# A Universal Representation for Quantum Commuting Correlations

aa r X i v : . [ qu a n t - ph ] F e b A UNIVERSAL REPRESENTATION FOR QUANTUM COMMUTINGCORRELATIONS

ROY ARAIZA, TRAVIS RUSSELL, AND MARK TOMFORDE

Abstract.

We explicitly construct an Archimedean order unit space whose state space is aﬃnelyisomorphic to the set of quantum commuting correlations. Our construction only requires funda-mental techniques from the theory of order unit spaces and operator systems. Our main results areachieved by characterizing when a ﬁnite set of positive contractions in an Archimedean order unitspace can be realized as a set of projections on a Hilbert space. Introduction

The past decade has witnessed a tremendous surge of interest in the theory of quantum corre-lations — probability distributions arising from independent measurements of entangled quantumsystems. About ten years ago, several authors (e.g., [9], [4], and [14]) uncovered deep connec-tions between the nearly ﬁfty-year-old Connes’ embedding problem [3] in operator algebras andTsirelson’s problem [18] concerning quantum correlations. In subsequent years, the literature onquantum correlations expanded rapidly as mathematicians, physicists, and computer scientistsworked in tandem to address Tsirelson’s problem. This ﬂurry of activity yielded several importantresults, including the non-closure of the set of ﬁnite-dimensional quantum correlations [17] and theequality of the complexity classes MIP ∗ and RE established in [8]. The latter of these two resultsimplies a negative solution to Connes’ embedding problem.Much of the literature on correlation sets focuses on distinguishing two classes of correlations:the class of ﬁnite-dimensional quantum correlations, denoted C q ( n, k ), and the class of quantumcommuting correlations, denoted C qc ( n, k ), where n and k are parameters that denote the number ofexperiments and the number of outcomes, respectively, in a measurement scenario. While these setsare known to be convex and satisfy C q ( n, k ) ⊆ C qc ( n, k ), a detailed description of their geometryhas only been obtained in certain restrictive scenarios (e.g., [6]). Indeed, a recent preprint [5] showsthat the problem of determining whether or not a given probability distribution belongs to the setof quantum correlations is undecidable. Consequently, new descriptions of the quantum correlationsets beyond their original deﬁnitions are valuable. One example of such a description is found in[13], which characterized the set C qc ( n, k ) as the set of probability distributions that can be certiﬁedby a certain inﬁnite hierarchy of semideﬁnite programs. Another example can be found in [12],where the set C qc ( n, k ) and the closure of the set C q ( n, k ) are separately identiﬁed with the statespaces of certain tensor products of ﬁnite-dimensional operator systems. These operator systemsarise as subsystems of non-amenable universal group C*-algebras.In this paper, we provide a new description for the set of quantum commuting correlations.Speciﬁcally, we construct a ﬁnite-dimensional ordered vector space whose state space is aﬃnelyisomorphic to the set C qc ( n, k ) of all quantum commuting correlations (see Theorem 5.14). Thisis achieved using techniques from the theory of operator systems. Moreover, our constructionproceeds without reference to either universal C*-algebras or to the hierarchy of operator system Mathematics Subject Classiﬁcation. ensor products from [11], relying instead on previous work of the ﬁrst two authors [1]. Thisconstruction is achieved by ﬁrst characterizing when a unital ordered vector space with a ﬁnite setof positive contractions { p , p , . . . , p n } can be realized as a subspace of the bounded operators ona Hilbert space H such that each vector in { p , p , . . . , p n } is a projection on H (see Theorem 4.18).We then construct, for each n and k , an Archimedean ordered vector space ( V ns , D ns , e ns ) whosestate space is aﬃnely isomorphic to the set of nonsignalling correlations C ns ( n, k ) (see Theorem5.10). Using our results on projections in ordered vector spaces, we complete the positive cone D ns of V ns to a positive cone D qc , which arises from projection-valued measures on a Hilbert space,yielding the following theorem: Theorem 1.1.

Let n, k ∈ N and let V ns denote the corresponding universal nonsignalling vectorspace with generators Q ( a, b | x, y ) where a, b ∈ { , . . . , k } , and x, y ∈ { , . . . , n } . If p = { p ( a, b | x, y ) } is a correlation, then p ∈ C qc ( n, k ) (respectively, p ∈ C ns ( n, k ) ) if and only if there exists a state φ : ( V ns , D qc , e ns ) → C (respectively, a state φ : ( V ns , D ns , e ns ) → C ) such that p ( a, b | x, y ) = φ ( Q ( a, b | x, y )) for each a, b ∈ { , . . . , k } and x, y ∈ { , . . . , n } . The organization of the paper is as follows: Section 2 covers preliminary material from the theoryof operator systems and Archimedean order unit spaces. Section 3 discusses Archimedean orderunit spaces containing a single non-trivial projection. Section 4 establishes results concerning ﬁnitesets of projections in Archimedean order unit spaces. We conclude in Section 5 with applicationsof our results to the theory of quantum correlation sets.2.

Preliminaries

Let N , R , and C denote the sets of natural, real and complex numbers, respectively. Given n ∈ N ,we let [ n ] := { , , . . . , n } . Throughout the paper we will be working in a variety of vector spaces,all of which may be assumed to be vector spaces over the ﬁeld of complex numbers unless speciﬁedotherwise. Given a vector space V , we let M n ( V ) denote the vector space M n ( C ) ⊗ V of n × n matrices with entries in V . In particular, M n ( C ) denotes the algebra of n × n matrices over C . Bya ∗ -vector space , we mean a vector space V equipped with a conjugate linear involution ∗ : V → V .The hermitian elements of V are those elements x satisfying x = x ∗ , and we let V h denote thereal vector space of hermitian elements. If A = ( a ij ) ∈ M n ( V ), then A ∗ will denote the conjugatetranspose of A ; i.e., the matrix whose ij entry is a ∗ ji . For each n ∈ N , we let I n ∈ M n ( C ) denotethe identity matrix and J n ∈ M n ( C ) denote the matrix with every entry equal to 1. If A ∈ M n ( C )and B ∈ M m ( C ), we let A ⊕ B ∈ M n + m ( C ) denote the direct sum; i.e., A ⊕ B := (cid:18) A B (cid:19) . We often use the canonical shuﬄe map ϕ : M n ( C ) ⊗ M m ( C ) → M m ( C ) ⊗ M n ( C ), which acts onelementary tensors via ϕ : A ⊗ B → B ⊗ A . This mapping is a ∗ -isomorphism of the C*-algebra M nm ( C ) and extends to the corresponding shuﬄe maps on M n ( V ) ⊗ M m ( C ) and M n ( C ) ⊗ M m ( V )via the identiﬁcations M n ( V ) ⊗ M m ( C ) ∼ = M nm ( C ) ⊗ V ∼ = M n ( C ) ⊗ M m ( V ).An ordered ∗ -vector space is a pair ( V , C ) where V is a ∗ -vector space and C ⊆ V h is a positivecone; i.e., C + C ⊆ C and R + C ⊆ C . A cone C is proper if C ∩ − C = { } , and when this occurswe call ( V , C ) a proper ordered ∗ -vector space . Given any ordered ∗ -vector space ( V , C ) we maydeﬁne J := span C ∩ − C and form the proper ordered ∗ -vector space V / J with the proper ordering C + J . For any proper ordered ∗ -vector space ( V , C ), the positive cone C induces a partial orderingon V h by declaring x ≤ y if and only if y − x ∈ C. An element e ∈ V h is called an order unit if forall v ∈ V h there exists t > te − v ∈ C. An Archimedean order unit is an order unti e that also satisﬁes the Archimedean property : if v ∈ V and if ǫe + v ∈ C for all ǫ >

0, then v ∈ C . Ashort argument shows that an Archimedean order unit must be an element of C . These properties mply that the positive cone majorizes the hermitian elements of V and that the positive cone C isArchimedean closed. Deﬁnition . An Archimedean order unit (AOU) space is a triple ( V , C, e ), where ( V , C ) is aproper ordered ∗ -vector space and e is an Archimedean order unit.If e is an Archimedean order unit for ( V , C ) but C fails to be proper, we may form J := span C ∩− C and then ( V / J , C + J , e + J ) is an AOU space (see Proposition 2.3). A linear map ϕ : ( V , C, e ) → ( W , D, f ) between AOU spaces will be called positive if ϕ ( C ) ⊆ D , and unital if ϕ ( e ) = f . If ϕ isa (unital) linear isomorphism such that ϕ and ϕ − are both positive then we call ϕ a (unital) orderisomorphism . Given an AOU space ( V , C, e ) there is a canonical norm k · k : V h → [0 , ∞ ) deﬁnedon the hermitian elements and given by k v k := inf { t > te ± v ∈ C} . We call this norm the ordernorm associated with e .A function system is deﬁned to be a self-adjoint unital subspace of C ( K ) for some compactHausdorﬀ space K . Due to the results stemming from [10, 16], every AOU space V may beidentiﬁed as a concrete function subsystem of the continuous functions on the state space of V . In particular, every AOU space may be identiﬁed with the continuous aﬃne functions on its statespace, and conversely if K is a compact convex subset of a locally convex space, and if A ( K ) denotesthe space of continuous aﬃne functions on K , then K is aﬃnely isomorphic to the state space of A ( K ).Let V be a ∗ -vector space. We deﬁne a matrix ordering to be a sequence C := {C n } n ∈ N , where C n is a cone in M n ( V ) h for each n ∈ N . We call C proper if each C n is a proper cone. The pair( V , C ) is a (proper) matrix ordered ∗ -vector space if V is a ∗ -vector space and C is a (proper) matrixordering. The matrix ordering C deﬁnes a partial ordering on M n ( V ) h for each n ∈ N by declaring x ≤ y if and only if y − x ∈ C n . An element e ∈ V h will be called a matrix order unit if I n ⊗ e isan order unit for the ordered ∗ -vector space ( M n ( V ) , C n ) for each n ∈ N . It is a fact that e ∈ C is an order unit if and only if e is a matrix order unit (see, e.g., [1, Proposition 2.4]). If I n ⊗ e isan Archimedean order unit for ( M n ( V ) , C n ) for each n ∈ N , then we call e an Archimedean matrixorder unit . This latter property ensures that each cone C n is Archimedean closed. Deﬁnition . An operator system is a triple ( V , C , e ), where V is a ∗ -vector space, C is a propermatrix ordering, and e is an Archimedean matrix order unit.When no confusion arises we will simply denote an operator system by V . If u : V → W is alinear map between operator systems, we deﬁne the n th -ampliﬁcation of u to be the map u n := I n ⊗ u : M n ( V ) → M n ( W ) deﬁned by P ij e i e ∗ j ⊗ v ij e i e ∗ j ⊗ u ( v ij ), where { e i } ni =1 ⊆ C n denotesthe canonical column basis vectors. Letting C and D be the respective proper matrix orderings on V and W , we call the map u completely positive if u n ( C n ) ⊆ D n for each n ∈ N . If u : V → W is a (unital) linear isomorphism such that both u and u − are completely positive then we saythat u is a (unital) complete order isomorphism. When u is a (unital) complete order isomorphismonto its range, we will sometimes call u a (unital) complete order embedding . We will identify twooperator systems if there exists a unital complete order isomorphism between them. A concreteoperator system is deﬁned to be a unital self-adjoint subspace of B ( H ). A fundamental result from[2, Theorem 4.4] states that for any operator system ( V , C , e ) then there exists a Hilbert space H and a concrete operator system e V ⊆ B ( H ) such that V is unital completely order isomorphic to e V . Similar to the case for AOU spaces, it follows that operator systems may be identiﬁed with whatare known as continuous matrix aﬃne functions acting on the matrix state space of that operatorsystem and the converse noncommutative analogue also holds. For the converse, one may identifyobjects known as compact matrix convex sets with the matrix state space of the continuous matrixaﬃne functions acting on that matrix convex set. This duality is known as Webster-Winkler dualityand was ﬁrst investigated in [19]. hroughout the manuscript we will be dealing with matrix orderings that are not a priori proper.For such a matrix ordering, we may always consider a natural quotient of the matrix ordered spacesuch that the quotient is necessarily a proper matrix ordered ∗ -vector space. In particular, given anymatrix ordered ∗ -vector space ( V , C ) we consider the quotient V / J , where J := span C ∩−C . Withthe natural involution deﬁned on cosets as ( v + J ) ∗ := v ∗ + J and letting C + J := {C n + M n ( J ) } n ∈ N ,it necessarily follows that ( V / J , C + J ) is a proper matrix ordered ∗ -vector space. Given any matrixordered ∗ -vector space ( V , C ) with Archimedean matrix order unit e , it also follows that e + J isan Archimedean matrix order unit for the proper matrix ordered ∗ -vector space ( V / J , C + J ) . Inparticular:

Proposition 2.3 ([1, Proposition 4.4]) . Given any matrix ordered ∗ -vector space ( V , C ) withArchimedean matrix order unit e , the triple ( V / J , C + J , e + J ) is an operator system. Given an operator system V , it was shown in [7] that there exists a C*-algebra A and a unitalcomplete order embedding j : V → A such that A = C ∗ ( j ( V )) and A satisﬁes the followinguniversal property: given a pair ( D , i ) where i : V → D is a unital complete order embedding and D is a (unital) C*-algebra generated by i ( V ), then there exists a unique surjective ∗ -homomorphism σ : D → A such that σ ◦ i = j. Thus we have the following commutative diagram:

DV A σi j

The C*-algebra A is called the C*-envelope of V and we will often denote it by C ∗ e ( V ) . Given anoperator system V , a pair ( D , i ) consisting of a C*-algebra D and a unital complete order embedding i : V → D is called a

C*-cover of V when D = C ∗ ( i ( V )) . Thus the C*-envelope of an operatorsystem may be viewed as the minimal C*-cover of that operator system.In [15], it was shown that given any AOU space V there exists a maximal operator systemstructure on V . Given an AOU space ( V , C, e ), for each n ∈ N we deﬁne D maxn := { a ∗ va : a ∈ M m,n and v = ⊕ mi =1 v i for some m ∈ N and for some v , . . . , v m ∈ C } . Then D max := { D maxn } ∞ n =1 is a proper matrix ordering on V with the property that if P is anyother matrix ordering on V with P = C , then D maxn ⊆ P n for each n ∈ N . In general, the matrixorder unit e may fail to be Archimedean for D max . Thus, for each n ∈ N we deﬁne C maxn := { v ∈ M n ( V ) h : ∀ ǫ > , v + ǫ ( I n ⊗ e ) ∈ D maxn } and form the proper matrix ordering C max := { C maxn } ∞ n =1 . (The process of going from D maxn to C maxn is called Archimedeanization ; see [16].) The triple ( V , C max , e ) is an operator system calledthe maximal operator system structure on V . For ease of notation, we shall denote the triple as V max when no confusion will arise.3. Abstract projections in operator systems and AOU spaces

Deﬁnition . Suppose that ( V , C , e ) is an operator system. Given a positive contraction p ∈ V ,we let C ( p ) denote the (generally non-proper) matrix ordering on M ( V ) deﬁned as follows: C ( p ) n := { x ∈ M n ( V ) h : ∀ ǫ > ∃ t > x + ǫI n ⊗ ( p ⊕ p ⊥ ) + tI n ⊗ ( p ⊥ ⊕ p ) ∈ C n } . Let J p := span C ( p ) ∩ − C ( p ) , and deﬁne π p : V → M ( V ) / J p by π p ( x ) = x ⊗ J + J p . We consider M ( V ) / J p with operator system structure ( M ( V ) / J p , C ( p ) + J p , I ⊗ e + J p ) . e remark that for each n ∈ N we identify M n ( M ( V ) / J p ) with M n ( V ) /M n ( J p ). Moreover,for each x ∈ M n ( V ), the n th -ampliﬁcation ( π p ) n : M n ( V ) → M n ( V ) /M n ( J p ) satisﬁes ( π p ) n ( x ) = x ⊗ J + M n ( J p ). Proposition 3.2.

Let ( V , C , e ) be an operator system, and suppose that p ∈ V is a positive con-traction. Then the map π p : V → M ( V ) / J p from Deﬁnition 3.1 is a unital completely positivemap.Proof. Unitality follows from [1, Lemma 5.6]. It remains to show complete positivity. Let x ∈ C n .Then x ⊗ J ∈ C n . It follows that for every ǫ > x ⊗ J + ǫI n ⊗ ( p ⊕ p ⊥ ) + ǫI n ⊗ ( p ⊥ ⊕ p ) ∈ C n . Therefore x ⊗ J ∈ C ( p ) n . It follows that( π p ) n ( x ) = x ⊗ J + M n ( J p ) ∈ C ( p ) n + M n ( J p ) . So π p is completely positive. (cid:3) While Proposition 3.2 implies that π p is always completely positive, it is not necessarily aninjective map in general, much less a complete order embedding. The importance of π p is illustratedby the following theorem. Theorem 3.3 ([1, Deﬁnition 5.4, Theorem 5.7, Theorem 5.8]) . Let ( V , C , e ) be an operator systemand suppose that p ∈ V is a positive contraction. Then the following statements are equivalent: (1) The map π p : V → M ( V ) / J p is a complete order embedding. (2) There exists a Hilbert space H and a unital complete order embedding π : V → B ( H ) suchthat π ( p ) is a projection in B ( H ) . (3) The element p is a projection in C ∗ e ( V ) .Deﬁnition . We say that a positive contraction p in an operator system ( V , C , e ) is an abstractprojection if p satisﬁes one (and hence all) of the conditions in Theorem 3.3. Remark . Our notion of an abstract projection above diﬀers slightly from the one in [1] in that werequire that p be a positive contraction, but do not require k p k ∈ { , } . However the requirementsthat p is a positive contraction and that π p is a complete order embedding are suﬃcient. Indeed,it can be shown that if 0 < k p k < π p ( p ) = 0 and therefore π p is not an order embedding.Notice that Condition (3) of Theorem 3.3 implies that whenever p and q are abstract projec-tions in an operator system V , there always exists a Hilbert space H and a unital complete orderembedding π : V → B ( H ) such that both π ( p ) and π ( q ) are projections on H . Indeed, this can beachieved by applying the Gelfand-Naimark Theorem to the C*-algebra C ∗ e ( V ).We now turn our attention to projections in AOU spaces. We would like to characterize when apositive contraction p in an AOU space ( V , C, e ) is an “abstract projection” in the sense that thereexists a Hilbert space H and a unital order embedding π : V → B ( H ) such that π ( p ) is a projectionin B ( H ). By Theorem 3.3 it suﬃces to characterize when there exists a matrix ordering C on V such that ( V , C , e ) is an operator system, C = C , and p is an abstract projection in ( V , C , e ). Lemma 3.6.

Let ( V , C, e ) be an AOU space. Suppose that C is a matrix ordering on V such that C = C and ( V , C , e ) is an operator system, and suppose that p ∈ V is a positive contraction. Deﬁnea matrix ordering T on V by T n := { x ∈ M n ( V ) : x ⊗ J ∈ C ( p ) n } . Then T ( p ) = C ( p ) . roof. First observe that T n = ( π p ) − n ( C ( p ) n + M n ( J p )), and hence T is a matrix ordering on V .By Proposition 3.2, we know that π p : V → M ( V ) / J p is completely positive. Therefore whenever x ∈ C n , we have ( π p ) n ( x ) ∈ C ( p ) n + M n ( J p ). It follows that C n ⊆ T n . Now suppose that x ∈ C ( p ) n .Then for every ǫ > t > x + ǫI n ⊗ ( p ⊕ p ⊥ ) + tI n ⊗ ( p ⊥ ⊕ p ) ∈ C n . But since C n ⊆ T n for all n we conclude that x ∈ T ( p ) n . Hence C ( p ) n ⊆ T ( p ) n .On the other hand, suppose that x ∈ T ( p ) n . Then for every ǫ > t > π p ) n ( x + ǫI n ⊗ ( p ⊕ p ⊥ ) + tI n ⊗ ( p ⊥ ⊕ p )) ∈ C ( p ) n + M n ( J p ) . Hence, for every ǫ >

0, there exist t, r > h x + ǫ I n ⊗ ( p ⊕ p ⊥ ) + tI n ⊗ ( p ⊥ ⊕ p ) i ⊗ J + ǫ I n ⊗ ( p ⊕ p ⊥ ) + rI n ⊗ ( p ⊥ ⊕ p ) ∈ C n . Applying the canonical shuﬄe ϕ : M n ⊗ M → M ⊗ M n to the expression above we obtain J ⊗ h x + ǫ I n ⊗ ( p ⊕ p ⊥ ) + tI n ⊗ ( p ⊥ ⊕ p ) i + ǫ p ⊕ p ⊥ ) ⊗ I n + r ( p ⊥ ⊕ p ) ⊗ I n or, in matrix form, (cid:18) x xx x (cid:19) + ǫ (cid:18) I n ⊗ ( p ⊕ p ⊥ ) I n ⊗ ( p ⊕ p ⊥ ) I n ⊗ ( p ⊕ p ⊥ ) I n ⊗ ( p ⊕ p ⊥ ) (cid:19) + t (cid:18) I n ⊗ ( p ⊥ ⊕ p ) I n ⊗ ( p ⊥ ⊕ p ) I n ⊗ ( p ⊥ ⊕ p ) I n ⊗ ( p ⊥ ⊕ p ) (cid:19) + ǫ (cid:18) I n ⊗ p I n ⊗ p ⊥ (cid:19) + r (cid:18) I n ⊗ p ⊥ I n ⊗ p (cid:19) . Let W be the 4 n × n scalar permutation matrix which exchanges the j th and ( j + 2 n ) th columnsfor j = 2 , , . . . , n . Then conjugating the ﬁnal expression above by W yields (cid:18) x xx x (cid:19) + ǫ (cid:18) I n ⊗ ( p ⊕ p ⊥ ) I n ⊗ ( p ⊕ p ⊥ ) I n ⊗ ( p ⊕ p ⊥ ) I n ⊗ ( p ⊕ p ⊥ ) (cid:19) + t (cid:18) I n ⊗ ( p ⊥ ⊕ p ) I n ⊗ ( p ⊥ ⊕ p ) I n ⊗ ( p ⊥ ⊕ p ) I n ⊗ ( p ⊥ ⊕ p ) (cid:19) + ǫ (cid:18) I n ⊗ ( p ⊕ p ⊥ ) 00 I n ⊗ ( p ⊥ ⊕ p ) (cid:19) + r (cid:18) I n ⊗ ( p ⊥ ⊕ p ) 00 I n ⊗ ( p ⊕ p ⊥ ) (cid:19) . Compressing to the upper left corner of this expression and using the compatibility of C , we seethat for every ǫ > t, r > x + ǫI n ⊗ ( p ⊕ p ⊥ ) + ( t + r ) I n ⊗ ( p ⊥ ⊕ p ) ∈ C n . Thus x ∈ C ( p ) n . Therefore T ( p ) n ⊆ C ( p ) n . We conclude that for every n ∈ N , T ( p ) n = C ( p ) n . (cid:3) Theorem 3.7.

Let ( V , C, e ) be an AOU space, and suppose that p ∈ V is a positive contraction.Then the following statements are equivalent: (1) There exists a matrix ordering C with C = C such that p is an abstract projection in theoperator system ( V , C , e ) . (2) The map π p : ( V , C ) → ( M ( V ) / J p , C max ( p ) + J p ) is an order embedding.Proof. Suppose that (1) holds and that π p ( x ) ∈ C max ( p ) + J p . Then for every ǫ > t > (cid:18) x xx x (cid:19) + ǫ ( p ⊕ p ⊥ ) + t ( p ⊥ ⊕ p ) ∈ C max ⊆ C . It follows that π p ( x ) ∈ C ( p ) + J p , and hence x ∈ C = C . On the other hand, if x ∈ C then π p ( x ) ∈ C max ( p ) + J p by Proposition 3.2. Hence (2) holds.Next suppose that (2) holds. Deﬁne a matrix ordering C on V by C n := ( π p ) − n ( C max ( p ) n + M n ( J p )) . ince π p is an order embedding, C = C . It remains to show that π p is a complete order embeddingon ( V , C ). By Lemma 3.6 we see that C ( p ) n = C max ( p ) n for each n . Thus, if ( π p ) n ( x ) ∈ C ( p ) n + M n ( J p ), then x ∈ C n , since C ( p ) n + M n ( J p ) = C max ( p ) n + M n ( J p ) for each n . Therefore π p is acomplete order embedding. (cid:3) Theorem 3.7 has an important limitation distinguishing it from Theorem 3.3. Suppose that p and q are elements of an AOU space ( V , C, e ) both of which satisfy Condition (2) of Theorem 3.7.Then there exist matrix orderings C and T such that ( V , C , e ) and ( V , T , e ) are operator systemswith C = T = C , p is an abstract projection in ( V , C , e ), and q is an abstract projection in( V , T , e ). However it is not necessary that C = T in general, nor is it clear how one could constructa new matrix ordering R with R = C for which p and q are both abstract projections in ( V , R , e ).Addressing this diﬃculty will be the subject of the next section.4. Multiple projections in AOU spaces

In Section 3 we described when a single contraction in an AOU space is a projection. In thissection we shall characterize when the elements of a ﬁnite set of contractions in an AOU space areprojections. To do this we ﬁrst consider ﬁnite sets of abstract projections in an operator system.

Deﬁnition . Let ( V , C , e ) be an operator system, and assume p , p , . . . , p N ∈ V are positivecontractions. For each k = 1 , , . . . , N deﬁne P Nk := I k − ⊗ ( p k ⊕ p ⊥ k ) ⊗ J N − k and Q Nk := I k − ⊗ ( p ⊥ k ⊕ p k ) ⊗ J N − k Similarly, we deﬁne b P Nk := J k − ⊗ ( p k ⊕ p ⊥ k ) ⊗ J N − k and b Q Nk := J k − ⊗ ( p ⊥ k ⊕ p k ) ⊗ J N − k . For each n ∈ N , we deﬁne C ( p , p , . . . , p N ) n to be the set of x ∈ M n N ( V ) such that x = x ∗ and for every ǫ , ǫ , . . . , ǫ N > t , t , . . . , t N > x + N X k =1 ǫ k I n ⊗ P Nk + N X k =1 t k I n ⊗ Q Nk ∈ C n N , with the property that if one replaces ǫ i with ǫ ′ i < ǫ i then there exists t ′ i > t i such that Equa-tion (1) still holds. We deﬁne b C ( p , p , . . . , p N ) n similarly with b P Nk and b Q Nk playing the role of P Nk and Q Nk . Note that for each n ∈ N the sets C ( p , . . . , p N ) n and b C ( p , . . . , p N ) n are nonemptysince if x ∈ M n N ( V ) + one may choose t i = ǫ i for i = 1 , . . . , N . Let C ( p , p , . . . , p N ) := {C ( p , p , . . . , p N ) n } ∞ n =1 , and b C ( p , p , . . . , p N ) := { b C ( p , p , . . . , p N ) n } ∞ n =1 .In the following we make use of the sequences of cones C ( p , . . . , p N ) and b C ( p , . . . , p N ). In thecase N = 1 these sequences coincide and are both equal to the non-proper matrix ordering C ( p ),which was used to characterize single projections in [1] (cf. Section 3). The cones b C ( p , . . . , p N ) areuseful because they are symmetric with respect to the order of the operators p , . . . , p N in the sensethat a rearrangement of these operators can be realized by simply applying a canonical shuﬄe tothe corresponding operators b P Ni . However we will focus primarily on the cones C ( p , p , . . . , p N ). Lemma 4.2. If N, n ∈ N , then b C ( p , p , . . . , p N ) n ⊆ C ( p , p , . . . , p N ) n . roof. First observe that for each k ∈ N we have I n ⊗ ˆ P Nk = I n ⊗ J k − ⊗ ( p k ⊕ p ⊥ k ) ⊗ J N − k ≤ k − I n ⊗ I k − ⊗ ( p k ⊕ p ⊥ k ) ⊗ J N − k = 2 k − I n ⊗ P Nk and similarly I n ⊗ ˆ Q Nk ≤ k − I n ⊗ Q Nk . Indeed, these inequalities hold because J ≤ I .Next suppose x ∈ b C ( p , p , . . . , p N ) n . Then for every ǫ , . . . , ǫ N > t , . . . , t N > x + N X k =1 ǫ k k − I n ⊗ b P Nk + N X k =1 t k I n ⊗ b Q Nk ≥ . Consequently x + N X k =1 ǫ k I n ⊗ P Nk + N X k =1 k − t k I n ⊗ Q Nk ≥ . We conclude that x ∈ C ( p , p , . . . , p N ) n . (cid:3) We will make use of the following result from [1].

Lemma 4.3 ([1, Theorem 5.3]) . Let

V ⊆ B ( H ) be an operator system. Suppose that p ∈ V is aprojection on H . Then for each x ∈ V we have that x ≥ if and only if for every ǫ > there exists t > such that J ⊗ x + ǫ ( p ⊕ p ⊥ ) + t ( p ⊥ ⊕ p ) ≥ . Proposition 4.4.

Let N ∈ N and V ⊆ B ( H ) be an operator system and suppose that p , p , . . . p N ∈V are all projections on H . For x ∈ M n ( V ) the following are equivalent: (1) x ∈ C n . (2) x ⊗ J N ∈ C ( p , p , . . . , p N ) n . (3) x ⊗ J N ∈ b C ( p , p , . . . , p N ) n .Proof. To see that (1) = ⇒ (3) = ⇒ (2) notice that if x ∈ C n , then x ⊗ J N ∈ b C ( p , . . . , p N ) n since C n ⊗ J N ⊆ b C ( p , . . . , p N ) n , which by Lemma 4.2 implies x ⊗ J N ∈ C ( p , . . . , p N ) n . We will prove (2) = ⇒ (1) by induction on N . The case N = 1 appears in Lemma 4.3. Supposethat the statement holds for N −

1, and consider the statement for N .Suppose that x ⊗ J N ∈ C ( p , p , . . . , p N ) n . Then for every ǫ , ǫ , . . . , ǫ N > t ,t , . . . , t N > x ⊗ J N + N − X k =1 ǫ k I n ⊗ P Nk + ǫ N I n N − ⊗ ( p N ⊕ p ⊥ N ) + N − X k =1 t k I n ⊗ Q Nk + t N I n N − ⊗ ( p ⊥ N ⊕ p N ) ∈ C n N . Applying the canonical shuﬄe map ϕ : M n ⊗ M N − ⊗ M → M ⊗ M n ⊗ M N − , we obtain J ⊗ " x ⊗ J N − + N − X k =1 ǫ k I n ⊗ P N − k + N − X k =1 t k I n ⊗ Q N − k + ǫ N ( p N ⊕ p ⊥ N ) ⊗ I n N − + t N ( p ⊥ N ⊕ p N ) ⊗ I n N − ∈ C n N . Since p N ⊗ I n N − is a projection in B ( H n N − ), we conclude that x ⊗ J N − + N − X k =1 ǫ k I n ⊗ P N − k + N − X k =1 t k I n ⊗ Q N − k ∈ C n N − . By the inductive hypothesis, x ∈ C n , and thus (1) holds. (cid:3) ur goal is to use Proposition 4.4 to abstractly characterize ﬁnite sets of projections in anoperator system and ultimately to abstractly characterize ﬁnite sets of projections in an AOUspace. We begin by showing that we can endow a quotient of M N ( V ) with a useful operatorsystem structure. Lemma 4.5.

Let ( V , C , e ) be an operator system and suppose p , p , . . . , p N ∈ V are positive con-tractions. Then C ( p , p , . . . , p N ) is a (non-proper) matrix ordering on the vector space M N ( V ) .Proof. The proof is similar to [1, Proposition 4.7]. To see that C ( p , . . . , p N ) is a matrix ordering,it suﬃces to check that it is closed under direct sums and conjugation by scalar matrices of theform α ⊗ I N (since the base vector space is M N ( V )). Suppose x ∈ C ( p , . . . , p N ) n and y ∈C ( p , . . . , p N ) m . If ǫ , . . . , ǫ N >

0, then there exist t , . . . , t N > r , . . . , r N > x + N X k =1 ǫ k I n ⊗ P Ni + N X k =1 t k I n ⊗ Q Nk ∈ C n N and y + N X k =1 ǫ k I m ⊗ P Nk + N X k =1 r k I m ⊗ Q Nk ∈ C m N . Let s k = max( t k , r k ) for each k . Then x ⊕ y + N X k =1 ǫ k I n + m ⊗ P Nk + N X k =1 s k I n + m ⊗ Q Nk ∈ C ( n + m )2 N . It follows that x ⊕ y ∈ C ( p , . . . , p N ) n + m .Next let x ∈ C ( p , . . . , p N ) n , and suppose that α ∈ M n,k . Let ǫ , . . . , ǫ N >

0. Then there exists t , . . . t N > x + N X i =1 ǫ i k α k I n ⊗ P Ni + N X i =1 t i I n ⊗ Q Ni ∈ C n N . Conjugating this expression by α ⊗ I N , we obtain( α ⊗ I N ) ∗ x ( α ⊗ I N ) + N X i =1 ǫ i k α k α ∗ α ⊗ P Ni + N X i =1 t i α ∗ α ⊗ Q Ni ∈ C k N . Since α ∗ α ≤ k α k I k we have( α ⊗ I N ) ∗ x ( α ⊗ I N ) + N X i =1 ǫ k I k ⊗ P Ni + N X i =1 k α k t i I k ⊗ Q Ni ∈ C k N . We conclude that ( α ⊗ I N ) ∗ x ( α ⊗ I N ) ∈ C ( p , . . . , p N ) k . (cid:3) Lemma 4.6.

Fix N ∈ N and suppose that ( V , C , e ) is an operator system and that p , p , . . . , p N ∈ V are positive contractions. Then I N ⊗ e is an Archimedean matrix-order unit for the non-propermatrix-ordered vector space ( M N ( V ) , C ( p , . . . , p N )) .Proof. To see that I N ⊗ e is a matrix-order unit, it suﬃces to show that I N ⊗ e is an order unitfor M N ( V ) with respect to the cone C ( p , . . . , p N ) (see, e.g., [1, Proposition 2.4]). To this end,suppose that x ∈ M N ( V ) and that x = x ∗ . Then because e is a matrix-order unit for V , thereexists t > x + tI N ⊗ e ∈ C N ⊆ C ( p , . . . , p N ) . Thus I N ⊗ e is an order unit. e now verify the Archimedean property. Let n ∈ N with x ∈ M n N ( V ) and suppose that x + ǫI n N ⊗ e ∈ C ( p , . . . , p N ) n for every ǫ >

0. Then for every ǫ , ǫ , . . . , ǫ N > t , t , . . . , t N > x + ǫ N I n N ⊗ e + N − X k =1 ǫ k I n ⊗ P Nk + ǫ N I n ⊗ P NN + N X k =1 t k I n ⊗ Q Nk ∈ C n N . Observe that I n N ⊗ e = I n ⊗ I N − ⊗ ( e ⊕ e ) = I n ⊗ P NN + I n ⊗ Q NN . Thus x + N X k =1 ǫ k I n ⊗ P Nk + N − X k =1 t k I n ⊗ Q Nk + (cid:16) t N + ǫ N (cid:17) I n ⊗ Q NN ∈ C n N . It follows that x ∈ C ( p , . . . , p N ) n . (cid:3) Proposition 4.7.

Let N ∈ N . Suppose that ( V , C , e ) is an operator system and p , p , . . . , p N ∈ V are positive contractions. Let J = C ( p , . . . , p N ) ∩ −C ( p , . . . , p N ) . Then ( M N ( V ) / J , {C ( p , . . . , p N ) n + M n ( J ) } ∞ n =1 , I N ⊗ e + J ) is an operator system.Proof. The result follows from Lemma 4.5 and Lemma 4.6. (cid:3)

Lemma 4.8.

Let N ∈ N and suppose that ( V , C , e ) is an operator system and p , p , . . . , p N ∈ V arepositive contractions. Let J = C ( p , . . . , p N ) ∩−C ( p , . . . , p N ) . Then the mapping x x ⊗ J N + J from V to M N ( V ) / J is unital. Moreover e ⊗ J N + J = I ⊗ e ⊗ J N − + J = I ⊗ e ⊗ J N − + J = · · · = I N ⊗ e + J . Proof.

The proof is similar to [1, Lemma 5.9]. We will show(3) I i − ⊗ e ⊗ J N − i +1 + J = P Ni + J , for i = 1 , . . . , N, and(4) I i ⊗ e ⊗ J N − i + J = P Ni + J , for i = 1 , . . . , N. We will ﬁrst show Equation (3). Fix i ∈ { , , . . . , N } . Since e = p i + p ⊥ i , we see that I i − ⊗ e ⊗ J N − i +1 + J = ( I i − ⊗ p i ⊗ J N − i +1 + J ) + ( I i − ⊗ p ⊥ i ⊗ J N − i +1 + J ) . We will show(5) I i − ⊗ p i ⊗ J N − i +1 + J = I i − ⊗ ( p i ⊕ ⊗ J N − i + J and(6) I i − ⊗ p ⊥ i ⊗ J N − i +1 + J = I i − ⊗ (0 ⊕ p ⊥ i ) ⊗ J N − i + J . Equation (3) can then be obtained by summing Equation (5) and Equation (6).In order to prove Equation (5) we rewrite the left hand side as I i − ⊗ (cid:18) p i p i p i p i (cid:19) ⊗ J N − i + J and the statement will follow by showing ± I i − ⊗ (cid:18) p i p i p i (cid:19) ⊗ J N − i ∈ C ( p , . . . , p N ) . o prove this, let ǫ , . . . , ǫ N >

0. Set t k = ǫ k for k = i . Let t i = 1 + ǫ i . Since (cid:18) p i p i p i (cid:19) + ǫ i (cid:18) p i p ⊥ i (cid:19) + t i (cid:18) p ⊥ i p i (cid:19) = (cid:18) ǫ i p i p i p i (1 + t i ) p i (cid:19) + (cid:18) t i p ⊥ i ǫ i p ⊥ i (cid:19) = (cid:18) ǫ i

11 2 + ǫ i (cid:19) ⊗ p i + (cid:18) ǫ i ǫ i (cid:19) ⊗ p ⊥ i ∈ C N and − (cid:18) p i p i p i (cid:19) + ǫ i (cid:18) p i p ⊥ i (cid:19) + t i (cid:18) p ⊥ i p i (cid:19) = (cid:18) ǫ i p i − p i − p i ( t i − p i (cid:19) + (cid:18) t i p ⊥ i ǫ i p ⊥ i (cid:19) = (cid:18) ǫ i − − ǫ i (cid:19) ⊗ p i + (cid:18) ǫ i ǫ i (cid:19) ⊗ p ⊥ i ∈ C N we conclude that ± I i − ⊗ (cid:18) p i p i p i (cid:19) ⊗ J N − i + ǫ i P Ni + t i Q Ni ∈ C N and hence ± I i − ⊗ (cid:18) p i p i p i (cid:19) ⊗ J N − i + N X k =1 ǫ k P Nk + N X k =1 t k Q Nk ∈ C N . Therefore ± I i − ⊗ (cid:18) p i p i p i (cid:19) ⊗ J N − i ∈ C ( p , . . . , p N ) , which proves Equation (5). A similar argument proves ± I i − ⊗ (cid:18) p ⊥ i p ⊥ i p ⊥ i (cid:19) ⊗ J N − i ∈ C ( p , . . . , p N ) , which proves Equation (6).Finally we show Equation (4). Let ǫ , . . . , ǫ N >

0. For k = i , set t k = ǫ k , and set t i = 1. Then ± Q Ni + X k ǫ k P Nk + X k t k Q Nk ∈ C N . Thus ± Q Ni ∈ C ( p , . . . , p N ) . It follows that P Ni + J = ( P Ni + Q Ni ) + J = I i ⊗ e ⊗ J N − i + J , which proves Equation (4). (cid:3) We now characterize when one has multiple projections in the operator system case. We pointout that when N = 1 this is done using the methods of [1]. Theorem 4.9.

Suppose that ( V , C , e ) is an operator system and p , p , . . . , p N ∈ V are positivecontractions. The following are equivalent: (1) Each p , . . . , p N is an abstract projection in V . (2) For each ≤ i ≤ N the map π p i : V → M ( V ) / J i , where J i := span C ( p i ) ∩ −C ( p i ) , is acomplete order embedding. (3) The map x x ⊗ J N + J , where J := span C ( p , . . . , p N ) ∩ −C ( p , . . . , p N ) , is a completeorder embedding from V to ( M N ( V ) / J , C ( p , . . . , p N ) + J , e ⊗ I N + J ) . roof. The equivalence of (1) and (2) follows from Theorem 3.3. Proposition 4.4 shows (1) implies(3). It remains to prove (3) implies (2).Suppose x x ⊗ J N + J is a complete order embedding. Fix i ∈ { , , . . . , N } . Since π p i isunital completely positive we need only show that π − p i : π p i ( V ) → V is completely positive. Let x ∈ M n ( V ) such that ( π p i ) n ( x ) ∈ C ( p i ) n + M n ( J i ) . Then for every ǫ i > t i > x ⊗ J + ǫI n ⊗ ( p i ⊕ p ⊥ i ) + t i I n ⊗ ( p ⊥ i ⊕ p i ) ∈ C n . By tensoring on the left by the positive matrix J N − i and tensoring on the right by the positivematrix J i − we obtain J N − i ⊗ ( x ⊗ J ) ⊗ J i − + ǫ i J N − i ⊗ I n ⊗ ( p i ⊕ p ⊥ i ) ⊗ J i − + t i J N − i ⊗ I n ⊗ ( p ⊥ i ⊕ p i ) ⊗ J i − ∈ C n N . Applying the canonical shuﬄe ϕ : M N − i ⊗ M n ⊗ M ⊗ M i − → M n ⊗ M N − i ⊗ M ⊗ M i − we have x ⊗ J N + ǫ i I n ⊗ b P Ni + t i b Q Ni ∈ C n N , where b P Ni and b Q Ni are from Deﬁnition 4.1. Let ǫ k > k = i and set t k = ǫ k for each k = i .Then x ⊗ J N + X k ǫ k I n ⊗ ˆ P Nk + X k t k I n ⊗ ˆ Q Nk ∈ C n N . Hence x ⊗ J N ∈ b C ( p , . . . , p N ) n , which by Lemma 4.2 implies x ⊗ J N ∈ C ( p , . . . , p N ) n . Since x x ⊗ J N + J is a complete order embedding, we conclude that x ∈ C n . (cid:3) Our next goal is to generalize Theorem 3.7 to the multiple projection case; i.e., to characterizeAOU spaces with multiple projections. In the single projection case, this was achieved usingLemma 3.6, which essentially says we can enlarge an arbitrary operator system structure ( V , C , e )by restricting the operator system structure( M ( V ) / J , {C ( p ) n + M n ( J ) } ∞ n =1 , e + J )to the subsystem V ⊗ J + J identiﬁed with V via x x ⊗ J + J . Deﬁnition . Let ( V , C , e ) be an operator system and let p , p , . . . , p N ∈ V be positive contrac-tions. Deﬁne P Ni and Q Ni as in Deﬁnition 4.1. For each L ∈ N we deﬁne P N,Li,k := I N ( k − ⊗ P Ni ⊗ J N ( L − k ) , ≤ i ≤ N and 1 ≤ k ≤ L. We deﬁne Q N,Li,k analogously. For each L ∈ N we deﬁne the non-proper matrix ordering C ( p , . . . , p N ) L := C ( p , . . . , p N , p , . . . , p N , . . . , p , . . . , p N | {z } L -times ) . More speciﬁcally, x ∈ C ( p , . . . , p N ) Ln if and only if x ∈ M n NL ( V ), x = x ∗ , and for each N × L matrix ( ǫ i,j ) of strictly positive real numbers there exists a corresponding N × L matrix ( t ij ) ofstrictly positive real numbers such that(7) x + X i,j ǫ ij I n ⊗ P N,Lij + X ij t ij I n ⊗ Q N,Lij ∈ C n NL , with the property that if one replaces ǫ ij with ǫ ′ ij < ǫ ij then there exists t ′ ij > t ij such thatEquation (7) still holds.Since the cones C ( p , . . . , p N ) Ln are simply the cones one obtains from Deﬁnition 4.1 when theprojections p , . . . , p N are repeated sequentially L times, the following proposition is an immediateconsequence of Proposition 4.7 and Theorem 4.9. roposition 4.11. Let ( V , C , e ) be an operator system, let p , . . . , p N ∈ V be positive contractions,and let L ∈ N . Then the following statements hold. (1) Setting J L = span C ( p , . . . , p N ) L ∩ −C ( p , . . . , p N ) L , the triple ( M NL ( V ) / J L , C ( p , . . . , p N ) L + J L , e + J L ) is an operator system, and the mapping x x ⊗ J NL + J L is unital for every L . (2) The operators p , . . . p N are each abstract projections in ( V , C , e ) if and only if the mapping x x ⊗ J NL + J L from V to M NL ( V ) / J L is a complete order embedding for every L .Deﬁnition . Let V be a ∗ -vector space and suppose that e is a nonzero hermitian element of V .Suppose that for every positive integer L there exists a matrix ordering C L = {C Ln } n ∈ N such that( V , C L , e ) is an operator system. We deﬁne the matrix orderings {C L } ∞ L =1 to be nested increasing if for each n ∈ N we have C Ln ⊆ C L +1 n for all L ∈ N .For a nested increasing sequence {C L } ∞ L =1 deﬁne C ∞ n to be the Archimedean closure of S ∞ L =1 C Ln ;that is, x ∈ C ∞ n if and only if for every ǫ > L ∈ N such that x + ǫI n ⊗ e ∈ C Ln . Wecall C ∞ := {C ∞ n } ∞ n =1 the inductive limit of the sequence {C L } ∞ L =1 . Lemma 4.13.

Let V be a ∗ -vector space and suppose that e is a nonzero hermitian element of V .Suppose that for every positive integer L there exists a matrix ordering C L such that ( V , C L , e ) is anoperator system and {C L } ∞ L =1 is a nested increasing sequence. Then C ∞ is a (possibly non-proper)matrix ordering for V and e is an Archimedean matrix-order unit for ( V , C ∞ ) .Proof. First set e C n := S ∞ L =1 C Ln for each n ∈ N . Suppose x ∈ e C n and y ∈ e C m . Then there exist L, L ′ > x ∈ C Ln and y ∈ C L ′ m . Therefore x ∈ C max( L,L ′ ) n and y ∈ C max( L,L ′ ) m . This implies x ⊕ y ∈ C max( L,L ′ ) n + m . Therefore x ⊕ y ∈ e C n + m . It is immediate that { e C n } ∞ n =1 is compatible sinceeach matrix ordering C L is compatible. Let x ∈ M n ( V ) h . Then for each L > t > x + tI n ⊗ e ∈ C Ln ⊆ e C n . Thus e is a matrix order unit for { e C n } ∞ n =1 . Since C ∞ n is theArchimedean closure of e C n , it follows that C ∞ is a (possibly non-proper) matrix ordering for V and e is an Archimedean matrix order unit for ( V , C ∞ ). (cid:3) Proposition 4.14.

Let ( V , C , e ) be an operator system let p , . . . , p N ∈ V be positive contractions.For L ∈ N let π L : V → M NL ( V ) denotes the mapping x x ⊗ J NL . Then { π − L ( C ( p , . . . , p N ) L ) } ∞ L =1 is a nested increasing sequence of matrix orderings on V .Proof. Fix n, L ∈ N and let x ∈ M n ( V ) . Suppose that x ⊗ J NL ∈ C ( p , . . . , p N ) Ln . Hence, for every N × L matrix ( ǫ i,k ) of strictly positive real numbers there exists a corresponding N × L matrix( t i,k ) of strictly positive real numbers such that x ⊗ J NL + N X i =1 L X k =1 ǫ i,k I n ⊗ P N,Li,k + N X i =1 L X k =1 t i,k I n ⊗ Q N,Li,k ∈ C n NL . Tensoring this expression on the right by J N we obtain x ⊗ J N ( L +1) + N X i =1 L X k =1 ǫ i,k I n ⊗ P N,Li,k ⊗ J N + N X i =1 L X k =1 t i,k I n ⊗ Q N,Li,k ⊗ J N ∈ C n N ( L +1) . Let ǫ ,L +1 , . . . , ǫ N,L +1 > t i,L +1 := ǫ i,L +1 . Then x ⊗ J N ( L +1) + N X i =1 L +1 X k =1 ǫ i,k I n ⊗ P N,L +1 i,k + N X i =1 L +1 X k =1 t i,k I n ⊗ Q N,L +1 i,k ∈ C n N ( L +1) . e conclude that x ⊗ J N ( L +1) ∈ C ( p , . . . , p N ) L +1 n , and hence( π L ) − n ( C ( p , . . . , p N ) Ln ) ⊆ ( π L ) − n ( C ( p , . . . , p N ) L +1 n ) . (cid:3) Deﬁnition . Let ( V , C , e ) be an operator system and let p , . . . , p N ∈ V be positive contractions.We deﬁne the matrix ordering C ( p , . . . , p N ) ∞ on V to be the inductive limit of the nested increasingsequence { π − L ( C ( p , . . . , p N ) L ) } ∞ L =1 on V where π L : V → M NL ( V ) is given by x x ⊗ J NL . Lemma 4.16.

Let C be a matrix ordering on a ∗ -vector space V , and suppose that C is proper.Then C n is proper for every n ∈ N .Proof. Suppose that x ∈ C n ∩ −C n . Then for each k = 1 , , . . . , n we have x kk = e ∗ k xe k ∈ C ∩ −C ,where e k ∈ C n is the k th standard unit vector. Hence x kk = 0 since C is proper. Let k, l ∈{ , , . . . , n } with k = l . Then ( e k + e l ) ∗ x ( e k + e l ) = x lk + x kl = 2 Re( x lk ) ∈ C ∩ −C . HenceRe( x lk ) = 0. Also ( e k − ie l ) ∗ x ( e k − ie l ) = i ( x lk − x kl ) = 2 i Im( x lk ) ∈ C ∩ −C . Hence Im( x lk ) = 0.Thus x lk = 0, and it follows that x = 0 and C n is proper. (cid:3) Proposition 4.17.

Let ( V , C , e ) be an operator system and let p , . . . , p N ∈ V be positive contrac-tions. If the cone C ( p , . . . , p N ) ∞ is proper then ( V , C ( p , . . . , p N ) ∞ , e ) is an operator system, andfor each k = 1 , . . . , N we have that p k is an abstract projection in ( V , C ( p , . . . , p N ) ∞ , e ) .Proof. Since V is an operator system and p , p , . . . , p N ∈ V are contractions, it follows from theﬁrst statement of Proposition 4.11 that π − L ( C ( p , . . . , p N ) L ) is a matrix ordering on V for each L ∈ N and that e is an Archimedean matrix order unit for each matrix ordered vector space( V , π − L ( C ( p , . . . , p N ) L )). Then by Lemma 4.13 and Deﬁnition 4.15 we conclude that the inductivelimit C ( p , . . . , p N ) ∞ is a matrix ordering on V and e is an Archimedean matrix order unit forthe matrix ordered ∗ -vector space ( V , C ( p , . . . , p N ) ∞ ). If C ( p , . . . , p N ) ∞ is proper, by Lemma 4.16 C ( p , . . . , p N ) ∞ is a proper matrix ordering. Therefore ( V , C ( p , . . . , p N ) ∞ , e ) is an operator system.Given x ∈ M n ( V ) suppose that for every ǫ , . . . , ǫ N > t , . . . , t N > x ⊗ J N + X i ǫ i I n ⊗ P Ni + X i t i I n ⊗ Q Ni ∈ C ( p , . . . , p N ) ∞ n N . Then for every ǫ > L ∈ N such that " x ⊗ J N + X i ǫ i I n ⊗ P Ni + X i t i I n ⊗ Q Ni + ǫI n N ⊗ e ⊗ J NL ∈ C ( p , . . . , p N ) Ln N , and in particular, " x ⊗ J N + X i ǫ i I n ⊗ P Ni + X i t i I n ⊗ Q Ni + ǫI n N ⊗ e ⊗ J NL + M n N ( J L ) ∈ C ( p , . . . , p N ) Ln N + M n NL ( J L ) . By Lemma 4.8 we see that I N ⊗ e ⊗ J NL + M N ( J L ) = e ⊗ J N ( L +1) + M N ( J L ). Hence I n N ⊗ e ⊗ J NL + M n N ( J L ) = I n ⊗ e ⊗ J N ( L +1) + M n N ( J L ) . This implies " x ⊗ J N + X i ǫ i I n ⊗ P Ni + X i t i I n ⊗ Q Ni + ǫI n ⊗ e ⊗ J N ⊗ J NL ∈ C ( p , . . . , p N ) Ln N . herefore for every N × L matrix ( δ i,k ) of strictly positive real numbers there exists an N × L matrix ( r i,k ) of strictly positive real numbers such that " x ⊗ J N + X i ǫ i I n ⊗ P Ni + X i t i I n ⊗ Q Ni + ǫI n ⊗ e ⊗ J N ⊗ J NL + X i,k δ i,k I n N ⊗ P N,Li,k + X i,k r ik I n N ⊗ Q N,Li,k ∈ C n N ( L +1) . Since I n N ⊗ P N,Li,k = I n ⊗ P N,L +1 i,k +1 and I n ⊗ P Ni ⊗ J NL = I n ⊗ P N,L +1 i, , by setting ǫ i = δ i,L +1 and t i = r i,L +1 for 1 ≤ i ≤ N , we conclude that( x + ǫI n ⊗ e ) ⊗ J N ( L +1) + N X i =1 L +1 X k =1 δ i,k I n ⊗ P N,L +1 i,k + N X i =1 L +1 X k =1 r i,k I n ⊗ Q N,L +1 i,k ∈ C n N ( L +1) , and hence ( x + ǫI n ⊗ e ) ⊗ J N ( L +1) ∈ C ( p , . . . , p N ) L +1 n . Since ǫ > x ∈ C ( p , . . . , p N ) ∞ n by the Archimedean property.Therefore x x ⊗ J N + J is a complete order embedding of ( V , C ( p , . . . , p N ) ∞ , e ) into thequotient operator system( M N ( V ) / J , C ( p , . . . , p N ) ∞ ( p , . . . , p N ) + J , e ⊗ I N + J )where J = span C ( p , . . . , p N ) ∞ ( p , . . . , p N ) ∩ −C ( p , . . . , p N ) ∞ ( p , . . . , p N ) . The statement then follows from Theorem 4.9. (cid:3)

We can now characterize multiple projections in an AOU space.

Theorem 4.18.

Let ( V , C, e ) be an AOU space and let p , p , . . . , p N be positive contractions in V .Then the following statements are equivalent: (1) There exists a Hilbert space H and a unital order embedding π : V → B ( H ) such that π ( p i ) is a projection on H for each i = 1 , . . . , N . (2) There exists a proper matrix ordering C with C = C such that ( V , C , e ) is an operator systemwith abstract projections p , p , . . . , p N . (3) C = C max ( p , . . . , p N ) ∞ .Proof. The equivalence of (1) and (2) follows from Theorem 4.9. We shall establish the equivalenceof (2) and (3).First suppose (2) holds. Note that C = C max where C max denotes the maximal matrix or-dering on the AOU space ( V , C, e ) . Since C max ⊆ C max ( p , . . . , p N ) L for each L ∈ N , we have C max ⊆ C max ( p , . . . , p N ) ∞ , and hence C ⊆ C max ( p , . . . , p N ) ∞ . For the reverse inclusion, if x ∈ C max ( p , . . . , p N ) ∞ , then for every ǫ > L ∈ N such that ( x + ǫe ) ⊗ J NL ∈ C max ( p , . . . , p N ) L . Since C max ( p , . . . , p N ) L ⊆ C ( p , . . . , p N ) L , for every ǫ > L ∈ N such that ( x + ǫe ) ⊗ J NL ∈ C ( p , . . . , p N ) L . Because p , . . . , p N are abstract projections relativeto C , it follows that x + ǫe ∈ C . Because this holds for all ǫ > C is Archimedean closed, itfollows that x ∈ C = C. Thus (3) holds.Next assume (3) holds. Since C = C max ( p , . . . , p N ) ∞ and C is a proper cone, Proposition 4.17implies ( V , C max ( p , . . . , p N ) ∞ , e ) is an operator system with abstract projections p , . . . , p N . Thusthe matrix ordering C max ( p , . . . , p N ) ∞ satisﬁes the conditions of (2). (cid:3) he following corollary is immediate from Proposition 4.17 and Theorem 4.18. Corollary 4.19.

Let ( V , C, e ) be an AOU space and let p , p , . . . , p N be positive contractions ∈ V .Suppose that the cone C max ( p , . . . , p N ) ∞ is proper. Then there exists a unital order embedding π : ( V , C max ( p , . . . , p N ) ∞ , e ) → B ( H ) such that π ( p k ) is a projection for each k = 1 , . . . , N .Proof. Since C max ( p , . . . , p N ) ∞ is proper then by Proposition 4.17 the triple( V , C max ( p , . . . , p N ) ∞ , e )is an operator system and each p i is an abstract projection in ( V , C max ( p , . . . , p N ) ∞ , e ). In par-ticular, ( V , C max ( p , . . . , p N ) ∞ , e ) is an AOU space and thus by Condition (2) and Condition (3)of Theorem 4.18 there exists a unital order embedding π : ( V , C max ( p , . . . , p N ) ∞ , e ) → B ( H )mapping each p i to a projection. (cid:3) Quantum commuting correlations as states on AOU spaces

Let n, k ∈ N . We call a tuple p = { p ( a, b | x, y ) : a, b ∈ [ k ] , x, y ∈ [ n ] } a correlation with n inputsand k outputs if for each a, b ∈ [ k ] and x, y ∈ [ n ], p ( a, b | x, y ) is a non-negative real number, and foreach x, y ∈ [ n ] we have k X a,b =1 p ( a, b | x, y ) = 1 . We let C ( n, k ) denote the set of all correlations with n inputs and k outputs. A correlation p iscalled nonsignalling if for each a, b ∈ [ k ] and x, y ∈ [ n ] the values p A ( a | x ) := X d p ( a, d | x, w ) and p B ( b | y ) := X c p ( c, b | z, y )are well-deﬁned, meaning that p A ( a | x ) is independent of the choice of w ∈ [ n ] and p B ( b | y ) isindependent of the choice of z ∈ [ n ]. We let C ns ( n, k ) denote the set of all nonsignalling correlationswith n inputs and k outputs.Much of the literature on correlation sets is focused on various subsets of the nonsignallingcorrelation sets. We mention three of these subsets here, namely the quantum commuting, quantum,and local correlations. A correlation p is a quantum commuting correlation with n inputs and k outputs if there exists a Hilbert space H , a pair of C*-algebras A , B ⊆ B ( H ) with z z = z z for all z ∈ A and z ∈ B , projection-valued measures { E x,a } ka =1 ⊆ A and { F y,b } kb =1 ⊆ B for each x, y ∈ [ n ], and a state φ : AB → C such that p ( a, b | x, y ) = φ ( E x,a F y,b ) for all a, b ∈ [ k ] and x, y ∈ [ n ].A quantum commuting correlation is called a quantum correlation if we require the Hilbert space H to be ﬁnite-dimensional. A quantum commuting correlation is called local if we require that theC*-algebras A and B are commutative. We let C qc ( n, k ) , C q ( n, k ), and C loc ( n, k ) denote the sets ofquantum commuting, quantum, and local correlations, respectively.It is well-known that for each input-output pair ( n, k ) each of the correlation sets mentionedabove are convex subsets of R n k and satisfy C loc ( n, k ) ⊆ C q ( n, k ) ⊆ C qc ( n, k ) ⊆ C ns ( n, k ) ⊆ C ( n, k ) . Moreover, each inclusion in the above sequence is proper for some choice of input n and output k .The set C q ( n, k ) is known to be non-closed for certain values n and k [17]. Let C qa ( n, k ) denote theclosure of C q ( n, k ) for each pair ( n, k ). Whether or not C qa ( n, k ) is equal to C qc ( n, k ) for every input n and output k is equivalent to Connes’ embedding problem from operator algebras (see [9], [4], and[14]). The recent preprint [8] implies that C qa ( n, k ) does not equal C qc ( n, k ) for some values of n and k , estimated to be approximately 10 each. It remains open whether or not C qa ( n, k ) diﬀers from C qc ( n, k ) for small values of n and k . Although it is well-known that C q (2 ,

2) = C qa (2 ,

2) = C qc (2 , C qa (3 ,

2) = C qc (3 ,

2) or C qa (3 ,

2) = C q (3 , et G ( n, k ) := ( Z /k Z ) ∗ · · · ∗ ( Z /k Z ) | {z } n times ;i.e., G ( n, k ) is the n -fold free product of Z /k Z with itself amalgamated over the group identity. Let A ( n, k ) := C ∗ ( G ( n, k )) denote the full group C*-algebra of G ( n, k ). Then A ( n, k ) is the universalC*-algebra generated by n noncommuting projection-valued measures { P ,i } ki =1 , . . . { P n,i } ki =1 (herewe take { P x,i } ki =1 to be the spectral projections for the generator of the x th factor Z /k Z of thefree product). Let S ( n, k ) ⊆ A ( n, k ) denote the operator system spanned by the projections { P x,i : x ∈ [ n ] , i ∈ [ k ] } for in A ( n, k ). Theorem 5.1 ([12]) . The following statements are equivalent: (1) p ∈ C ns ( n, k ) (resp. C qc ( n, k ) , C qa ( n, k ) ). (2) There exists a state φ on S ( n, k ) ⊗ max S ( n, k ) (resp. S ( n, k ) ⊗ c S ( n, k ) , S ( n, k ) ⊗ min S ( n, k ) )such that p ( a, b | x, y ) = φ ( P x,a ⊗ P y,b ) for each a, b ∈ [ k ] and x, y ∈ [ n ] . A corollary to Theorem 5.1 is that there exist AOU spaces ( V r , V + r , e ) for r = ns, qc, qa eachgenerated by positive operators Q ( a, b | x, y ) such that p ( a, b | x, y ) ∈ C r ( n, k ) if and only if thereexists a state φ : V r → C such that p ( a, b | x, y ) = φ ( Q ( a, b | x, y )). Hence, determining the geometryof the positive cones V + r is equivalent to determining the geometry of the set C r ( n, k ) by Kadison’sTheorem [10].We recall the following deﬁnition from [1]. Deﬁnition . Let n, k ∈ N . We call an operator system ( V , C , e ) a nonsignalling operator system if V = span { Q ( a, b | x, y ) } a,b ∈ [ k ]; x,y ∈ [ n ] where Q ( a, b | x, y ) ∈ C for each a, b ∈ [ k ] and x, y ∈ [ n ], k X a,b =1 Q ( a, b | x, y ) = e for each x, y ∈ [ n ], and the marginal vectors E ( a | x ) := k X c =1 Q ( a, c | x, z ) and F ( b | y ) := k X d =1 Q ( d, b | w, y )are well-deﬁned. We call V a quantum commuting operator system if it is a nonsignalling operatorsystem with the additional condition that each Q ( a, b | x, y ) is an abstract projection in ( V , C , e ).The terminology in Deﬁnition 5.2 is justiﬁed by the following theorem. Theorem 5.3 ([1, Thoerem 6.3]) . A correlation p ∈ C ( n, k ) is nonsignalling (resp. quantumcommuting) if and only if there exists a nonsignalling (resp. quantum commuting) operator system V with generators { Q ( a, b | x, y ); a, b ∈ [ k ] , x, y ∈ [ n ] } and a state φ on V such that p ( a, b | x, y ) = φ ( Q ( a, b | x, y )) for each a, b ∈ [ k ] and x, y ∈ [ n ] . Our goal is to construct a pair of AOU spaces universal with respect to nonsignalling andquantum commuting correlations, in the sense that p is a nonsignalling (resp. quantum commuting)correlation if and only if p ( a, b | x, y ) = φ ( Q ( a, b | x, y )) for some state φ , where { Q ( a, b | x, y ) : a, b ∈ [ k ] , x, y ∈ [ n ] } contains certain canonical elements of these AOU spaces. We begin by generalizingDeﬁnition 5.2. eﬁnition . We call a ﬁnite-dimensional vector space V a nonsignalling vector space on n inputsand k outputs if V is spanned by vectors { Q ( a, b | x, y ) : a, b ∈ [ k ] , x, y ∈ [ n ] } satisfying X a,b ∈ [ k ] Q ( a, b | x, y ) = e for some ﬁxed nonzero vector e , which we call the unit of V , and such that the vectors E ( a | x ) := k X c =1 Q ( a, c | x, z ) and F ( b | y ) := k X d =1 Q ( d, b | w, y )are well-deﬁned. When V is nonsignalling, we write n ( V ) and k ( V ) for the number of inputs andfor the number of outputs, respectively; i.e., V = span { Q ( a, b | x, y ) : a, b ∈ [ k ( V )] , x, y ∈ [ n ( V )] } . We use the notation n ( V ) and k ( V ) to speak of nonsignalling vector spaces without referenceto the input and and output parameters n and k , only referencing them when needed. However,it should be understood that any nonsignalling vector space V has unique associated input andoutput parameters n ( V ) and k ( V ) whether they are mentioned or not.The following example will play a role in several proofs hereafter. Example . Let n, k ∈ N . Let D k denote the set of diagonal k × k matrices. Let E a denote thediagonal matrix with 1 for its a th diagonal entry and zeroes elsewhere. Let V ⊆ D ⊗ nk denote thevector space spanned by the operators { Q ( a, b | x, y ) : a, b ∈ [ k ] , x, y ∈ [ n ] } deﬁned by Q ( a, b | x, y ) := I ⊗ x − k ⊗ E a ⊗ I ⊗ n − xk ⊗ I ⊗ y − k ⊗ E b ⊗ I ⊗ n − yk where I k denotes the k × k identity matrix and I ⊗ nk denotes the n -fold tensor product of I k with itself(understanding I ⊗ k = 1). Then V is a nonsignalling vector space with generators { Q ( a, b | x, y ) } .Moreover dim( V ) = ( n ( k −

1) + 1) . To see this, ﬁrst observe that for each a, b ∈ [ k ] and x, y ∈ [ n ] E ( a | x ) = I ⊗ x − k ⊗ E a ⊗ I ⊗ n − xk ⊗ I ⊗ nk and F ( b | y ) = I ⊗ nk ⊗ I ⊗ y − k ⊗ E b ⊗ I ⊗ n − yk . Moreover Q ( a, b | x, y ) = E ( a | x ) F ( b | y ), where the product is taken in the algebra D ⊗ nk . Hence V = V A V B where V A denotes span { E ( a | x ) : a ∈ [ k ] , x ∈ [ n ] } and V B denotes span { F ( b | y ) : b ∈ [ k ] , y ∈ [ n ] } . It follows that V A is spanned by the set S = { E ( a | x ) : a ∈ [ k − , x ∈ [ n ] } ∪ { I ⊗ nk } since E ( k | x ) = I ⊗ nk − ( P k − a =1 E ( a | x )). It follows that S is linearly independent and hence a basisfor V A . Thus dim( V A ) = n ( k −

1) + 1. A similar observation shows that dim( V B ) = n ( k −

1) + 1.Hence dim( V ) = dim( V A ) dim( V B ) = ( n ( k −

1) + 1) . For the remainder of this section, we establish the existence and describe the structure of universalnonsignalling and quantum commuting operator systems and AOU spaces. Indeed, the existence ofsuch spaces is already implied by Theorem 5.1 as remarked above, so the details of the constructionswill be more important than the observation that such spaces exist. Each of these structures willhave a common underlying vector space, which we describe in the next proposition.

Proposition 5.6.

For each n, k ∈ N , there exists a nonsignalling vector space V ns with n ( V ns ) = n , k ( V ns ) = k , and generators { Q ns ( a, b | x, y ) } satisfying the following universal property: if W isanother nonsignalling vector space with n ( W ns ) = n , k ( W ns ) = k , and generators { Q ( a, b | x, y ) } ,then there exists a linear map φ : V ns → W satisfying φ ( Q ns ( a, b | x, y )) = Q ( a, b | x, y ) . Moreover dim( V ns ) = ( n ( k −

1) + 1) . roof. Let W be any nonsignalling vector space with generators { Q ( a, b | x, y ) } satisfying n ( W ns ) = n and k ( W ns ) = k . Let e V := C n k and denote the canonical basis elements as { e Q ( a, b | x, y ) : a, b ∈ [ k ] , x, y ∈ [ n ] } . Let φ : e V → W be the linear map that takes e Q ( a, b | x, y ) Q ( a, b | x, y ). Deﬁne thevectors F ( x, y | x ′ , y ′ ) := X a,b e Q ( a, b | x, y ) − X a,b e Q ( a, b | x ′ , y ′ ) ,G ( a | x, z, w ) := X c e Q ( a, c | x, z ) − X c e Q ( a, c | x, w ) , and H ( b | y, z, w ) := X d e Q ( d, b | z, y ) − X d e Q ( d, b | w, y ) . Let J be the subspace spanned by the vectors { F ( x, y | x ′ , y ′ ) , G ( a | x, z, w ) , H ( b | y, z, w ) } . Then J ⊆ ker φ . Set V ns := e V /J and Q ns ( a, b | x, y ) := e Q ( a, b | x, y ) + J ∈ V ns . Then φ descends to alinear map e φ : V ns → W taking Q ns ( a, b | x, y ) Q ( a, b | x, y ). Moreover V ns is a nonsignalling vectorspace with unit e := P a,b Q ( a, b | x, y ). Since e φ is linear and since W is arbitrary, V ns satisﬁes theconditions of the theorem.We now calculate dim( V ns ). It suﬃces to show that the set B = { e ns , Q ns ( a, b | x, y ) , E ns ( a | x ) , F ns ( b | y ) : a, b ∈ [ k − , x, y ∈ [ n ] } is a basis for V ns (where E ns ( a | x ) and F ns ( b | y ) are the marginal vectors and e ns is the unit of V ns ),because this set contains n ( k − + 2 n ( k −

1) + 1 = ( n ( k −

1) + 1) elements. To see that B spans V ns , we only need to show that span B contains the vectors Q ns ( k, b | x, y ) , Q ns ( a, k | x, y ), and Q ns ( k, k | x, y ) for each a, b ∈ [ k −

1] and x, y ∈ [ n ]. To this end, ﬁx b ∈ [ k ] and x, y ∈ [ n ]. Then F ns ( b | y ) − k − X a =1 Q ns ( a, b | x, y ) = k X a =1 Q ns ( a, b | x, y ) − k − X a =1 Q ns ( a, b | x, y ) = Q ns ( k, b | x, y ) . Thus Q ns ( k, b | x, y ) ∈ span B . A similar observation shows that Q ns ( a, k | x, y ) ∈ span B for each a ∈ [ k ] and x, y ∈ [ n ]. Finally let x, y ∈ [ n ]. Then e ns − k − X a,b =1 Q ns ( a, b | x, y ) − k − X b =1 Q ns ( k, b | x, y ) − k − X a =1 Q ns ( a, k | x, y ) = Q ns ( k, k | x, y ) . Thus Q ns ( k, k | x, y ) ∈ span B . We conclude that dim( V ns ) ≤ | B | = ( n ( k −

1) + 1) . To show thatdim( V ns ) ≥ | B | , let V denote the nonsignalling vector space from Example 5.5. By the universalproperty of V ns , there exists a linear surjection from V ns to V . Hence dim( V ns ) ≥ dim( V ) =( n ( k −

1) + 1) . Therefore dim( V ns ) = ( n ( k −

1) + 1) = | B | , and thus B is a basis for V ns . (cid:3) The observation that dim( V ns ) = ( n ( V ns )( k ( V ns ) −

1) + 1) implies that V ns is isomorphic to thenonsignalling vector space from Example 5.5, since vector spaces are unique up to dimension.Having established the structure of the underlying vector space V ns , we seek to construct positivecones D ns and D qc making ( V ns , D ns , e ns ) and ( V ns , D qc , e ns ) universal AOU spaces with respect tononsignalling and quantum commuting correlations, respectively. We will establish the necessaryuniversal properties by constructing corresponding matrix orderings D ns and D qc with ( D ns ) = D ns and ( D qc ) = D qc so that ( V ns , D ns , e ns ) and ( V ns , D qc , e ns ) are universal nonsignalling and quantumcommuting operator systems, respectively. We begin with the nonsignalling case. Deﬁnition . Let n, k ∈ N , and let V ns denote the universal nonsignalling vector space with n ( V ns ) = n and k ( V ns ) = k . We deﬁne( V ns ) h := nX t ( a, b | x, y ) Q ns ( a, b | x, y ) : t ( a, b | x, y ) ∈ R for all a, b ∈ [ k ] , x, y ∈ [ n ] o nd D ns := nX t ( a, b | x, y ) Q ns ( a, b | x, y ) : t ( a, b | x, y ) ≥ a, b ∈ [ k ] , x, y ∈ [ n ] o so that D ns is the smallest cone generated by the vectors { Q ns ( a, b | x, y ) } . Given an arbitrary ele-ment z = P r ( a, b | x, y ) Q ns ( a, b | x, y ) with r ( a, b | x, y ) ∈ C , deﬁne z ∗ := P r ( a, b | x, y ) Q ns ( a, b | x, y ).Let D ns denote the maximal matrix ordering D maxns , so that ( D maxns ) = D ns .We ﬁrst establish that ( V ns , D ns , e ns ) is an AOU space. It will follow that ( V ns , D ns , e ns ) is anoperator system. Proposition 5.8.

Let n, k ∈ N , and let V ns denote the universal nonsignalling vector space with n ( V ns ) = n and k ( V ns ) = k . Then ( V ns , D ns , e ns ) is an AOU space.Proof. First, observe that the map ∗ : V ns → V ns from Deﬁnition 5.7 is a well-deﬁned involutionon V ns . Indeed, this involution coincides with the canonical involution on the nonsignalling vectorspace V from Example 5.5. It is clear that ( V ns ) h = { x ∈ V ns : x = x ∗ } and that D ns ⊆ ( V ns ) h . Tocomplete the proof, we only need to establish that e ns is an interior point of the cone D ns . Sincethe extreme points of D ns are precisely the rays { tQ ns ( a, b | x, y ) : t ∈ [0 , ∞ ) } and since e ns can beexpressed as a convex combination of nonzero points along these rays, for example as e ns = 1 n k X a,b ∈ [ k ] ,x,y ∈ [ n ] k Q ns ( a, b | x, y ) , we see that e ns is an interior point for D ns . Therefore ( V ns , D ns , e ns ) is an AOU space. (cid:3) The next proposition shows that the operator system ( V ns , D ns , e ns ) is universal with respect toall nonsignalling operator systems. Proposition 5.9.

Let n, k ∈ N , and let V ns denote the universal nonsignalling vector space with n ( V ns ) = n and k ( V ns ) = k . Suppose that ( W , C , e ) is a nonsignalling operator system with n ( W ) = n , k ( W ) = k , and generators { Q ( a, b | x, y ) } . Then the map π : ( V ns , D ns , e ns ) → ( W , C , e ) deﬁnedby π ( Q ns ( a, b | x, y )) = Q ( a, b | x, y ) is unital completely positive.Proof. Since W is a nonsignalling vector space, the map π is well-deﬁned, linear, and unital byProposition 5.6. By Deﬁnition 5.2, each operator Q ( a, b | x, y ) is positive, and thus X t ( a, b | x, y ) Q ( a, b | x, y ) ≥ t ( a, b | x, y ) ≥ a, b ∈ [ k ] and x, y ∈ [ n ]. It follows that π is a unital positive mapbetween the AOU spaces ( V ns , D ns , e ns ) and ( W , C , e ). Hence π is a unital positive map on theoperator system ( V ns , D ns , e ns ). Since D ns = D maxns , π is completely positive. (cid:3) The following theorem shows that the AOU space ( V ns , D ns , e ns ) is precisely the aﬃne dual ofthe convex set C ns ( n, k ) when n ( V ns ) = n and k ( V ns ) = k . Theorem 5.10.

Let n, k ∈ N , let V ns denote the universal nonsignalling vector space with n ( V ns ) = n and k ( V ns ) = k , and let { Q ns ( a, b | x, y ) } be the set of generators of V ns . If p = { p ( a, b | x, y ) } is acorrelation, then p ∈ C ns ( n, k ) if and only if there exists a state φ : ( V ns , D ns , e ns ) → C such that p ( a, b | x, y ) = φ ( Q ns ( a, b | x, y )) for each a, b ∈ [ k ] , x, y ∈ [ n ] .Proof. First suppose p ∈ C ns ( n, k ). Then by Theorem 5.3, there exists a nonsignalling operatorsystem ( W , C , e ) with generators { Q ( a, b | x, y ) } and a state φ : W → C such that p ( a, b | x, y ) = φ ( Q ( a, b | x, y )). By Proposition 5.9, the mapping π : V ns → W given by π ( Q ns ( a, b | x, y )) = Q ( a, b | x, y ) is a unital positive map. Hence φ ◦ π : V ns → C is a state satisfying p ( a, b | x, y ) = φ ◦ π ( Q ns ( a, b | x, y )). n the other hand, suppose φ : ( V ns , D ns , e ns ) → C is a state. Then φ is also a state onthe operator system ( V ns , D ns , e ns ). Since this operator system is nonsignalling, it follows fromTheorem 5.3 that p ( a, b | x, y ) := φ ( Q ns ( a, b | x, y )) deﬁnes a nonsignalling correlation. (cid:3) We will now pursue the analogous results for quantum commuting operator systems and quantumcommuting correlations.

Deﬁnition . Let n, k ∈ N and let V ns denote the universal nonsignalling vector space with n ( V ns ) = n , and k ( V ns ) = k . We deﬁne D qc := D ns ( p , . . . , p N ) ∞ and D qc := D ns ( p , . . . , p N ) ∞ = ( D qc ) , where N = n k and { p , p , . . . , p N } is some enumeration of the generators { Q ns ( a, b | x, y ) } .A priori, the cone D qc and the matrix ordering D qc depend on a choice of ordering { p , p , . . . , p N } for the generators { Q ( a, b | x, y ) } . However we will see in Corollary 5.13 that the D qc and D qc arethe same regardless of which ordering { p , p , . . . , p N } is chosen.In the following proof, we let P Ni ( x ) := I i − ⊗ ( x ⊕ x ⊥ ) ⊗ J N − i and P N,Li,j ( x ) := I N ( j − ⊗ P Ni ( x ) ⊗ J N ( L − j ) for any positive contraction x , where x ⊥ := e − x , i ∈ [ N ], and j ∈ [ L ]. Proposition 5.12.

Let n, k ∈ N , and let H be a Hilbert space. Suppose that { E x,a } ka =1 , { F y,b } kb =1 ⊆ B ( H ) are projection-valued measures and E x,a F y,b = F y,b E x,a for each a, b, ∈ [ k ] , and x, y ∈ [ n ] .Let W := span { E x,a F y,b } . Then the map π : V ns → W deﬁned by π ( Q ( a, b | x, y )) = E x,a F y,b isunital completely positive.Proof. First observe that W is a quantum commuting operator system with generators E x,a F y,b and unit I . By Proposition 5.9, the mapping π : ( V ns , D ns , e ns ) → W , deﬁned by π ( Q ( a, b | x, y )) = E x,a F y,b is unital completely positive with respect to the matrix ordering D ns . It remains to showthat π is unital completely positive with respect to the matrix ordering D qc .Let N := n k , and for each L > P N,Li,j := P N,Li,j ( p i ) and Q N,Li,j := Q N,Li,j ( p i ), where { p , p , . . . , p N } is the enumeration of the positive contractions { Q ( a, b | x, y ) } used in Deﬁnition 5.11.Likewise, let e P N,Li,j := P N,Li,j ( e p i ) and e Q N,Li,j := Q N,Li,j ( e p i ), where { e p , . . . , e p N } denotes the correspond-ing enumeration of the n k projections { E x,a F y,b } in W .Let x ∈ ( D qc ) n . By the deﬁnition of D qc , we see that for every ǫ > L > N × L matrix { ǫ i,j } of strictly positive real numbers there exists a corresponding N × L matrix { t i,j } of strictly positive real numbers such that( x + ǫI n ⊗ e ns ) ⊗ J NL + X i,j ǫ i,j I n ⊗ P N,Li,j + X i,j t i,j I n ⊗ Q N,Li,j ∈ ( D ns ) n NL . By applying π n NL to this expression, Proposition 5.9 implies that( π n ( x ) + ǫI n ⊗ I H ) ⊗ J NL + X i,j ǫ i,j I n ⊗ e P N,Li,j + X i,j t i,j I n ⊗ e Q N,Li,j ∈ B ( H n NL ) + . Since each E x,a F y,b is a projection, it follows from Proposition 4.4 that π n ( x ) + ǫI n ⊗ I H ≥

0. Itthen follows from the Archimedean property that π n ( x ) ≥

0. Hence π is unital completely positivewith respect to D qc . (cid:3) We have not yet established that D qc is a proper cone. Using Proposition 5.12, we can nowobtain this result as a corollary. orollary 5.13. Let n, k ∈ N and let V ns denote the universal nonsignalling vector space with n ( V ns ) = n and k ( V ns ) = k . Then D qc is a proper cone. Hence ( V ns , D qc , e ns ) is an AOU space,and ( V ns , D qc , e ns ) is a quantum commuting operator system. Moreover, the matrix ordering D qc is independent of the choice of enumeration { p , p , . . . , p N } of the generators { Q ns ( a, b | x, y ) } usedin Deﬁnition 5.11.Proof. We ﬁrst establish that ( V ns , D qc , e ns ) is a quantum commuting operator system. By Corol-lary 4.19 we see that ( V ns , D qc , e ns ) is a quantum commuting operator system if and only if the cone D qc is proper. To show that D qc is proper, it suﬃces to show that there exists a quantum commut-ing operator system W ⊆ B ( H ) with generators { Q ( a, b | x, y ) } such that dim( W ) = ( n ( k −

1) + 1) .Indeed, by Proposition 5.12 the map π : V ns → W deﬁned by π ( Q ns ( a, b | x, y )) = Q ( a, b | x, y ) ispositive, and hence π must map D qc ∩ − D qc to { } . However dim( V ns ) = dim( W ), so π is injec-tive, and hence D qc ∩ − D qc = { } . For an example of such an operator system, one can take thenonsignalling vector space from Example 5.5 equipped with the canonical unit and order structureinherited from its natural embedding into B ( C k n ). Since ( V ns , D qc , e ns ) is an operator system, itfollows that ( V ns , D qc , e ns ) is an AOU space.Finally, let { p , . . . , p N } and { q , . . . , q n } be two enumerations of the set of generators of V ns .Let D qc and D ′ qc denote the corresponding matrix orderings, as described in Deﬁnition 5.11. ByProposition 5.12, the identity map is unital completely positive whether regarded as a map from( V ns , D qc , e ns ) to ( V ns , D ′ qc , e ns ) or from ( V ns , D ′ qc , e ns ) to ( V ns , D qc , e ns ). It follows that D qc = D ′ qc . (cid:3) We are now prepared to C qc ( n, k ) is aﬃnely isomorphic to the state space of ( V ns , D qc , e ns ) when n ( V ns ) = n and k ( V ns ) = k . Theorem 5.14.

Let n, k ∈ N and let V ns denote the universal nonsignalling vector space with n ( V ns ) = n and k ( V ns ) = k . If p = { p ( a, b | x, y ) } is a correlation, then p ∈ C qc ( n, k ) if andonly if there exists a state φ : ( V ns , D qc , e ns ) → C such that p ( a, b | x, y ) = φ ( Q ( a, b | x, y )) for each a, b ∈ [ k ] , x, y ∈ [ n ] .Proof. First suppose that p ∈ C qc ( n, k ). Then by Theorem 5.3, there exists a quantum com-muting operator system ( W , C , e ) with generators { Q ( a, b | x, y ) } and a state φ : W → C suchthat p ( a, b | x, y ) = φ ( Q ( a, b | x, y )). By Proposition 5.12, the mapping π : V ns → W given by π ( Q ns ( a, b | x, y )) = Q ( a, b | x, y ) is a unital positive map with respect to the cone D qc . Hence φ ◦ π : V ns → C is a state on ( V ns , D qc , e ns ) satisfying p ( a, b | x, y ) = φ ◦ π ( Q ns ( a, b | x, y )).Conversely, suppose that φ : ( V ns , D qc , e ns ) → C is a state. Then φ is also a state on theoperator system ( V ns , D qc , e ns ). Since this operator system is quantum commuting, it follows fromTheorem 5.3 that p ( a, b | x, y ) := φ ( Q ns ( a, b | x, y )) deﬁnes a quantum commuting correlation. (cid:3) Remark . We conclude with some remarks on generalizations of the above ideas that imme-diately follow from our work. First, our notions of nonsignalling vector spaces and nonsignallingoperator systems are readily generalized to the multipartite situation, where one considers correla-tions of the form p ( a , a , . . . , a n | x , x , . . . , x n ). These correlations describe the scenario where n spacially distinct parties each perform a measurement on their respective quantum system. One maythen consider quantum commuting correlations arising from n mutually commuting C*-algebras ina common Hilbert space. To describe these correlations using our ideas, we redeﬁne nonsignallingoperator systems to be generated by operators { Q ( a , . . . , a n | x , . . . , x n ) } satisfying X a ,...,a n Q ( a , . . . , a n | x , . . . , x n ) = e nd having well-deﬁned marginal operators E i ( a i | x i ) = X a j ,j = i Q ( a , . . . , a n | x , . . . , x n ) . Requiring each generator to be an abstract projection yields a quantum commuting operator system.The constructions of universal nonsignalling and quantum commuting operator systems proceeds inthe same manner as the bipartite case. Our work also readily generalizes to the setting of matricialcorrelation sets, as described in [14]. Indeed it is straightforward to see that the matrix aﬃnedual of the matricial nonsignalling and quantum commuting correlations are precisely the operatorsystems ( V ns , D ns , e ns ) and ( V ns , D qc , e ns ), respectively, using Webster-Winkler duality [19]. References [1] Roy Araiza and Travis Russell. An abstract characterization for projections in operator sys-tems. arXiv:2006.03094 , 2020.[2] Man-Duen Choi and Edward G Eﬀros. Injectivity and operator spaces.

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Department of Mathematics, Purdue University, West Lafayette, IN 47907

Email address : [email protected] Army Cyber Institute, United States Military Academy, West Point, NY 10996

Email address : [email protected] Department of Mathematics, University of Colorado, Colorado Springs, CO 80918, USA

Email address : [email protected]@uccs.edu