one-dimensional and ψ Q ( I ) = 0, there exists a positive linear functional χ on B ( H ) which annihilates K ( H ) and satisfies χ ( I ) = ψ Q ( I ) while χ = ψ Q . Then,the state τ + χ agrees with ψ on B , yet it is distinct from ψ .Third, assume that ψ = ψ Q . We claim that ψ restricts to be pure on B . To seethis, put ρ = ψ | B and suppose that there are states ϕ , ϕ on B with the propertythat ρ = 12 ( ϕ + ϕ ) . Then, we have ϕ ( K ) = − ϕ ( K ) for every K ∈ K ( H ). Since ϕ and ϕ are positive,we conclude that ϕ ( K ) = ϕ ( K ) = 0whenever K ∈ K ( H ) is positive. Using the Schwarz inequality for states [33, Propo-sition 3.3], we see that | ϕ ( K ) | ≤ ϕ ( K ∗ K ) = 0 , K ∈ K ( H ) . Hence ϕ annihilates K ( H ), and so does ϕ by the same argument. Since ϕ ( I ) = ϕ ( I ) = 1 we must have ϕ = ϕ = ρ . Therefore, ρ is pure.Finally, assume that ψ = τ . Then, ψ has the unique extension property withrespect to B by virtue of Corollary 4.4. Also, it is readily seen from Lemma 3.9that ψ restricts to be pure on B if and only if X ψ has rank one (i.e. ψ is a vectorstate). (cid:3) Unperforated pairs of subspaces in a C ∗ -algebra In the previous section, we focused on the unique extension property for states,partly because it provides a means to produce a family of states on a C ∗ -algebrawith the pure restriction property (see Theorem 3.10 and its proof). In this section,we explore a different path and introduce a concept, which, under appropriateconditions, also leads to the identification of an abundance of states that restrictto be pure.Let A be a unital C ∗ -algebra. Let S and T be self-adjoint subspaces of A . Wesay that the pair ( S , T ) is unperforated if for every pair of self-adjoint elements a ∈ S , b ∈ T such that a ≤ b , we can find another self-adjoint element b ′ ∈ T with the property that k b ′ k ≤ k a k and a ≤ b ′ ≤ b . Clearly, the pair ( S , T ) isautomatically unperforated if S ⊂ T . NPERFORATED PAIRS AND HYPERRIGIDITY 17
We provide now an example of an unperforated pair ( S , T ) for which there areself-adjoint elements a ∈ S , b ∈ T with a ≤ b such that no element b ′ ∈ T can bechosen to satisfy a ≤ b ′ ≤ b and k b ′ k = k a k . Example 5.1.
Let M denote the 3 × s = − − − ∈ M and t = − ∈ M . Let S = C s and T = C t , which are both self-adjoint subspaces of M . Let a = αs and b = βt for some α ∈ C and β ∈ C . Assume that a ≤ b , so that − α − α
00 0 − α ≤ β − β
00 0 β . This is equivalent to the inequalities − α ≤ β, − α ≤ β ≤ α/ . In particular, we see that α ≥ | β | ≤ α . Thus, k b k = 2 | β | ≤ α = k a k . We conclude that the pair ( S , T ) is unperforated. In fact, it has an additionalnoteworthy property. Choose α = 1 and β = 1 /
2. Then, we trivially have that − α ≤ β, − α ≤ β ≤ α/ a ∈ S and b ∈ T satisfy a ≤ b as seen above. If λ ∈ R satisfies a ≤ λt ≤ b , then − − − ≤ λ − λ
00 0 λ ≤ / − / which forces λ = 1 /
2. We infer that k λt k = 1 < k a k . (cid:3) We will give further examples of unperforated pairs below (see Proposition 5.4).In the meantime, we illustrate their usefulness for our purposes by leveraging theirdefining property.
Theorem 5.2.
Let A be a unital C ∗ -algebra, let S ⊂ A be a self-adjoint subspaceand let T ⊂ A be a separable operator system. Assume that the pair ( S , T ) isunperforated. Then, for every self-adjoint element s ∈ S , there is a state ψ on A which restrict to be pure on T and such that | ψ ( s ) | = k s k . In particular, there is afamily of states on A which separate S and restrict to be pure on T .Proof. Fix a self-adjoint element s ∈ S . It is no loss of generality to assume that k s k = 1. Upon replacing s with − s , we can find a state θ on A with the propertythat θ ( s ) = 1. Since T is assumed to be separable we may invoke Theorem 2.2 tofind a Borel probability measure µ concentrated on S p ( T ) with the property that θ ( t ) = Z S p ( T ) ω ( t ) dµ ( ω ) , t ∈ T . Assume on the contrary that for each pure state χ on T , we have thatmax ψ ∈E ( χ, A ) | ψ ( s ) | < . We will derive a contradiction by showing that µ ( S p ( T )) = 0. To see this, first useLemma 2.3. We infer that for every pure state χ on T we have thatinf { χ ( t ) : t ∈ T , t ≥ s } < , and thus there is a self-adjoint element t χ ∈ T such that t χ ≥ s and χ ( t χ ) <
1. Sincethe pair ( S , T ) is unperforated, there is t ′ χ ∈ T such that k t ′ χ k ≤ s ≤ t ′ χ ≤ t χ .In particular, we note that χ ( t ′ χ ) <
1. Consider now the weak- ∗ open set A χ = { ω ∈ S p ( T ) : ω ( t ′ χ ) < } . Then, χ ∈ A χ and we see that S p ( T ) = ∪ χ ∈S p ( T ) A χ . Moreover, since k t ′ χ k ≤ θ ( s ) ≤ θ ( t ′ χ ) = Z S p ( T ) ω ( t ′ χ ) dµ ( ω )we find µ ( A χ ) = 0.By assumption, T is separable and thus so is the subset Q = { t ′ χ : χ ∈ S p ( T ) } . Accordingly let { χ n } n ∈ N be a countable subset of S p ( T ) such that { t ′ χ n } n ∈ N is densein Q . Let χ ∈ S p ( T ) and ω ∈ A χ , so that ω ( t ′ χ ) = 1 − ε for some ε >
0. There is N ∈ N such that k t ′ χ N − t ′ χ k < ε/ ω ( t ′ χ N ) < ω ( t ′ χ ) + ε/ − ε/ ω ∈ A χ N . This shows that S p ( T ) = ∪ χ ∈S p ( T ) A χ = ∪ n ∈ N A χ n . Since µ ( A χ n ) = 0 for every n ∈ N , we conclude that µ ( S p ( T )) = 0. This contradictsthe fact that µ has total mass 1. (cid:3) Based on Theorem 3.2, we can now relate unperforated pairs and hyperrigidity.
Corollary 5.3.
Let S be a separable operator system and let A = C ∗ ( S ) . Assumethat every irreducible ∗ -representation of A is a boundary representation for S .Let π : A → B ( H ) be a unital ∗ -representation and let Π : A → B ( H ) be aunital completely positive extension of π | S . Then, the pair ((Π − π )( A ) , π ( A )) isunperforated if and only if Π = π .Proof. If Π = π , then (Π − π )( A ) = { } ⊂ π ( A ) so that the pair ((Π − π )( A ) , π ( A ))is trivially unperforated. Conversely, assume that the pair ((Π − π )( A ) , π ( A )) isunperforated. By Theorem 5.2, there is a family of states on C ∗ (Π( A )) whichseparate (Π − π )( A ) and restrict to be pure on π ( A ). Then Π = π by virtue ofTheorem 3.2. (cid:3) Next, we exhibit a non-trivial condition which ensures that a pair ( S , T ) isunperforated. NPERFORATED PAIRS AND HYPERRIGIDITY 19
Proposition 5.4.
Let A be a unital C ∗ -algebra. Let S and T be self-adjoint sub-spaces of A such that T is unital. Assume that S commutes with T . Then, the pair ( S , C ∗ ( T )) is unperforated.Proof. Let a ∈ S , b ∈ C ∗ ( T ) be self-adjoint elements such that a ≤ b . Define acontinuous function f : R → R as f ( t ) = t if | t | ≤ k a kk a k if t > k a k−k a k if t < −k a k . Observe that f ( a ) = a and that k f ( b ) k ≤ k a k by choice of f . Now, since a ≤ b we must have −k a k I ≤ b and thus the spectrum of b is contained in [ −k a k , k b k ].Furthermore, we have that f ( t ) ≤ t for every t ≥ −k a k . These two observationstogether show that f ( b ) ≤ b .We claim that a ≤ f ( b ). To see this, we note that C ∗ ( T ) commutes with C ∗ ( S )since S and T are self-adjoint, whence the unital C ∗ -algebra C ∗ ( a, b, I ) is commu-tative. Therefore, there is a compact Hausdorff space Ω and a unital ∗ -isomorphismΦ : C ∗ ( a, b, I ) → C (Ω). Put ϕ a = Φ( a ) , ϕ b = Φ( b ). Recalling that a = f ( a ), theclaim is equivalent to the fact that f ◦ ϕ a ≤ f ◦ ϕ b on Ω. Since we have that a ≤ b ,it follows that ϕ a ≤ ϕ b . The function f is non-decreasing, whence f ◦ ϕ a ≤ f ◦ ϕ b on Ω and the claim is established. Finally, the proof is completed by choosing b ′ = f ( b ). (cid:3) In particular, we single out the following noteworthy consequence.
Corollary 5.5.
Let S be a separable operator system and let A = C ∗ ( S ) . Assumethat every irreducible ∗ -representation of A is a boundary representation for S . Let π : A → B ( H ) be a unital ∗ -representation and let Π : A → B ( H ) be a unitalcompletely positive extension of π | S . Assume that (Π − π )( A ) and π ( A ) commute.Then, Π = π .Proof. Simply combine Proposition 5.4 with Corollary 5.3. (cid:3)
In trying to verify that a general pair ( S , C ∗ ( T )) is unperforated, one may hopeto proceed as in the proof of Proposition 5.4 and use the functional calculus to“truncate” b inside of C ∗ ( T ) to have norm at most k a k . However, in general it isnot clear that this truncation should still dominate a . Indeed, the non-decreasingfunction f defined in the proof is not operator monotone. In fact, there are manysimple instances of non-unperforated pairs. Example 5.6.
Let T = C ⊕ C and let S ⊂ M be the self-adjoint subspacegenerated by the matrix (cid:20) (cid:21) . Then, the pair ( S , T ) is not unperforated. Indeed, consider a = (cid:20) (cid:21) ∈ S , b = (cid:20) (cid:21) ∈ T and note that b − a = (cid:20) − − (cid:21) is positive, whence a ≤ b . Let b ′ = (cid:20) x y (cid:21) ∈ T be self-adjoint such that a ≤ b ′ and k b ′ k ≤ k a k = 2. Then, b ′ − a = (cid:20) x − − y (cid:21) ≥ . In particular, we see that x ≥ , y ≥ xy ≥
4. Since k b ′ k ≤
2, we concludethat max { x, y } ≤
2. Hence, x = y = 2 so that b ′ = (cid:20) (cid:21) . But then b − b ′ = (cid:20) − (cid:21) is not positive. (cid:3) In view of this difficulty, a pressing question emerges: how common are unper-forated pairs? We saw in Proposition 5.4 that they can be found easily in thepresence of some form of commutativity, but Example 5.6 indicates the situationmay be bleak in general. Accordingly we aim to introduce flexibility in the defin-ing condition for a pair to be unperforated. The key property we require is thefollowing.A C ∗ -algebra A is said to have the weak expectation property [29] if for everyinjective ∗ -representation π : A → B ( H π ), there is a unital completely positive map E π : B ( H π ) → π ( A ) ′′ satisfying Eπ ( a ) = π ( a ) for every a ∈ A (see for instance [10] for details). The nextdevelopment shows that if B ⊂ A are unital C ∗ -algebras, then the weak expectationproperty for B may be viewed as a variation on the fact that the pair ( A , B ) isunperforated. Interestingly, this fact uses (albeit indirectly) some recent technologyfrom the theory of tensor products of operator systems. Theorem 5.7.
Let A be a unital C ∗ -algebra and let B ⊂ A be a unital separable C ∗ -subalgebra with the weak expectation property. Let a ∈ A be a self-adjoint elementand let ε > . Then, there is a sequence ( β n ) n of self-adjoint elements in B withthe following properties. (1) We have k β n k ≤ (1 + ε ) k a k for every n ∈ N and lim sup n →∞ k β n k ≤ k a k . (2) We have lim sup n →∞ ψ ( β n ) ≤ inf { ψ ( b ) : b ∈ B , b ≥ a } and sup { ψ ( c ) : c ∈ B , c ≤ a } ≤ lim inf n →∞ ψ ( β n ) for every state ψ on B . NPERFORATED PAIRS AND HYPERRIGIDITY 21
Proof.
Assume that B ⊂ A ⊂ B ( H ). Consider the sets U a = { b ∈ B : b ≥ a } , L a = { c ∈ B : c ≤ a } . Since B is separable, so are U a and L a . Thus, there are countable dense subsets { u n } n ∈ N ⊂ U a , { ℓ n } n ∈ N ⊂ L a . Because B has the weak expectation property, itfollows from [25, Theorem 7.4] that it has the so-called tight Riesz interpolationproperty in B ( H ). Noting that B is unital and that − (1 + εn − ) k a k I < a < (1 + εn − ) k a k I, n ∈ N this interpolation property guarantees that for each n ∈ N we can find a self-adjointelement β n ∈ B satisfying − (1 + εn − ) k a k I < β n < (1 + εn − ) k a k I and ℓ j − n − I < β n < u k + n − I for every 1 ≤ j, k ≤ n . In particular, we note that k β n k ≤ (1 + ε ) k a k , n ∈ N and lim sup n →∞ k β n k ≤ k a k . Moreover it follows from the construction of the sequence ( β n ) n that if ψ is a stateon B then sup m ∈ N ψ ( ℓ m ) ≤ lim inf n →∞ ψ ( β n ) ≤ lim sup n →∞ ψ ( β n ) ≤ inf m ∈ N ψ ( u m ) . On the other hand, we have thatinf m ∈ N ψ ( u m ) = inf { ψ ( b ) : b ∈ B , b ≥ a } and sup m ∈ N ψ ( ℓ m ) = sup { ψ ( c ) : c ∈ B , c ≤ a } by density. Hence, lim sup n →∞ ψ ( β n ) ≤ inf { ψ ( b ) : b ∈ B , b ≥ a } and sup { ψ ( c ) : c ∈ B , c ≤ a } ≤ lim inf n →∞ ψ ( β n ) . (cid:3) Of course, the weak expectation property arises naturally without the need forany kind of commutativity, so that Properties (1) and (2) from Theorem 5.7 con-stitute a flexible substitute for the fact that the pair ( A , B ) is unperforated. Wesubstantiate this claim in what follows. We start with a concrete observation. Example 5.8.
Let H be an infinite dimensional separable Hilbert space. The unitalseparable C ∗ -algebra B = K ( H ) + C I is nuclear since K ( H ) is nuclear [10, Exercise2.3.5]. In particular, it has the weak expectation property [10, Exercise 2.3.14].Next, let θ be a state on B ( H ) which has the unique extension property withrespect to B . By Example 4.5 we conclude that there is a positive trace classoperator X θ ∈ B ( H ) with tr( X θ ) = 1 and such that θ ( a ) = tr( aX θ ) , a ∈ B ( H ) . Upon applying the spectral theorem to X θ , we may find a sequence of positivenumbers ( t n ) n ∈ N and a sequence of orthonormal vectors ( ξ n ) n ∈ N such that θ ( a ) = tr( aX θ ) = ∞ X n =1 t n h aξ n , ξ n i for every a ∈ B ( H ). In particular, we see that P ∞ n =1 t n = 1. Fix now a self-adjointelement a ∈ B ( H ). A moment’s thought reveals that there must be ξ ∈ { ξ n } n ∈ N with the property that |h aξ, ξ i| ≥ | θ ( a ) | . Furthermore, if we denote by χ the vector state on B ( H ) corresponding to ξ , wesee from Example 4.5 that χ restricts to be pure on B . This is a manifestation ofa general phenomenon, as we show next. (cid:3) Theorem 5.9.
Let A be a unital C ∗ -algebra and let B ⊂ A be a unital separable C ∗ -subalgebra with the weak expectation property. Let θ be a state on A whichhas the unique extension property with respect to B . Then, for every self-adjointelement a ∈ A there is a state ψ on A which restricts to be pure on B and suchthat | ψ ( a ) | ≥ | θ ( a ) | .Proof. Fix a self-adjoint element a ∈ A , which we may assume is non-zero withoutloss of generality. The desired conclusion is unchanged if we replace a by − a , so wemay assume that θ ( a ) ≥
0. We argue by contradiction. Assume on the contrarythat for each pure state ω on B we havemax ψ ∈E ( ω, A ) | ψ ( a ) | < θ ( a ) . Then, we infer from Lemma 2.3 thatinf { ω ( b ) : b ∈ B , b ≥ a } < θ ( a ) . Now, by Theorem 5.7 there is a sequence ( β n ) n ∈ N of self-adjoint elements in B with k β n k ≤ k a k for every n ∈ N and such thatsup { θ ( c ) : c ∈ B , c ≤ a } ≤ lim inf n →∞ θ ( β n )and lim sup n →∞ ω ( β n ) ≤ inf { ω ( b ) : b ∈ B , b ≥ a } < θ ( a )for every pure state ω on B . Since θ is assumed to have the unique extensionproperty with respect to B , by Lemma 2.3 we find θ ( a ) ≤ lim inf n →∞ θ ( β n ) . On the other hand, since B is assumed to be separable we may invoke Theorem2.2 to find a Borel probability measure µ concentrated on S p ( B ) with the propertythat θ ( b ) = Z S p ( B ) ω ( b ) dµ ( ω ) , b ∈ B . Upon applying Fatou’s lemma to the sequence of non-negative continuous functions ω k a k − ω ( β n ) , ω ∈ S p ( B ) NPERFORATED PAIRS AND HYPERRIGIDITY 23 a simple calculation yieldslim sup n →∞ Z S p ( B ) ω ( β n ) dµ ( ω ) ! ≤ Z S p ( B ) (cid:18) lim sup n →∞ ω ( β n ) (cid:19) dµ ( ω ) . Consequently lim sup n →∞ θ ( β n ) = lim sup n →∞ Z S p ( B ) ω ( β n ) dµ ( ω ) ! ≤ Z S p ( B ) (cid:18) lim sup n →∞ ω ( β n ) (cid:19) dµ ( ω ) < θ ( a ) . But this implies that θ ( a ) ≤ lim inf n →∞ θ ( β n ) ≤ lim sup n →∞ θ ( β n ) < θ ( a )which is absurd. (cid:3) We mention a noteworthy consequence of Theorem 5.9 which is related to hy-perrigidity.
Corollary 5.10.
Let S be a separable operator system and let A = C ∗ ( S ) . Assumethat every irreducible ∗ -representation of A is a boundary representation for S . Let π : A → B ( H ) be a unital ∗ -representation such that π ( A ) has the weak expectationproperty, and let Π : A → B ( H ) be a unital completely positive extension of π | S .Then, π = Π if and only if there is a family of states on C ∗ (Π( A )) which separate (Π − π )( A ) and have the unique extension property with respect to π ( A ) .Proof. Assume that there is a family of states on C ∗ (Π( A )) which separate (Π − π )( A ) and have the unique extension property with respect to π ( A ). We may applyTheorem 5.9 to the inclusion π ( A ) ⊂ C ∗ (Π( A )) (see (1)) to find a (potentiallydifferent) family of states on C ∗ (Π( A )) which separate (Π − π )( A ) and restrict tobe pure on π ( A ). Consequently, π = Π by virtue of Theorem 3.2. The converse istrivial. (cid:3) We draw the reader’s attention to the main point of Corollary 5.10: unlike inTheorem 3.10, the separating family is not assumed to consist of pure states.We finish by mentioning that it would be of interest to obtain a version ofTheorem 5.2 or Corollary 5.3 based on Theorem 5.7. It is not clear to us how thiscan be achieved at present. The promise of such an application of Theorem 5.7is the reason why we chose to state it in the context of B being separable. Thereader will notice that this condition can be removed at the cost of obtaining a netrather than a sequence. We opted for the current version as sequences seem moreappropriate for arguments relying on integration techniques. References [1] Francesco Altomare,
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