An abstract characterization for projections in operator systems
aa r X i v : . [ m a t h . OA ] J un An abstract characterization for projections in operator systems
Roy Araiza and Travis Russell Department of Mathematics, Purdue University, West Lafayette, IN Army Cyber Institute, United States Military Academy, West Point, NY
Abstract
We show that the set of projections in an operator system can be detected using only the abstract data of the operatorsystem. Speciο¬cally, we show that if π is a positive contraction in an operator system ξ which satisο¬es certain order-theoretic conditions, then there exists a complete order embedding of ξ into π΅ ( π» ) mapping π to a projection operator.Moreover, every abstract projection in an operator system ξ is an honest projection in the C*-envelope of ξ . Usingthis characterization, we provide an abstract characterization for operator systems spanned by two commuting familiesof projection-valued measures and discuss applications in quantum information theory. Beginning with the work of Choi-Eο¬ros in [3], an abstract characterization for self-adjoint unital subspaces of thebounded operators on a Hilbert space was given. More recently the abstract theory of operator systems progressedfurther with the development of the theory of tensors. In particular it was shown in [10, 11] that if classes of operatorsystems satisο¬ed certain nuclearity properites then it must follow that πΆ β ( πΉ β ) had Lanceβs weak expectation property,i.e. particular nuclearity properties of operator systems were proven to be equivalent to Kirchbergβs conjecture. Kirch-berg showed in [12] that if the local lifting property for C*-algebras implied the weak expectation property then Connesβembedding conjecture, originally appearing in [4], must hold. In [9], [5], and [14], an equivalence between Kirchbergβsconjecture and what is known as Tsirelsonβs problem was established. Tsirelsonβs problem asks if for all pairs of natu-ral numbers ( π, π ) the equality πΆ ππ ( π, π ) = πΆ ππ ( π, π ) holds, where πΆ ππ ( π, π ) denotes the closure of the set of quantumcorrelations , and πΆ ππ ( π, π ) denotes the set of quantum commuting correlations. Introducing new ideas coming fromcomputer science, a recent preprint [8] demonstrates the existence of integers π and π such that πΆ ππ ( π, π ) β πΆ ππ ( π, π ) ,simultaneously refuting the long-standing conjectures of Tsirelson, Kirchberg, and Connes. Sharp estimates on theordered pairs ( π, π ) for which πΆ ππ ( π, π ) β πΆ ππ ( π, π ) are not known. Estimating these values could shed more light onthe failure of the conjectures of Kirchberg and Connes, which would be useful in the study of operator algebras. Thepurpose of this paper is to better understand the role played by projections in operator systems, since projections playan outsized role in Tsirelsonβs problem. Our principle motivation is the hope that new insights about the structure ofoperator systems may be useful in the study of Tsirelsonβs problem.In this paper, we provide an abstract characterization for the set of projections in an operator system. Given anabstract operator system ξ and a positive element π β ξ of unit norm (as induced by the order unit and positive coneon ξ ), we consider a collection of cones { πΆ ( π π )} π induced by π and prove that the quotient β -vector space ξ β π½ π isan operator system with order unit π + π½ π and matrix ordering { πΆ ( π π ) + π π ( π½ π )} π , where π½ π = span πΆ ( π ) β© β πΆ ( π ) (Theorem 4.9). When ξ is a concrete operator system in π΅ ( π» ) and π is a projection, we show that ξ β π½ π is completelyorder isomorphic to the compression operator system π ξ π β π΅ ( ππ» ) (Corollary 4.10). We call a positive element π β ξ an abstract projection if π has unit norm and the mapping π π βΆ ξ β π ( ξ )β π½ πβπ , π₯ β¦ ( π₯ π₯π₯ π₯ ) + π½ πβπ is a complete order isomorphism, where π = π β π and π is the unit of ξ (Deο¬nition 5.4). Our main result is that apositive element π β ξ is an abstract projection if and only if there exists a Hilbert space π» and a unital complete order1mbedding π βΆ ξ β π΅ ( π» ) such that π ( π ) is a projection (Theorem 5.7), establishing a one-to-one correspondencebetween abstract and concrete projections. Similar to a result of Blecher and Neal appearing in [1, Lemma 2.3], wethen prove in Theorem 5.8 that π β ξ is an abstract projection if and only if π is a projection in the C*-envelope πΆ β π ( ξ ) . These observations lead us quickly to a new characterization of the set of quantum commuting correlations πΆ ππ ( π, π ) in terms of abstract operator systems and abstract projections. Speciο¬cally, we show that { π ( π, π | π₯, π¦ )} β πΆ ππ ( π, π ) ifand only if π ( π, π | π₯, π¦ ) = π ( π ( π, π | π₯, π¦ )) where π is a state on an operator system ξ spanned by abstract projections { π ( π, π | π₯, π¦ )} satisfying certain operator non-signalling conditions, namely that β π,π π ( π, π | π₯, π¦ ) = π for all π₯, π¦ (where π is the unit of ξ ) and that the marginal operators πΈ ( π | π₯ ) βΆ= β π π ( π, π | π₯, π¦ ) and πΉ ( π | π¦ ) βΆ= β π π ( π, π | π₯, π¦ ) are well deο¬ned (Theorem 6.3).We conclude this introduction by pointing out some related results in the literature. The idea of abstractly char-acterizing certain types of operators in operator spaces was investigated extensively by Blecher and Neal [1, 2], whostudied the abstract (linear-metric) structure of an operator space that contains a unit. More precisely, given a pair ( ξ± , π’ ) where ξ± is an abstract operator space and π’ β ξ± , they characterized when there exists a completely isometricembedding of ξ± into π΅ ( π» ) such that π’ is mapped to a unitary. They also showed that given a unitary π’ β ξ± , then π’ is necessarily a unitary in the ternary envelope π ( ξ± ) of ξ± . Other operator system characterizations for correlation setscan be found in the literature, for example in Theorem 3.1 of [13] and Theorem 2.4 of [7]. These papers characterizecorrelations as having the form π ( π, π | π₯, π¦ ) = π ( πΈ π₯,π β πΈ π¦,π ) where { πΈ π₯,π } π are projection-valued measures spanninga certain canonical operator system π and π is a state on π β π‘ π , where β π‘ denotes the various operator system tensorproducts of [10] depending on which type of correlation one intends to construct. Our results diο¬er in that we do notappeal to the hierarchy of operator system tensor products or make use of any canonical operator system. We do not,however, have any result analogous to theirs for quantum approximate or local correlations.Our paper is organized as follows. In Section 2, we provide some elementary background in operator systems andoperator spaces, including some preliminary remarks that will be useful throughout the paper. In Section 3 we studythe compression of a concrete operator system ξ by a projection π β ξ , i.e., the operator system π ξ π β π΅ ( ππ» ) .The observations of this section motivate the results of Section 4, where we provide an abstract deο¬nition for thecompression of an abstract operator system by a positive contraction π which is not a priori a projection. In Section 5,we combine the results of Section 3 and Section 4 and some additional observations to prove the main results of thepaper. We conclude with Section 6 where we present our applications to the theory of quantum correlation sets. Though we assume some familiarity with general operator system and operator space theory we will review somedeο¬nitions and constructions that appear throughout the manuscript.
Deο¬nition 2.1.
Given a Hilbert space π» a concrete operator system is a self-adjoint unital subspace ξ of π΅ ( π» ) . An abstract operator system is deο¬ned to be the triple ( ξ , { πΆ π } π β β , π ) where ξ is a complex vector space with a conjugate-linear involution β (i.e. a β -vector space ), { πΆ π } π is a proper matrix ordering on ξ and π is an Archimedean matrix orderunit. By a matrix ordering on ξ , we mean a sequence { πΆ π } π which satisο¬es the following two properties:1. πΆ π β π π ( ξ ) β is a cone whose elements are invariant under the involution β ;2. Given any π, π β β and πΌ β π π,π then πΌ β πΆ π πΌ β πΆ π . A matrix ordering { πΆ π } π is called proper if in additon πΆ π β© β πΆ π = {0} for each π . A element π β ξ β is called an Archimedean matrix order unit for a matrix ordering { πΆ π } π if given any π₯ β π π ( ξ ) β there exists π > such that if π π βΆ= πΌ π β π, then ππ π β π₯ β π π ( ξ ) + and if ππ π + π₯ β π π ( ξ ) + for all π > then π₯ β π π ( ξ ) + . matrix ordering . Our use of the term matrix ordering diο¬ersfrom some authors (e.g. [10]) where the term matrix ordering is used synonymously with proper matrix ordering. Wenote that the family { πΆ π } π with πΆ π β π π ( ξ ) β is a matrix ordering if πΆ π β πΆ π β πΆ π + π and πΌ β πΆ π πΌ β πΆ π for all πΌ β π π,π , π, π β β . When no confusion will arise we will simply denote an operator system by ξ . The morphisms in use between matrixordered β -vector spaces will be completely positive maps, and the morphisms between operator systems will be theunital completely positive maps. Given a linear map π βΆ ξ β ξ between operator systems, then for each π β β thereis an induced linear map πΌ π β π βΆ= π π βΆ π π ( ξ ) β π π ( ξ ) deο¬ned by β π,π β€ π | π β©β¨ π | β π₯ ππ β¦ β π,π β€ π | π β©β¨ π | β π ( π₯ ππ ) . The map π π is called the nth ampliο¬cation of π. Here we have let { | π β© } π denote column vectors in β π with in the π th position. We say that π is completely positive if π π ( π π ( ξ ) + ) β π π ( ξ ) + for every π , and is a complete orderisomorphism if π is invertible with π and π β1 both completely positive. A map π βΆ ξ β ξ which is not necessarilysurjective is called a complete order embedding if it is completely positive, injective and π β1 is completely positiveon π ( ξ ) β ξ . We will identify two operator systems ξ and ξ if there exists a (unital) complete order isomorphism π between the two and we will denote this by ξ β ξ . A classical result due to Choi and Eο¬ros shows that there is aone-to-one correspondence between concrete and abstract operator systems.
Theorem 2.2 ([3]) . Given an abstract operator system ξ then there is a Hilbert space π» and a concrete operatorsystem ξ β π΅ ( π» ) such that ξ β ξ . Conversely, every concrete operator system is an abstract operator system. A well-known fact is that given a matrix-ordered β -vector space, i.e., a pair ( ξ , { πΆ π } π ) consisting of a β -vectorspace ξ and a matrix-ordering { πΆ π } π , then an element π β ξ + is an order unit for ξ if and only if it is a matrix orderunit. We provide a brief proof below for completeness. Lemma 2.3 ([16]) . Given an β -vector space ξ then π π ( ξ ) β = ( π π ) β β ξ β . Proposition 2.4.
Given a matrix ordered β -vector space ξ then π β ξ is an order unit if and only if it is a matrix orderunit.Proof. Of course we need only show the forward direction. Let π₯ β π π ( ξ ) β such that π₯ = β π β€ π π΄ π βπ₯ π β ( π π ) β β ξ β .For each π write π΄ π = π π β π π , where π π , π π β π + π . Choose π > such that ππ Β± π₯ π β ξ + for each π . We then see π ( β π β€ π π π + π π ) β π β π₯ = β π β€ π π π β ( ππ β π₯ π ) + β π β€ π π π β ( ππ + π₯ π ) β π π ( ξ ) + . Simply choose Μπ such that ΜππΌ π β₯ π ( β π β€ π π π + π π ) which proves the claim. Remark 2.5 (The Canonical Shuο¬e) . Throughout the manuscript we will implement the βcanonical shuο¬eβ (see [15,Chapter 8]). For example, given an operator system ξ β π΅ ( π» ) , we use this in Lemma 3.9 to identify π π ( ξ β ξ ) with π π ( ξ ) β π π ( ξ ) , where the direct sum is in the π β sense, i.e., we are realizing ξ β ξ as the diagonal matriceswith entries from ξ . In particular, consider π΄ β π π ( π π ( ξ )) . We then write π΄ = β ππ | π β©β¨ π | β π΄ ππ , π΄ ππ β π π ( ξ ) . (1)It follows π΄ = β ππ | π β©β¨ π | β π΄ ππ = β ππ | π β©β¨ π | β β ππ | π β©β¨ π | β π ππππ , π ππππ β ξ . (2)Here { | π β©β¨ π | } π,π β€ π denotes the matrix units of π π and { | π β©β¨ π | } π,π β€ π denotes the matrix units of π π . We see β ππ | π β©β¨ π | β π΄ ππ = β ππ | π β©β¨ π | β β ππ | π β©β¨ π | β π ππππ = β ππ | π β©β¨ π | β β ππ | π β©β¨ π | β π ππππ = β ππ | π β©β¨ π | β π΅ ππ = π΅, (3)3here π΅ ππ = β ππ | π β©β¨ π | β π ππππ , and π΅ β π π ( π π ( ξ )) . This canonical map is a β -isomorphism between the ambient πΆ β -algebras, i.e., betweeen π π ( π π ( π΅ ( π» ))) and π π ( π π ( π΅ ( π» ))) . One may also view this using the commutativityof the operator system minimal tensor product, i.e., π π ( π π ( ξ )) β π π β min π π β min ξ β π π β min π π β min ξ β π π ( π π ( ξ )) . Given an operator system ξ consider now a pair ( π , ξ ) where π βΆ ξ β ξ is a unital complete order embedding,and ξ is a C*-algebra. We will call the pair a C*-extension of ξ if ξ is generated by the image π ( ξ ) as a C*-algebra.In particular, there exists a minimal such extension satisfying a universal property. Given an operator system ξ andtwo C*-extensions, ( π , ξ ) , ( π , ξ ) , we say that the extensions are ξ -equivalent if there exists a β -isomorphism π βΆ ξ β ξ such that ππ = π . Theorem 2.6 (Arveson-Hamana) . Given an operator system ξ then there exists a C*-extension ( π , π΄ ) satisfying thefollowing universal property: given any other C*-extension ( π, π΅ ) of ξ then there exists a unique β -epimorphism π βΆ π΅ β π΄ such that π β¦ π = π . Deο¬nition 2.7.
Given an operator system ξ then the C*-envelope of ξ will be any C*-extension satisfying Theorem2.6. Such a C*-extension is unique up to ξ -equivalence and we denote it by πΆ β e ( ξ ) . Given a Hilbert space π» a concrete operator space is a closed subspace ξ± β π΅ ( π» ) . An abstract operator space will be the pair ( ξ± , { πΌ π } π ) where ξ± is a linear space and { πΌ π } π is a sequence of matrix norms on ξ± , that is πΌ π βΆ π π ( ξ± ) β [0 , β) for all π β β , satisfying Ruanβs axioms. This is to say that the following two properties are satisο¬ed:1. πΌ π + π ( π₯ β π¦ ) = max{ πΌ π ( π₯ ) , πΌ π ( π¦ )} for all π₯ β π π ( ξ± ) , π¦ β π π ( ξ± ) ;2. πΌ π ( ππ₯π ) β€ β π β πΌ π ( π₯ ) β π β for all π, π β π π . When no confusion will arise we will simply denote an operator space by ξ± . Given a linear map π βΆ ξ± β ξ² betweenoperator spaces, then we say π is completely bounded if β π β cb βΆ= sup π ββ π π ββ < β , and π will be called completelyisometric if π π is an isometry for all π β β . We identify two operator spaces if there exists a completely isometricbijection between the two. This is to say that ξ± is completely isometric to ξ² if there exists a linear map π βΆ ξ± β ξ² which is invertible and both π and π β1 are completely contractive. We will thus say that ξ± is completely isometric to ξ² . Due to a result of Ruan there is a one-to-one correspondence between concrete and abstract operator spaces.
Theorem 2.8 ([18]) . Given an abstract operator space ξ± then there exists a Hilbert space π» and a concrete operatorspace ξ² β π΅ ( π» ) such that ξ± and ξ² are completely isometric. Conversely, any concrete operator space is an abstractoperator space. It is natural to consider the structural properties of an operator space that contains a βunitβ.
Deο¬nition 2.9 ([1]) . A unital operator space is a pair ( ξ± , π’ ) where ξ± is an operator space and π’ β ξ± is such that thereexists a completely isometric embedding π βΆ ξ± β π΅ ( π» ) such that π ( π’ ) = πΌ π» . We will call π’ a unitary of ξ± if thereexists a completely isometric embedding which maps π’ to a unitary of some π΅ ( π» ) .Analogous to the discussion preceding Theorem 2.6 and the theorem itself, we may talk about ternary extensionsof unital operator spaces. Consider a ternary ring of operators (TRO) π β π΅ ( πΎ, π» ) . This is to say that π is a closedsubspace of π΅ ( πΎ, π» ) and ππ β π β π.
Then an element π’ β π is called a C*-unitary if for all π§ β π we have π§π’ β π’ = π§ and π’π’ β π§ = π§. Given two TROs π and π , a ternary morphism is a linear map π βΆ π β π such that π ( π₯π¦ β π§ ) = π ( π₯ ) π ( π¦ ) β π ( π§ ) for each π₯, π¦, π§ β π . Given an operator space ξ± , a ternary envelope for ξ± is a TRO π ( ξ± ) generated by ξ± satisfying the following universal property: if π is some other ternary envelope generated by ξ± , thenthere exists a ternary morphism π βΆ π β π ( ξ± ) extending the identity map on ξ± . The existence of the ternary envelopewas established in [6]. Theorem 2.10 ([1]) . Given an operator space ξ± . the following are equivalent for an element π’ β ξ± :1. π’ is a unitary in ξ± .2. There exists a TRO π containing ξ± completely isometrically such that π’ is a πΆ β -unitary in π . . π’ is a πΆ β -unitary in the ternary envelope π ( ξ± ) . Remark 2.11.
Throughout the rest of the manuscript there will be results where we will be considering ξ as alreadysitting in some π΅ ( π» ) . In particular, we will consider projections in a concrete operator system in Section 3 and projec-tions in an abstract operator system in Section 4. Thus we will interchange between the use of π or πΌ as the Archimedeanorder unit dependent on whether our operator system is abstract or concrete. The Archimedean order unit will alwaysbe clear from context. In this section we present the motivation for Sections 4 and 5. Starting with a concrete operator system ξ β π΅ ( π» ) we show that an element π β ξ + , which is also a projection on π΅ ( π» ) induces a natural collection of cones whichcorrespond to the hermitian elements of ξ whose compression by π is positive. These cones form a matrix orderingand in particular, will form a proper matrix ordering on a certain quotient β -vector space. It will follow that image of π in the quotient is an Archimedean matrix order unit for the space.Let ξ β π΅ ( π» ) be an operator system and let π β ξ be a projection (as an operator in π΅ ( π» ) ). Then letting π π βΆ= πΌ π β π we consider the collection of sets { πΆ ( π π )} π where for each ππΆ ( π π ) = { π₯ β π π ( ξ ) β βΆ π π π₯π π β π΅ ( π» π ) + } , where π» π denotes the π -fold Hilbertian direct sum. We will show that the sequence { πΆ ( π π )} π is a (not necessarilyproper) matrix ordering on ξ . Proposition 3.1.
Let ξ β π΅ ( π» ) be an operator system and suppose that π β ξ where π is a projection in π΅ ( π» ) . Thesequence of sets { πΆ ( π π )} π is a matrix ordering on ξ . Furthermore If π β€ π β€ πΌ then π is an Archimedean matrix orderunit for ( ξ , { πΆ ( π π )} π ) . Proof.
By deο¬nition πΆ ( π π ) β = πΆ ( π π ) for all π β β . The compression by the projection π is a linear map and thus ππΆ ( π π ) β πΆ ( π π ) for all π β β , π > and πΆ ( π π ) + πΆ ( π π ) β πΆ ( π π ) . Finally, let πΌ β π π,π , π₯ β πΆ ( π π ) . Then π π πΌ β π₯πΌπ π = π π ββββ[β ππ πΌ ππ π₯ ππ πΌ ππ ] ππ ββββ π π = [ π (β ππ πΌ ππ π₯ ππ πΌ ππ ) π ] ππ = [β ππ πΌ ππ ππ₯ ππ ππΌ ππ ] ππ = πΌ β ( π π π₯π π ) πΌ, and πΌ β ( π π π₯π π ) πΌ β π΅ ( π» π ) + . This proves the ο¬rst statement.We now show that if π β€ π β€ πΌ then π is an Archimedean matrix order unit for the pair ( ξ , { πΆ ( π π )} π ) . We ο¬rstmake some observations. It is immediate that given the projection π β ξ then π is an Archimedean matrix order unitfor operators of the form ππ§π, π§ β ξ . Simply notice that for any π β β and π₯ β π π ( ξ ) , then πΌ π β₯ π π and if for π > we have ππ π + π π π₯π π β π΅ ( π» π ) + then ππΌ π + π π π₯π π β₯ ππ π + π π π₯π π β π΅ ( π» π ) + . Since πΌ is an Archimedean matrix order unit for the operator system π΅ ( π» ) we have π π π₯π π β π΅ ( π» π ) + and thus π₯ β πΆ ( π π ) . By the assumption that π β€ π it follows that π β€ πππ . (In fact since π β€ πΌ we have πππ = π since it also follows π β πππ = π ( πΌ β π ) π β₯ ). Let π₯ = π₯ β β ξ and let π > such that ππ β ππ₯π β₯ . Then π ( ππ β π₯ ) π = ππππ β ππ₯π β₯ ππ β ππ₯π β₯ . (4)Thus ππ β π₯ β πΆ ( π ) implying π is an order unit. If for all π > , ππ + π₯ β πΆ ( π ) then ππππ + ππ₯π β₯ for all π > andthus ππ + ππ₯π β₯ ππππ + ππ₯π β₯ , β π > , ππ₯π β₯ ( and thus π₯ β πΆ ( π ) ) by our earlier remarks on π , and thus π is an Archimedean order unit.The same observations hold for π π and π π for all π β β and thus the same method will show that π is an Archimedeanmatrix order unit for ( ξ , { πΆ ( π π )} π ) .The next corollary, which follows from Proposition 3.1, states that the cones πΆ ( π π ) are closed in the order seminorminduced by the projection π . Corollary 3.2.
Given an operator system ξ β π΅ ( π» ) and projection π β ξ then if { πΆ ( π π )} π denotes the matrix orderingas in Lemma 3.1 then for each π β β , πΆ ( π π ) is closed in the order seminorm.Proof. For this proof let πΌ π βΆ π π ( ξ ) β [0 , β) denote the order seminorm deο¬ned for each π β β by πΌ π ( π₯ ) βΆ= inf { π > ( ππ π π₯π₯ β ππ π ) β πΆ ( π π )} . Let { π₯ π } π β πΌ β πΆ ( π π ) denote a net such that π₯ = πΌ π - lim π π₯ π ( π₯ is the limit of π₯ π relative to the norm πΌ π ). π₯ is necessarily β -hermitian. Let π > and let π π β πΌ such that πΌ π ( π₯ π β π₯ ) < π for π π βͺ― π. It follows that ( ππ π π₯ β π₯ π π₯ β π₯ π ππ π ) β πΆ ( π π ) and thus compression by ( ) implies ππ π + π₯ β π₯ π β πΆ ( π π ) and therefore ππ π + π₯ β πΆ ( π π ) . Since π is an Archimedeanmatrix order unit we have π₯ β πΆ ( π π ) as desired. Remark 3.3.
Used in Proposition 3.1 we make the point of saying that if π βΆ ξ β π΅ ( π» ) is the map deο¬ned by π ( π£ ) = ππ£π then π π ( π£ ) = π π π£π π , π£ β π π ( ξ ) . First note that compression by the projection is linear completelypositive. Notice, π π π£π π = ( πΌ π β π )( β ππ | π β©β¨ π | β π£ ππ )( πΌ π β π ) = β ππ | π β©β¨ π | β ππ£ ππ π = β ππ | π β©β¨ π | β π ( π£ ππ ) = π π ( π£ ) , Lemma 3.4.
Given an operator system ξ β π΅ ( π» ) let π β ξ be a projection as above. Let π½ π = span πΆ ( π ) β© β πΆ ( π ) , and π½ π π = span πΆ ( π π ) β© β πΆ ( π π ) . Then π π ( π½ π ) = π½ π π .Proof. If π₯ β π π ( π½ π ) then we write π₯ = β ππ | π β©β¨ π | β π₯ ππ , π₯ ππ β π½ π . For each π, π we then write π₯ ππ = β π π πππ π πππ where π πππ β β , π πππ β πΆ ( π ) β© β πΆ ( π ) . We then see π π π₯π π = π π ( β πππ π πππ | π β©β¨ π | β π πππ ) π π = β πππ π πππ | π β©β¨ π | β ππ πππ π = 0 , since Β± ππ πππ π β π΅ ( π» ) + for all π, π, π and therefore is equal to 0. Thus π₯ β πΆ ( π π ) β© β πΆ ( π π ) which yields the ο¬rstinclusion. Conversely, let π₯ β π½ π π and write π₯ = β π π π π₯ π , π π β β , π₯ π β πΆ ( π π ) β© β πΆ ( π π ) . We write π₯ π = β ππ | π β©β¨ π | β π πππ β π π ( ξ ) . By the assumption that π π π₯ π π π = 0 we see that π π π₯ π π π = β ππ | π β©β¨ π | β ππ πππ π = 0 , for all π, π, π, which in turn yields that ππ πππ π = 0 and thus π πππ β πΆ ( π ) β© β πΆ ( π ) . This then yields π₯ = β πππ | π β©β¨ π | β π π π πππ β π π ( π½ π ) , ο¬nishing the proof. 6 emma 3.5. Given an operator system ξ β π΅ ( π» ) with projection π β ξ , then for all π β β , π½ π π β π π ( ξ ) is β -closedand if πΌ β π π,π then πΌ β π½ π π πΌ β π½ π π . In particular π π π½ π π π π β π½ π π . Proof.
Let π₯ β π½ π π . Then π₯ = β πππ | π β©β¨ π | β π π π πππ with, as we saw in the proof of Lemma 3.4, π πππ β πΆ ( π ) β© β πΆ ( π ) for all π, π, π, and therefore π β πππ = π πππ . Thus π₯ β = β πππ | π β©β¨ π | β π π π πππ β π½ π π . It is immediate that for any π₯ β π½ π π that π π π₯π π β π½ π π since if we write π₯ = β π π π π₯ π , with π₯ π β πΆ ( π π ) β© β πΆ ( π π ) then π ( ππ₯ π π ) π = ππ₯ π π β π΅ ( π» π ) + . This holdsfor all π and thus π π π½ π π π π β π½ π π . Finally if π₯ β π½ π π and πΌ β π π,π we have Β± π π πΌ β π₯πΌπ π = πΌ β (Β± π π π₯π π ) πΌ β π΅ ( π» π ) + . Thus, πΌ β π₯πΌ β π½ π π . We now wish to consider the vector space ξ β π½ π where ξ β π΅ ( π» ) is an operator system, and π β ξ is a projectionin π΅ ( π» ) . The vector space operations and involution are deο¬ned in the natural way and are well-deο¬ned by Lemma3.5. We consider the family { ΜπΆ ( π π )} π where for each π β β we have ΜπΆ ( π π ) βΆ= {( π₯ ππ + π½ π ) β π π ( ξ β π½ π ) βΆ π₯ = ( π₯ ππ ) β πΆ ( π π )} . (5)Note that π β π½ π , for π a nonzero projection in ξ . This is immediate since if π = β π π π π π , Β± π π β πΆ ( π ) β© β πΆ ( π ) then it would follow that π = πππ = π ( β π π π π π ) π = β π π π ππ π π, (6)and since Β± ππ π π β π΅ ( π» ) + for all π implies ππ π π = 0 . Thus, π = 0 . Theorem 3.6.
The triple ( ξ β π½ π , { ΜπΆ ( π π )} π , π + π½ π ) is an operator system.Proof. We begin by showing that { ΜπΆ ( π π )} π is a proper matrix ordering. Let ( π₯ ππ + π½ π ) β π π ( ξ β π½ π ) . If π > then π ( π₯ ππ + π½ π ) = ( ππ₯ ππ + π½ π ) and ππ₯ β ππΆ ( π π ) β πΆ ( π π ) , thus π ( π₯ ππ + π½ π ) β ΜπΆ ( π π ) . If ( π₯ ππ + π½ π ) , ( π¦ ππ + π½ π ) β ΜπΆ ( π π ) then ( π₯ ππ + π½ π ) + ( π¦ ππ + π½ π ) = ( π₯ ππ + π¦ ππ ) + π½ π β ΜπΆ ( π π ) since π₯ + π¦ β πΆ ( π π ) . If π β β and ( π₯ ππ + π½ π ) β ΜπΆ ( π π ) and πΌ β π π,π then πΌ β ( π₯ ππ + π½ π ) πΌ = β ππππ | π β©β¨ π | β ( πΌ ππ π₯ ππ πΌ ππ + π½ π ) = ( β ππ πΌ ππ π₯ ππ πΌ ππ + π½ π ) ππ , and πΌ β π₯πΌ β πΆ ( π π ) which implies πΌ β ( π₯ ππ + π½ π ) πΌ β ΜπΆ ( π π ) . If ( π₯ ππ + π½ π ) β ΜπΆ ( π π ) then ( π₯ ππ + π½ π ) β = ( π₯ β ππ + π½ π ) and since π₯ β πΆ ( π π ) implies π₯ β β πΆ ( π π ) which implies ( π₯ ππ + π½ π ) β = ( π₯ β ππ + π½ π ) β ΜπΆ ( π π ) . Finally, suppose that ( π₯ ππ + π½ π ) β ΜπΆ ( π π ) β© β ΜπΆ ( π π ) . This implies that Β± π₯ β πΆ ( π π ) . Then it follows π₯ β π½ π π = π π ( π½ π ) by Lemma 3.4 andthus π₯ ππ β π½ π for all π, π. In other words ( π₯ ππ + π½ π ) = (0 + π½ π ) implying that our cones are proper, and thus the matrixordering { ΜπΆ ( π π )} π on ξ β π½ π is proper. It remains to show that π + π½ π is an Archimedean matrix order unit. Considera β -hermitian element ( π₯ ππ + π½ π ) β π π ( ξ β π½ π ) . Let π > be such that ππ π β π₯ β πΆ ( π π ) (see Proposition 3.1). It thenfollows π ( πΌ π β ( π + π½ π )) β ( π₯ ππ + π½ π ) = ( πΌ π β ( ππ + π½ π )) β ( π₯ ππ + π½ π ) = (( π ( π π ) ππ β π₯ ππ ) + π½ π ) β ΜπΆ ( π π ) , since ππ π β π₯ β πΆ ( π π ) . Finally if for all π > we have ( π ( π π ) ππ + π₯ ππ + π½ π ) β ΜπΆ ( π π ) then ππ π + π₯ β πΆ ( π π ) which implies π₯ β πΆ ( π π ) (seeProposition 3.1) and thus ( π₯ ππ + π½ π ) β ΜπΆ ( π π ) . This ο¬nishes our proof.7 eο¬nition 3.7. Given an operator system ξ β π΅ ( π» ) with π β π΅ ( π» ) a projection, we call the set π ξ π , regarded aslinear operators on the Hilbert space ππ» , the concrete compression operator system .Thus, we have seen how to regard the compression of an operator system by a single projection as a quotient of theoriginal operator system. We now wish to explore the compression of the operator system π ( ξ ) relative to the cones { πΆ ( π π β π π )} π where π β ξ is a projection with π = πΌ β π, and πΆ ( π π β π π ) βΆ= { π₯ β π π ( ξ ) βΆ π₯ = π₯ β , ( π π β π π ) π₯ ( π π β π π ) β π΅ ( π» π ) + } . (7)We ο¬rst prove the following quick lemma: Lemma 3.8.
Given an operator system ξ β π΅ ( π» ) and projections π, π β ξ then πΆ ( π π ) β πΆ ( π π ) β πΆ ( π π β π π ) for all π β β .Proof. Given π₯ β πΆ ( π π ) and π¦ β πΆ ( π π ) then we see ( π π β π π )( π₯ β π¦ )( π π β π π ) = ( π π π₯π π β π π π¦π π ) and for any ( π , π ) β π» π β π» π we see β¨ ( π π π₯π π β π π π¦π π )( π , π ) | ( π , π ) β© = β¨ π π π₯π π π | π β© + β¨ π π π¦π π π | π β© β β + and thus π₯ β π¦ β πΆ ( π π β π π ) which proves the result.We will use the βCanonical shuο¬eβ (see Remark 2.5) as presented in the preliminary section. Lemma 3.9.
Given an operator system ξ β π΅ ( π» ) , with projection π β ξ , π = πΌ β π , consider the operator π βΆ π ( ξ ) β π΅ ( π» ) deο¬ned by π ( π₯ ) = ( π β π ) π₯ ( π β π ) . Then πΌ π β ( π β π ) β ( π π β π π ) under the β -isomorphism π π ( π ( ξ )) β π ( π π ( ξ )) . This is to say that the nth-ampliο¬cation of π is given by π π β π π under the canonicalshuο¬e.Proof. Fix π β π π ( π ( ξ )) . We then write π = β ππ | π β©β¨ π | β π ππ , π ππ β π ( ξ ) and see π π ( β ππ | π β©β¨ π | β π ππ ) = β ππ | π β©β¨ π | β π ( π ππ ) = β ππ | π β©β¨ π | β ( π β π )( π ππ )( π β π )= β ππ | π β©β¨ π | β [ | β©β¨ | β ππ ππ π + | β©β¨ | β ππ ππ π + | β©β¨ | β ππ ππ π + | β©β¨ | β ππ ππ π ] , where { | π β©β¨ π | } π,π in the bracket above denote the matrix units in π . It then follows β ππ | π β©β¨ π | β [ | β©β¨ | β ππ ππ π + | β©β¨ | β ππ ππ π + | β©β¨ | β ππ ππ π + | β©β¨ | β ππ ππ π ]= | β©β¨ | β β ππ | π β©β¨ π | β ππ ππ π + | β©β¨ | β β ππ | π β©β¨ π | β ππ ππ π + | β©β¨ | β β ππ | π β©β¨ π | β ππ ππ π + | β©β¨ | β β ππ | π β©β¨ π | β ππ ππ π. Let π ππ = β ππ | π β©β¨ π | β π ππππ . We ο¬nally get | β©β¨ | β β ππ | π β©β¨ π | β ππ ππ π + | β©β¨ | β β ππ | π β©β¨ π | β ππ ππ π + | β©β¨ | β β ππ | π β©β¨ π | β ππ ππ π + | β©β¨ | β β ππ | π β©β¨ π | β ππ ππ π = ( π π β π π )( β ππ | π β©β¨ π | β π ππ )( π π β π π ) which proves the result. 8 emma 3.10. Given an operator system ξ β π΅ ( π» ) with π β ξ a projection. Then { πΆ ( π π β π π )} π is a matrix orderingon ξ β ξ .Proof. We begin with an observation. By Lemma 3.8 we know πΆ ( π π ) β πΆ ( π π ) β πΆ ( π π β π π ) . If we consider π₯ β π¦ β π π ( ξ ) β π π ( ξ ) then if π₯ β π¦ β πΆ ( π π β π π ) we see ( π π β π π )( π₯ β π¦ )( π π β π π ) = ( π π π₯π π π π π¦π π ) β π΅ ( π» π ) + . Compression by ( ) implies that π π π₯π π β π΅ ( π» π ) + and therefore π₯ β πΆ ( π π ) . Similarly, compression by ( ) impliesthat π π π¦π π β π΅ ( π» π ) + . Thus π₯ β π¦ β πΆ ( π π ) β πΆ ( π π ) . Thus, we have πΆ ( π π ) β πΆ ( π π ) = πΆ ( π π β π π ) when restricting toelements of the form π₯ β π¦. β -closed follows by deο¬nition. Fix π β β . It is immediate that given any π > that ππΆ ( π π β π π ) β πΆ ( π π β π π ) . Let π, π β πΆ ( π π β π π ) . Since the compression by the projection π π β π π is linear, we have that ( π π β π π )( π + π )( π π β π π ) =( π π β π π ) π ( π π β π π ) + ( π π β π π ) π ( π π β π π ) β π΅ ( π» π ) + + π΅ ( π» π ) + β π΅ ( π» π ) + . We now check compatibilty. Let π β β with πΌ β π π, π and consider πΌ β ( π₯ β π¦ ) πΌ β π ( π π ( ξ )) , where π₯ β π¦ β πΆ ( π π β π π ) . Let π βΆ π ( ξ ) β π΅ ( π» ) bedeο¬ned as in Lemma 3.9. Then ( π π β π π ) πΌ β ( π₯ β π¦ ) πΌ ( π π β π π ) = π π ( πΌ β ( π₯ β π¦ ) πΌ ) = πΌ β π π ( π₯ β π¦ ) πΌ = πΌ β ( π π β π π )( π₯ β π¦ )( π π β π π ) πΌ β πΌ β π΅ ( π» π ) + πΌ β π΅ ( π» π ) + . which proves the result.We consider the family { ΜπΆ ( π π β π π )} π where for each π β β , ΜπΆ ( π π β π π ) = {(( π₯ ππ β π¦ ππ ) + π½ πβπ ) β π π (( ξ β ξ )β π½ πβπ ) βΆ ( π₯ β π¦ ) β πΆ ( π π β π π ) , π₯ = ( π₯ ππ ) , π¦ = ( π¦ ππ )} . Here we have let ξ β ξ denote the β -vector subspace of π ( ξ ) consisting of all diagonal matrices over ξ . Proposition 3.11.
Given an operator system ξ β π΅ ( π» ) and a projection π β ξ . Let π = πΌ π» β π. Then ( ξ β ξ )β π½ πβπ is an operator system.Proof. By Lemma 3.10 it is immediate to show that { ΜπΆ ( π π β π π )} π is a matrix ordering. Another direct consequenceof the deο¬nition is that given any (( π₯ ππ β π¦ ππ ) + π½ πβπ ) ππ β πΆ ( π π β π π ) β© β πΆ ( π π β π π ) then π₯ ππ β πΆ ( π ) β© β πΆ ( π ) and π¦ ππ β πΆ ( π ) β©β πΆ ( π ) for all π, π which, along with Lemma 3.8, proves the cones are proper. The claim that π βπ + π½ πβπ isan Archimedean order unit follows from the results that π is an Arhimedean order unit for { πΆ ( π π )} π , π is an Archimedeanorder unit for { πΆ ( π π )} π and then applying Lemma 3.8 and the observation in Lemma 3.10. We leave the details to thereader.More generally it will follow that ( π β π ) π ( ξ )( π β π ) is an operator system. We begin with a lemma: Lemma 3.12.
Given an operator system ξ β π΅ ( π» ) with projections π, π β ξ then { πΆ ( π π β π π )} π is a matrix orderingon π ( ξ ) . Proof.
This proof is the same as Lemma 3.10 but we will make some remarks for completeness. Once again, β -closed,and positive homogeniety of the family is immediate. Let π βΆ π ( ξ ) β π΅ ( π» ) once again denote the linear mapgiven by π₯ β¦ ( π β π ) π₯ ( π β π ) . Then compatibility is immediate from Lemma 3.9. This ο¬nishes the proof.
Theorem 3.13.
Given an operator system ξ β π΅ ( π» ) and a projection π β ξ , with π = πΌ β π then ( π ( ξ )β π½ π β π ) , { ΜπΆ ( π π β π π )} π , ( π β π ) + π½ πβπ ) is an operator system. roof. The fact that π½ πβπ is a subspace along with Lemma 3.12 immediately gives that the family { ΜπΆ ( π π β π π )} π is amatrix ordering. Given any ( π₯ ππ ) ππ β π π ( π ( ξ )) one can show that for each π, π that π₯ ππ = ( π₯ ππ π₯ ππ π₯ ππ π₯ ππ ) = ( π₯ ππ π₯ ππ ) + ( π₯ ππ π₯ ππ ) β π½ πβπ + π½ πβπ β π½ πβπ . (8)Thus ΜπΆ ( π π β π π ) β© β ΜπΆ ( π π β π π ) has trivial intersection which implies the family { ΜπΆ ( π π β π π )} π is proper. An exercisein matrix mechanics shows that π β π + π½ πβπ is an Archimedean matrix order unit. This ο¬nishes the proof. Corollary 3.14.
Consider a family of projections { π π } ππ =1 β ξ , and let π π = πΌ β π π for all π . Let π βΆ= β π π π , π βΆ= β π π π .Then π π ( ξ )β π½ π βπ is an operator system.Proof.
This is immediate since
π , π β π π ( ξ ) are projections (as operators in π΅ ( π» π ) ) with of course πΌ π β π = β π πΌ β π π = β π π π = π, and after making the identiο¬cation π π ( ξ ) = π ( π π ( ξ )) . Thus we see ( π β π ) π π ( ξ )( π β π ) = (
π β π ) π ( π π ( ξ ))( π β π ) is an operator system by Theorem 3.13. Motivated by Section 3 we now wish to consider compression operator systems in an abstract sense. We begin byconsidering a β -vector space with structure similar to that of an operator system, but lacking proper cones. We willshow that a natural quotient of such a β -vector space is in fact an operator system. We will then consider a particularcase of a β -vector space with such a matrix ordering induced by positive contractions. Finally we will show that thisstructure coincides with the structure of an operator system compressed by a projection, as studied in Section 3.Let ξ be a β -vector space, and let { πΆ π } π be a matrix ordering on ξ . For every π β β , let π½ π βΆ= span πΆ π β© β πΆ π . Lemma 4.1.
Suppose that π₯ β π½ π . Then π₯ = π + ππ for some π, π β πΆ π β© β πΆ π .Proof. If π₯ β π½ π , then π₯ = β π π π π π where each π π β πΆ π β© β πΆ π and each π π β β . Assume π π = π π + ππ π for π π , π π β β .Then π₯ = ( β π π π π π ) + π ( β π π π π π ) . So π = ( β π π π π π ) β πΆ π β© β πΆ π and π = ( β π π π π π ) β πΆ π β© β πΆ π .For convenience, set π½ βΆ= π½ . Compare the next result with Lemma 3.4. Lemma 4.2.
For every π β β , π π ( π½ ) = π½ π .Proof. We ο¬rst show that π π ( π½ ) β π½ π . It is necessary and suο¬cient to show that for every π, π β€ π and π₯ β π½ we have | π β©β¨ π | β π₯ β π½ π . Let π₯ β π½ . By the previous lemma, π₯ = π + ππ for π, π β πΆ β© β πΆ . Then | π β©β¨ π | β π, | π β©β¨ π | β π β πΆ π β© β πΆ π since { πΆ π } is a matrix cone. Hence Β± | π β©β¨ π | β π₯ β π½ π . Similarly, ( | π β© + | π β© )( β¨ π | + β¨ π | ) β π₯ β π½ π and ( | π β© + π | π β© )( β¨ π | β π β¨ π | ) β π₯ β π½ π . So (( | π β© + | π β© )( β¨ π | + β¨ π | ) β | π β©β¨ π | β | π β©β¨ π | ) β π₯ = ( | π β©β¨ π | + | π β©β¨ π | ) β π₯ β π½ π and π (( | π β© + π | π β© )( β¨ π | β π β¨ π | ) β | π β©β¨ π | β | π β©β¨ π | ) β π₯ = ( | π β©β¨ π | β | π β©β¨ π | ) β π₯ β π½ π . It follows that | π β©β¨ π | β π₯ β π½ .Next we show π½ π β π π ( π½ ) . Let π₯ = ( π₯ ππ ) β π½ π . Then it is necessary and suο¬cient to show that π₯ ππ β π½ forevery π, π . By the previous lemma, π₯ = π + ππ for π, π β π½ π , so it suο¬ces to show that π ππ and π ππ are elementsof π½ . We will show that π ππ β π½ , and the proof for π ππ is identical. Since π β πΆ π β© β πΆ π and { πΆ π } π is a matrixcone, we have π ππ = β¨ π | π | π β© , π ππ = β¨ π | π | π β© β π½ . Also ( β¨ π | + β¨ π | ) π ( | π β© + | π β© ) = π ππ + π ππ + π ππ + π ππ β π½ , and ( β¨ π | + π β¨ π | ) π ( | π β© β π | π β© ) = π ππ β ππ ππ + ππ ππ + π ππ β π½ . It follows that π ππ + π ππ and π ππ β π ππ are elements of π½ andhence π ππ β π½ . Similarly π ππ β π½ and hence π₯ ππ β π½ . 10emma 4.2 allows us to identify the vector spaces π π ( ξ β π½ ) , π π ( ξ )β π π ( π½ ) and π π ( ξ )β π½ π . Deο¬ne ΜπΆ π βΆ= {( π₯ ππ + π½ ) β π π ( ξ β π½ ) βΆ π₯ = ( π₯ ππ ) β πΆ π } . Lemma 4.3.
The sequence { ΜπΆ π } is a matrix ordering on ξ β π½ .Proof. That π ΜπΆ π β ΜπΆ π and ΜπΆ π + ΜπΆ π are immediate. To check compatibility, ο¬x π β β and let πΌ β π π,π and ( π₯ ππ + π½ ) β ΜπΆ π . Then πΌ β ( π₯ ππ + π½ ) πΌ = β π,π | π β©β¨ π | β β π,π πΌ ππ ( π₯ ππ + π½ ) πΌ ππ = β π,π | π β©β¨ π | β ( β π,π πΌ ππ π₯ ππ πΌ ππ + π½ )= ( β π,π πΌ ππ π₯ ππ πΌ ππ + π½ ) ππ , and πΌ β π₯πΌ β πΌ β πΆ π πΌ β πΆ π . This ο¬nishes the proof.Suppose { πΆ π } π is a (not necessarily proper) matrix ordering on a β -vector space ξ . Recall that π β ξ a matrixorder unit for ( ξ , { πΆ π } π ) if for every π₯ β π π ( ξ ) with π₯ β = π₯ , there exists π > such that π₯ + ππ π β πΆ π . If π₯ + ππ π β πΆ π for all π > implies that π₯ β πΆ π , we call π Archimedean . Proposition 4.4.
Suppose that ξ is a β -vector space with matrix ordering { πΆ π } π and an Archimedean matrix orderunit π . Then ( ξ β π½ , { ΜπΆ π } π , π + π½ ) is an operator system.Proof. By Lemma 4.3 we need only show that the family { ΜπΆ π } π is proper and that π + π½ is an Archimedean matrixorder unit. If ( π₯ ππ + π½ ) β ΜπΆ π β© β ΜπΆ π then Β± π₯ β πΆ π which implies π₯ β π½ π = π π ( π½ ) by Lemma 4.2. Thus, π₯ ππ β π½ forall π, π and therefore ( π₯ ππ + π½ ) = (0 + π½ ) . It remains to show that π + π½ is an Archimedean matrix order unit. Let ( π₯ ππ + π½ ) β π π ( ξ β π½ ) be β -hermitian andchoose π > such that ππ π β π₯ β πΆ π . Then π ( πΌ π β ( π + π½ )) β ( π₯ ππ + π½ ) = ( πΌ π β ( ππ + π½ )) β ( π₯ ππ + π½ ) = ( π ( π π ) ππ + π½ ) β ( π₯ ππ + π½ ) = ( π ( π π ) ππ β π₯ ππ + π½ ) β ΜπΆ π since ππ π β π₯ β πΆ π . Finally, if ππΌ π β ( π + π½ ) + ( π₯ ππ + π½ ) β ΜπΆ π for all π > then ( π ( π π ) ππ + π₯ ππ + π½ ) β ΜπΆ π for all π > and by deο¬nition ππ π + π₯ β πΆ π giving us π₯ β πΆ π and thus ( π₯ ππ + π½ ) β ΜπΆ π . This ο¬nishes the proof.We now brieο¬y return to the structural properties induced by projections in a concrete operator system ξ β π΅ ( π» ) . The following lemma will motivate Deο¬nition 4.6 below for a matrix ordering { πΆ ( π π )} when π is only a positivecontraction in an abstract operator system. Lemma 4.5.
Let ξ β π΅ ( π» ) be an operator system and suppose that π β ξ is a projection. Then for any π₯ β ξ with π₯ = π₯ β , we have that ππ₯π β₯ in π΅ ( π» ) if and only if for every π > there exists a π‘ > such that π₯ + ππ + π‘ ( πΌ β π ) β₯ . Proof.
First assume that for every π > there exists a π‘ > such that π¦ = π₯ + ππ + π‘ ( πΌ β π ) β₯ . Then the compressionof π¦ by π is positive. Hence ππ¦π = ππ₯π + ππ β₯ . Since this holds for all π > and since π is an Archimedean orderunit for the set of operators of the form ππ§π (see Proposition 3.1) it follows that ππ₯π β₯ .Now assume that ππ₯π β₯ , and let π > . Let π = πΌ β π . It follows that if we write π» = ππ» β ππ» , we see that (byexercise 3.2(i) in [15]) an operator π is positive if and only if ππ π β₯ , ππ π β₯ , and for every β β ππ» and π β ππ» we have that |β¨ ππ ππ | β β©| β€ β¨ ππ πβ | β β©β¨ ππ ππ | π β© . π‘ > β π₯ β such that π ( π‘ β β π₯ β ) > β π₯ β and consider π = π₯ + ππ + π‘π . Then ππ π = ππ₯π , ππ π = ππ₯π + ππ β₯ ,and ππ π = π‘π + ππ₯π β₯ π‘π β β π₯ β π β₯ . Moreover |β¨ ππ ππ | β β©| β€ β π₯ β β β β β π β β€ π ( π‘ β β π₯ β ) β β β β π β β€ β¨ ππ πβ | β β©β¨ ππ ππ | π β© , since π β β β = β¨ ππβ | β β© β€ β¨ ππ πβ | β β© and ( π‘ β β π₯ β ) β π β = β¨ ( π‘π β β π₯ β π ) π | π β© β€ β¨ ππ ππ | π β© . So π₯ + ππ + π‘π β₯ .This lemma thus relates positivity of the compression by π to positivity in the operator system ξ . This motivatesus to make the following deο¬nition.
Deο¬nition 4.6.
Let ( ξ , { πΆ π } π , π ) be an operator system, and suppose that π β ξ with β€ π β€ π , i.e., let π β ξ be apositive contraction of ξ . For each π β β and let π π = πΌ π β π . We deο¬ne the positive cone relative to π π , denoted πΆ ( π π ) , to be πΆ ( π π ) βΆ = { π₯ β π π ( ξ ) βΆ π₯ = π₯ β , for all π > there exists π‘ > such that π₯ + ππ + π‘ ( π β π ) β πΆ π } . (9)An immediate consequence of Lemma 4.5 is that if a positive contraction π β ξ is a projection, then for each π β β the positive cone relative to π π becomes πΆ ( π π ) = { π₯ β π π ( ξ ) βΆ π₯ = π₯ β , π π π₯π π β π΅ ( π» π ) + } , (10)We now prove a similar string of results mirroring those of Section 3. Proposition 4.7.
Let ξ be an operator system and β€ π β€ π . Then the sequence { πΆ ( π π )} π is a matrix ordering for ξ ,and π is an Archimedean matrix order unit for { πΆ ( π π )} π .Proof. We ο¬rst check that { πΆ ( π π )} π is a matrix ordering. As noted in the preliminaries, it suο¬ces to check that For each π, π β β , πΆ ( π π ) β πΆ ( π π ) β πΆ ( π π + π ) , and for each π, π β β and πΌ β π π,π , πΌ β πΆ ( π π ) πΌ β πΆ ( π π ) . Suppose π₯ β πΆ ( π π ) and π¦ β πΆ ( π π ) . Let π > . Then there exist π , π > such that π₯ + ππ π + π ( π π β π π ) β₯ and π¦ + ππ π + π ( π π β π π ) β₯ .It follows that ( π₯ β π¦ ) + ππ π + π + max( π , π )( π π + π β π π + π ) β₯ . Now, let π > and suppose that π₯ β πΆ ( π π ) . Alsoassume πΌ β . Then there exists π > such that π₯ + π β πΌ β π π + π ( π π β π π ) β₯ . It follows that πΌ β π₯πΌ + π β πΌ β πΌ β π π πΌ + ππΌ β ( π π β π π ) πΌ β₯ . However, since πΌ β π π πΌ β€ β πΌ β π π and πΌ β ( π π β π π ) πΌ β€ β πΌ β ( π π β π π ) we have πΌ β π₯πΌ + ππ π + ( π β πΌ β )( π π β π π ) β₯ . It follows that πΌ β π₯πΌ β πΆ ( π π ) .We now show that π is an Archimedean matrix order unit for { πΆ ( π π )} π . We verify the relevant properties for thecase for π = 1 and for π > the proofs are similar. Choose π > such that π₯ + ππ β₯ . Let π > . Then ( π₯ + ππ ) + ππ + π ( π β π ) = π₯ + ππ + ππ β₯ . It follows that π₯ + ππ β πΆ ( π ) . Finally, assume π₯ + πΏπ β πΆ ( π ) for all πΏ > and let π > . Then there exists π > suchthat ( π₯ + π β2 π ) + π β2 π + π ( π β π ) β₯ . It follows that π₯ + ππ + π ( π β π ) β₯ . So π₯ β πΆ ( π ) . 12e now come to the main results of this section. Similarly to our notation in the Section 3, given an operator system ξ , and a positive contraction π β ξ we consider the matrix ordering { πΆ ( π π )} π and we let π½ π = span πΆ ( π ) β© β πΆ ( π ) . We recall the following deο¬nition. Given an Archimedean order unit space ξ then the minimal order norm πΌ π on ξ is deο¬ned for π₯ β ξ by πΌ π ( π₯ ) = sup{ | π ( π₯ ) | βΆ π β ξΏ ( ξ )} (11)where ξΏ ( ξ ) denotes the set of states on ξ . It is not diο¬cult to show that if πΌ π βΆ ξ β β [0 , β) denotes the order norminduced by π given by πΌ π ( π₯ ) = inf{ π‘ > π‘π Β± π₯ β ξ + } , (12)then πΌ π = πΌ π when restricted to ξ β . We refer the interested reader to [17, Section 4] for the details.
Proposition 4.8.
Let ξ an operator system and let π β ξ be a nonzero positive contraction. Let πΌ π βΆ ξ β [0 , β) denote the minimal order norm induced by π . Then πΌ π ( π ) = 1 if and only if π β π½ π . Proof.
By the assumption that π is a positive contraction we know that πΌ π ( π ) = inf{ π‘ > π‘π β π β ξ + } β€ . Theassumption that πΌ π ( π ) = 1 implies πΌ π ( π ) = 1 . We ο¬rst show that π β β πΆ ( π ) . Suppose the contrary. Then by deο¬nitionfor all π > there exists π‘ > such that β π + ππ + π‘ ( π β π ) β ξ + . In other words it must follow π β€ π‘ π‘ β π π. If π < then for all π‘ > we have π‘ π‘ β π < which contradicts the assumption that πΌ π ( π ) = 1 . Thus π β β πΆ ( π ) . Nowsuppose that π β π½ π . Since π β = π we have π β πΆ ( π ) β© β πΆ ( π ) , a contradiction.As in Proposition 4.4 we will deο¬ne the family of sets { ΜπΆ ( π π )} π where for each π β β we have ΜπΆ ( π π ) = {( π₯ ππ + π½ π ) β π π ( ξ β π½ π ) βΆ π₯ = ( π₯ ππ ) β πΆ ( π π )} . (13)We now have the abstract analogue to Theorem 3.6. Theorem 4.9.
Given an operator system ξ and positive contraction π β ξ such that πΌ π ( π ) = 1 , the triple ( ξ β π½ π , { ΜπΆ ( π π )} π β β , π + π½ π ) is a non-trivial operator system.Proof. By Proposition 4.7 we already have that { πΆ ( π π )} π is a matrix ordering and π is an Archimedean matrix orderunit for the pair ( ξ , { πΆ ( π π )} π ) . By Proposition 4.4, we deduce that ( ξ β π½ π , { ΜπΆ ( π π )} π , π + π½ π ) is an operator system.Moreover, this operator system is non-trivial by Proposition 4.8, since πΌ π ( π ) = 1 .Combining Theorem 4.9, Lemma 4.5 and Theorem 3.6 yields the following key observation. Corollary 4.10.
Suppose that ξ β π΅ ( π» ) is an operator system and that π β ξ is a projection in π΅ ( π» ) . Then theabstract compression ( π β π½ π , { ΜπΆ ( π π )} π β β , π + π½ π ) is completely order isomorphic to the concrete compression π ξ π . Corollary 4.10 justiο¬es the terminology we will use in the following deο¬nition.
Deο¬nition 4.11.
Given an operator system ξ and a positive contraction π β ξ such that πΌ π ( π ) = 1 then we call theoperator system ( ξ β π½ π , { ΜπΆ ( π π )} π β β , π + π½ π ) the abstract compression operator system and denote it by ξ β π½ π . In the next section of the paper we will make use of the structure of the abstract compression operator system π ( ξ )β π½ πβπ where π is a positive contraction and π = π β π . We denote the positive cones relative to the positivecontraction π β π by { πΆ (( π β π ) π )} π where for each π β β πΆ (( π β π ) π ) = { π₯ β π π ( ξ ) βΆ π₯ = π₯ β , β π > π‘ > such that π₯ + π (( π β π ) π ) + π‘ (( π β π ) π ) β πΆ π } , (14)and π½ πβπ = span πΆ ( π β π ) β© β πΆ ( π β π ) . The concrete analogue of the following corollary was stated in Corollary 3.14.
Corollary 4.12.
Given an operator system ξ and a ο¬nite family of positive contractions { π π } ππ =1 β ξ + such that πΌ π ( π π ) = 1 for some π , with π π = π β π π for all π, , let π = β π π π , and π = βπ π . Then π π ( ξ )β π½ π βπ is an operatorsystem. Projections in operator systems
In this section we will develop an abstract characterization for projections in operator systems. We start with thefollowing useful observation.
Lemma 5.1.
Let ξ β π΅ ( π» ) be an operator system and suppose that π β ξ is a projection (when viewed as an operatoron π» ). If π = πΌ β π and π, π, π β ξ with π β = π and π β = π , then πππ + πππ + ππ β π + πππ β π΅ ( π» ) + if and only if ( π ππ β π ) β πΆ ( π β π ) . Proof.
First suppose that ( π ππ β π ) β πΆ ( π β π ) . Then by Lemma 4.5 we know that ( π π ) ( π ππ β π ) ( π π ) β₯ . Conjugating this matrix by the scalar matrix ( ) yields the expression πππ + πππ + ππ β π + πππ and hence πππ + πππ + ππ β π + πππ β₯ .Now suppose that πππ + πππ + ππ β π + πππ β₯ . Again, by Lemma 4.5 it suο¬ces to prove that the operator π = ( π π ) ( π ππ β π ) ( π π ) is positive. To this end, let β, π β π» . Deο¬ne β = πβ and β = ππ . Note that β¨ β | β β© = 0 since π and π are orthogonalprojections. Let Μβ = β + β . Then β¨ π ( β β π ) | ( β β π ) β© = β¨( π ππ β π ) ( β β ) | ( β β )β© = β¨ ( πππ + πππ + ππ β π + πππ ) Μβ | Μβ β© β₯ . We conclude that ( π ππ β π ) β πΆ ( π β π ) . In Section 3, we observed that { πΆ ( π π β π π )} was a matrix ordering on π ( ξ ) . This is due to Lemma 3.9 whichshows that πΆ (( π β π ) π ) can be identiο¬ed with πΆ ( π π β π π ) via the canonical shuο¬e map. The next Lemma is an abstractvariation on the same result. Lemma 5.2.
Let π βΆ π π ( π ( ξ )) β π ( π π ( ξ )) denote the canonical shuο¬e map. Then π ( πΆ (( π β π ) π ) = πΆ ( π π β π π ) and hence π₯ β πΆ (( π β π ) π ) if and only if π ( π₯ ) β πΆ ( π π β π π ) .Proof. As noted in Remark 2.5, the canonical shuο¬e can be written as π = π β ππ βΆ π π ( π ) β ξ β π ( π π ) β ξ where π βΆ π π ( π ) β π ( π π ) is a β -isomorphism and ππ βΆ ξ β ξ is the identity map. Hence it is a complete orderembedding. The statement follows from the observation that π ( π₯ + π ( π β π ) π + π‘ ( π β π ) π ) = π ( π₯ ) + ππ π β π π + π‘π π β π π . Our abstract characterization for projections is based on the following Theorem.14 heorem 5.3.
Let ξ β π΅ ( π» ) be an operator system and π β ξ be a projection. Set π = πΌ β π . Then for every π β β and π₯ β π π ( ξ ) we have that π₯ β πΆ π if and only if ( π₯ π₯π₯ π₯ ) β πΆ ( π π β π π ) . Proof.
Suppose that π₯ β πΆ π which implies that π π π₯π π , π π π₯π π β₯ . Applying Lemma 5.1 we have ( π₯ π₯π₯ π₯ ) β πΆ ( π π βπ π ) if and only if π π π₯π π + π π π₯π π + π π π₯π π + π π π₯π π β₯ . It follows π π π₯π π + π π π₯π π + π π π₯π π + π π π₯π π = π π π₯ ( π π + π π ) + π π π₯ ( π π + π π ) = ( π π + π π ) π₯ ( π π + π π ) = π₯ β₯ . This proves one direction. Conversely, suppose that ( π₯ π₯π₯ π₯ ) β πΆ ( π π β π π ) . Once again by Lemma 5.1 this implies β€ π π π₯π π + π π π₯π π + π π π₯π π + π π π₯π π = ( π π + π π ) π₯ ( π π + π π ) = π₯. Deο¬nition 5.4.
Let ( ξ , { πΆ π } π , π ) be an abstract operator system and suppose that β€ π β€ π for some π β ξ + and πΌ π ( π ) = 1 . Set π = π β π . We call π an abstract projection if the map π π βΆ ξ β π ( ξ )β π½ πβπ deο¬ned by π π βΆ π₯ β¦ ( π₯ π₯π₯ π₯ ) + π½ πβπ is a complete order isomorphism.Theorem 5.3, together with Corollary 4.10 and Lemma 5.2 imply that the map π π is a complete order isomorphismwhenever π is a projection in a concrete operator system. In other words, every concrete projection is an abstract projec-tion. It remains to show that every abstract projection is a concrete projection under some complete order embeddingof its containing operator system. We proceed by ο¬rst showing that matrix-valued ucp maps on π ( ξ )β π½ πβπ can bemodiο¬ed to build new matrix-valued ucp maps sending π β π½ πβπ and β π + π½ πβπ to projections. Proposition 5.5.
Suppose that ξ is an operator system and that π is an abstract projection in ξ . Let π = π β π . Thenfor every ucp map π βΆ π ( ξ )β π½ πβπ β π π there exists a π β β and a ucp map π βΆ π ( ξ )β π½ πβπ β π π such that π ( π β π½ πβπ ) and π (0 β π + π½ πβπ ) are projections and satisfying the property that π π ββββββββββ π π π β π βββββ + π π ( π½ πβπ ) βββββ = βββββ π π (( π
00 0 ) + π π ( π½ πβπ ) ) π π (( π ) + π π ( π½ πβπ ) ) π π (( π β ) + π π ( π½ πβπ ) ) π π (( π ) + π π ( π½ πβπ ) )βββββ β₯ if and only if π π (( π ππ β π ) + π π ( π½ πβπ ) ) β₯ for all π, π, π β π π ( ξ ) .Proof. To simplify notation, we will let Μπ₯ denote the coset π₯ + π π ( π½ πβπ ) for each π₯ β π π ( ξ ) throughout the proof.Suppose π βΆ π ( ξ )β π½ πβπ β π π is ucp. Since π ( Μπ β
0) = πΌ π β π ( Μ β π ) , we have that π ( Μπ β commutes with π ( Μ β π ) . Thus we may ο¬nd a common orthonormal basis for β π such that π ( Μπ β and π ( Μ β π ) are both diagonal.By reordering this basis, we may assume π ( Μπ β
0) = Μπ = ββββββ πΌ π π₯ π +1 β± π₯ π β² π β π β² ββββββ , π ( Μ β π ) = Μπ = ββββββ π π¦ π +1 β± π¦ π β² πΌ π β π β² ββββββ β€ π β€ π β² β€ π and π₯ π + π¦ π = 1 for each π = π + 1 , β¦ , π β² and π₯ π , π¦ π β (0 , . Deο¬ne rectangular matrices π = ββββββ πΌ π π₯ β1β2 π +1 β± π₯ β1β2 π β² ββββββ β π π β² ,π , π = ββββββ π¦ β1β2 π +1 β± π¦ β1β2 π β² πΌ π β π β² ββββββ β π π β π,π . Thus
π Μπ π β = πΌ π β² and π Μππ β = πΌ π β π . We may now deο¬ne π βΆ π ( π )β π½ πβπ β π π β² + π β π via π ( π ππ π ) = ( π π ) ββββββ π Μ ( π
00 0 ) π Μ ( π ) π Μ ( π ) π Μ ( π )ββββββ ( π β π β ) . Then π is ucp, π ( Μπ β
0) = πΌ π β² β π β π and π ( Μ β π ) = 0 π β² β πΌ π β π .It remains to check the ο¬nal statement of the proposition. To show this, it suο¬ces to show that the non-zero entriesof π ( Μπ β lie in its upper left π β² Γ π β² corner, the non-zero entries of π ( Μ β π ) lie in its lower right ( π β π ) Γ ( π β π ) corner, and the non-zero entries of π Μ ( π ) lie in its upper right π β² Γ ( π β π ) corner. Indeed, when these statements hold, the map π is simply the compression ofthe matrix ββββββ π Μ ( π
00 0 ) π Μ ( π ) π Μ ( π β ) π Μ ( π )ββββββ to the ( π β² + π β π ) Γ ( π β² + π β π ) submatrix upon which it is supported, followed by conjugation by the invertiblematrix ββββββββββββ πΌ π π₯ β1β2 π +1 β± π₯ β1β2 π β² π¦ β1β2 π +1 β± π¦ β1β2 π β² πΌ π β π β² ββββββββββββ . We ο¬rst consider the coset of the matrix with π in its upper right corner and zeroes elsewhere, where π β ξ . Since Μπ β π is an order unit for π ( ξ )β π½ πβπ , we may assume (by rescaling π if necessary) that ( π ππ β π ) β πΆ ( π β π ) . This implies that βββββ π π π β π βββββ β πΆ ( π β π ) . π implies that ββββββ Μπ π Μ ( π ) π Μ ( π β ) Μπ ββββββ β₯ . The claim follows.Next we consider π ( π β for π = π β . By again rescaling π as necessary, we may assume that ( π Β± π π ) β πΆ ( π β π ) . By the deο¬nition of πΆ ( π β π ) and compressing to the upper left corner, this implies that for every π > there exists a π‘ > such that π Β± π + ππ + π‘π β₯ . Using the deο¬nition of πΆ ( π β π ) again, we see that this implies ( π Β± π
00 0 ) β πΆ ( π β π ) . By the positivity of π , this means that Μπ Β± π Μ ( π
00 0 ) β₯ . It follows that the non-zero entries of π ( Μπ β lie in its upper left π β² Γ π β² corner as claimed. A similar proof showsthat π ( Μ β π ) has its non-zero entries in its lower right ( π β π ) Γ ( π β π ) corner whenever π = π β .We prove one ο¬nal lemma before arriving at the main result of this section. This lemma ensures that the map π π inDeο¬nition 5.4 is unital. Lemma 5.6.
For any positive contraction π in an operator system ξ we have ( π ππ π ) + π½ πβπ = ( π
00 0 ) + π½ πβπ and ( π ππ π ) + π½ πβπ = ( π ) + π½ πβπ where π = π β π . Consequently the map π π is unital.Proof. It suο¬ces to show that Β± ( ππ π ) , Β± ( π ππ ) β πΆ ( π β π ) . Let π > . Then ( ππ π ) + π ( π π ) + 1 π ( π π ) = ( ππ ππ (1 + π ) π ) + ( π π ππ ) = ( π
11 1 + π ) β π + ( π π ) β π β₯ . Also, ( π β π β π ) + π ( π π ) + (1 + 1 π ) ( π π ) = ( ππ β π β π π π ) + ( (1 + π ) π ππ ) = ( π β1β1 π ) β π + ( π π ) β π β₯ . Β± ( π ππ ) β πΆ ( π β π ) is similar. For the ο¬nal statement, we observe that π π ( π ) = ( π ππ π ) + π½ πβπ = ( π ππ π ) + π½ πβπ + ( π ππ π ) + π½ πβπ = ( π π ) + π½ πβπ . Noting that π β π + π½ πβπ is the unit of π ( ξ )β π½ πβπ concludes the proof Theorem 5.7.
Suppose that ξ is an operator system and that π β ξ is an abstract projection. Then there exists aunital complete order embedding π βΆ ξ β π΅ ( π» ) such that π ( π ) is a projection in π΅ ( π» ) .Proof. Since the abstract compression of π ( π ) by π β π is an operator system, the direct sum π = β¨ π, where the direct sum is taken over all ucp π βΆ π ( ξ )β π½ πβπ β π π and all π β β , is a unital complete order embedding(see [3] as well as Chapter 13 of [15]). Replacing each π with a corresponding π as in Proposition 5.5 we obtain aunital completely positive map π β² of π ( ξ )β π½ πβπ into π΅ ( π» ) mapping π β π½ πβπ to a projection. In fact, π β² is acomplete order embedding. To see this, suppose that ( π ππ β π ) + π π ( π½ πβπ ) β π π ( π ( ξ )β π½ πβπ ) is non-positive. Then it is necessarily the case that βββββ π π π β π βββββ + π π ( π½ πβπ ) is also non-positive. It follows that there exists a unital completely positive map π βΆ π ( ξ )β π½ πβπ β π π such that π π ββββββββββ π π π β π βββββ + π π ( π½ πβπ ) βββββ is a non-positive matrix (for example, see the proof of Theorem 13.1 in [15]). It follows from Proposition 5.5 that thecorresponding map π βΆ π ( ξ )β π½ πβπ β π π β² has the property that π ( π ππ β π ) is non-positive. We conlcude that π β² is a unital complete order embedding. We complete the proof by precomposing π β² with the complete order embedding π π so that π = π β² β¦ π π is the desired unital complete order embedding.Theorem 5.7 shows that when π β ξ is an abstract projection, we can build a complete order embedding of ξ into π΅ ( π» ) mapping π to an βhonestβ projection. Of course a given operator system may contain many abstractprojections, and a representation making π into a projection may not map other abstract projections to projections. Thenext theorem shows that there is always one complete order embedding of ξ which maps all abstract projections toconcrete projections. The reader should compare the following with Theorem 2.10. Theorem 5.8.
Let ξ be an operator system, and suppose that π β ξ is an abstract projection. Then π is a projectionin its C*-envelope πΆ β π ( ξ ) . roof. Suppose that π is an abstract projection in ξ , and let π βΆ ξ β πΆ β π ( ξ ) denote the inclusion map. By Theorem5.7, there exists a unital complete order embedding π βΆ ξ β π΅ ( π» ) with the property that π ( π ) is a projection. Let ξ βΆ= πΆ β ( π ( ξ )) . By the universal property of the C*-envelope, there exists a β -epimorphism π βΆ ξ β πΆ β π ( ξ ) satisfying π ( π ( π₯ )) = π ( π₯ ) for all π₯ β ξ . Consequently π ( π ) = π ( π ( π )) = π ( π ( π ) ) = π ( π ( π )) = π ( π ) . Since π ( π ) = π ( π ) β , we conclude that π ( π ) is a projection in πΆ β π ( ξ ) . We conclude with a brief application of our results to the theory of correlation sets in quantum information theory. Wemust ο¬rst brieο¬y recall some deο¬nitions.Let π, π β β be positive integers. We call a tuple { π ( π, π | π₯, π¦ ) βΆ π, π β {1 , , β¦ , π } , π₯, π¦ β {1 , , β¦ , π }} a correlation if π ( π, π | π₯, π¦ ) β₯ for each π, π β€ π and π₯, π¦ β€ π and if, for each π₯, π¦ β€ π , we have β π,π π ( π, π | π₯, π¦ ) = 1 .These conditions ensure that for each choice of π₯ and π¦ , the matrix { π ( π, π | π₯, π¦ )} π,π β€ π constitutes a joint probabilitydistribution. We say that a correlation { π ( π, π | π₯, π¦ )} is non-signalling if for each π₯ β€ π and π β€ π the quantity π π΄ ( π | π₯ ) βΆ= β π π ( π, π | π₯, π¦ ) is well-deο¬ned (i.e. independent of the choice of π¦ ), and that similarly for each π¦ β€ π and π β€ π the quantity π π΅ ( π | π¦ ) βΆ= β π π ( π, π | π₯, π¦ ) is well-deο¬ned. We refer to the integer π as the number of experiments and the integer π as the number of outcomes .We let πΆ ππ ( π, π ) denote the set of non-signalling correlations with π experiments and π outcomes.Correlations model a scenario where two parties, typically named Alice and Bob, are performing probabilisticexperiments. Suppose Alice and Bob each have π experiments, and that each experiment has π possible outcomes.Then the quantity π ( π, π | π₯, π¦ ) denotes the probability that Alice performs experiment π₯ and obtains outcome π whileBob performs experiment π¦ and obtains outcome π . Whenever Alice and Bob perform the experiments independentlywithout communicating to one another, the resulting correlation is non-signalling. It is well-known (and easy to see)that the set πΆ ππ ( π, π ) of non-signalling correlations is a convex polytope when regarded as a subset of β π π in theobvious way.Let π» be a Hilbert space. We call a set { π , π , β¦ , π π } β π΅ ( π» ) a projection-valued measure if each π π is aprojection on π» and β π π π = πΌ . A correlation { π ( π, π | π₯, π¦ )} β πΆ ππ ( π, π ) is called a quantum commuting correlation ifthere exists a Hilbert space π» , a unit vector π β π» , and projection valued measures { πΈ π₯,π } ππ =1 , { πΉ π¦,π } ππ =1 β π΅ ( π» ) foreach π₯, π¦ β€ π satisfying the conditions that πΈ π₯,π πΉ π¦,π = πΉ π¦,π πΈ π₯,π for all π₯, π¦ β€ π and π, π β€ π and π ( π, π | π₯, π¦ ) = β¨ π | πΈ π₯,π πΉ π¦,π π β© . The set of all quantum commuting correlations with π experiments and π outcomes is denoted by πΆ ππ ( π, π ) . It iswell-known that πΆ ππ ( π, π ) is a closed convex subset of πΆ ππ ( π, π ) and that it is not a polytope for any π β₯ or π β₯ .If we modify the deο¬nition of the quantum commuting correlations by requiring the Hilbert space π» to be ο¬nitedimensional, we obtain a quantum correlation. The set of quantum correlations with π experiments and π outcomesare denoted by πΆ π ( π, π ) . The set πΆ π ( π, π ) is known to be a convex subset of πΆ ππ ( π, π ) . It was shown by William Slofstrain [19] that for some values of π and π , πΆ π ( π, π ) is non-closed, and hence πΆ π ( π, π ) is a proper subset of πΆ ππ ( π, π ) .The recent preprint [8] shows that for some ordered pair ( π, π ) the closure of πΆ π ( π, π ) is a proper subset of πΆ ππ ( π, π ) .Precisely which ordered pairs ( π, π ) satisfy this relation remains unknown.One reason questions about πΆ π ( π, π ) and πΆ ππ ( π, π ) are diο¬cult to answer is that these sets are deο¬ned by applyingarbitrary vector states to arbitrary projection-valued measures acting on arbitrary Hilbert spaces. In principle, it maybe easier to understand the sets πΆ π ( π, π ) and πΆ ππ ( π, π ) if there were an equivalent deο¬nition which was independent ofHilbert spaces and Hilbert space operators. The following proposition indicates that such a characterization is, in somesense, possible. 19 roposition 6.1. Let π and π be positive integers. Then the following statements are equivalent.1. { π ( π, π | π₯, π¦ )} β πΆ ππ ( π, π ) (resp. { π ( π, π | π₯, π¦ )} β πΆ π ( π, π ) ).2. There exists a (resp. ο¬nite dimensional) C*-algebra ξ , projection valued measures { πΈ π₯,π } ππ =1 , { πΉ π¦,π } ππ =1 β ξ for each π₯, π¦ β€ π satisfying πΈ π₯,π πΉ π¦,π = πΉ π¦,π πΈ π₯,π for all π₯, π¦ β€ π and π, π β€ π , and a state π βΆ ξ β β such that π ( π, π | π₯, π¦ ) = π ( πΈ π₯,π πΉ π¦,π ) .3. There exists an operator system ξ β π΅ ( π» ) (resp. for a ο¬nite dimensional Hilbert space π» ), projection valuedmeasures { πΈ π₯,π } ππ =1 , { πΉ π¦,π } ππ =1 for each π₯, π¦ β€ π satisfying πΈ π₯,π πΉ π¦,π β ξ and πΈ π₯,π πΉ π¦,π = πΉ π¦,π πΈ π₯,π for all π₯, π¦ β€ π and π, π β€ π , and a state π βΆ ξ β β such that π ( π, π | π₯, π¦ ) = π ( πΈ π₯,π πΉ π¦,π ) .Proof. The equivalence of 1 and 3 are obvious, taking ξ to be the linear span of the operator products πΈ π₯,π πΉ π¦,π . Theequivalence of 1 and 2 is an application of the GNS construction, taking ξ to be the C*-algebra generated by the set { πΈ π₯,π πΉ π¦,π } .In principle, statement 2 of Proposition 6.1 provides an abstract characterization of correlation sets in that it isindependent of Hilbert space representation. However, C*-algebras are themselves complex structures, so it is notclear that statement 2 of Proposition 6.1 is a signiο¬cant improvement over the deο¬nitions of πΆ ππ ( π, π ) and πΆ π ( π, π ) .Moreover, the correlation is generated by applying a state on the C*-algebra ξ to operators of the form πΈ π₯,π πΉ π¦,π , whichspan only a linear subspace of ξ . Thus it seems that one can get by with signiο¬cantly less data than the C*-algebra ξ has to oο¬er. These observations make statement 3 of Proposition 6.1 seem more appealing, except that we have insistedthat the operator system ξ be concretely represented so that we can enforce the relations πΈ π₯,π πΉ π¦,π = πΉ π¦,π πΈ π₯,π and thateach πΈ π₯,π and πΉ π¦,π are projections. Theorem 5.8 provides us with the tools to ensure that πΈ π₯,π and πΉ π¦,π are projectionsin an abstract operator system. It remains to show that the condition πΈ π₯,π πΉ π¦,π = πΉ π¦,π πΈ π₯,π can also be enforced in anabstract operator system. Deο¬nition 6.2.
Let π, π β β . We call an operator system ξ a non-signalling operator system if it is the linear spanof positive operators { π ( π, π | π₯, π¦ ) βΆ π, π β€ π, π₯, π¦ β€ π } β ξ , called the generators or ξ , with the properties that β π,π π ( π, π | π₯, π¦ ) = π for each choice of π₯, π¦ β€ π and that the operators πΈ ( π | π₯ ) βΆ= β π π ( π, π | π₯, π¦ ) and πΉ ( π | π¦ ) βΆ= β π π ( π, π | π₯, π¦ ) are well-deο¬ned (i.e. πΈ ( π | π₯ ) is independent of the choice of π¦ and πΉ ( π | π¦ ) is independent to the choice of π₯ ). We callan operator system ξ a quantum commuting operator system if it is a non-signalling operator system with the propertythat each generator π ( π, π | π₯, π¦ ) is an abstract projection in ξ .The next theorem justiο¬es the choice of terminology in Deο¬nition 6.2. Theorem 6.3.
A correlation { π ( π, π | π₯, π¦ )} is non-signalling (resp. quantum commuting) if and only if there exists anon-signalling (resp. quantum commuting) operator system ξ with generators { π ( π, π | π₯, π¦ )} and a state π βΆ ξ β β such that π ( π, π | π₯, π¦ ) = π ( π ( π, π | π₯, π¦ )) for each π, π, π₯, π¦ .Proof. We ο¬rst verify the equivalence for non-signalling correlations. Suppose that ξ is a non-signalling operatorsystem with generators π ( π, π | π₯, π¦ ) . Let π βΆ ξ β β be a state, and deο¬ne π ( π, π | π₯, π¦ ) βΆ= π ( π ( π, π | π₯, π¦ )) . Then foreach π₯, π¦, π, π we have π ( π ( π, π | π₯, π¦ ) β₯ since π is positive, and for each π₯ and π¦ we have β π,π π ( π ( π, π | π₯, π¦ )) = π ( β π,π π ( π, π | π₯, π¦ )) = π ( π ) = 1 . So { π ( π, π | π₯, π¦ )} is a correlation. Similarly, the quantities π π΄ ( π | π₯ ) and π π΅ ( π | π¦ ) are well-deο¬ned for each π, π, π₯, π¦ with π π΄ ( π | π₯ ) = π ( πΈ ( π | π₯ )) and π π΅ ( π | π¦ ) = π ( πΉ ( π | π¦ )) . So { π ( π, π | π₯, π¦ )} is a non-signalling correlation.20n the other hand, suppose that { π ( π, π | π₯, π¦ )} is a non-signalling correlation. Let π» = β regarded as a one-dimensional Hilbert space. Set π ( π, π | π₯, π¦ ) = π ( π, π | π₯, π¦ ) for each π, π, π₯, π¦ . Clearly if ξ = span{ π ( π, π | π₯, π¦ )} = β = π΅ ( π» ) then ξ is a non-signalling operator system with generators { π ( π, π | π₯, π¦ )} , since π ( π, π | π₯, π¦ ) is a non-signallingcorrelation. Let π βΆ ξ β β be the state π ( π ) = π . Then π ( π ( π, π | π₯, π¦ )) = π ( π, π | π₯, π¦ ) for each π, π, π₯, π¦ .We now consider the equivalence for quantum commuting correlations. The forward direction is immediate fromProposition 6.1. We show the converse. Suppose that ξ is a quantum commuting operator system with generators { π ( π, π | π₯, π¦ )} , and let π βΆ ξ β β be a state. Since ξ is quantum commuting, each π ( π, π | π₯, π¦ ) is an abstract projection.Therefore by Theorem 5.8 each π ( π, π | π₯, π¦ ) is a projection when regarded as an element of ξ βΆ= πΆ β π ( ξ ) . Since β π,π π ( π, π | π₯, π¦ ) = π , it follows that π ( π, π | π₯, π¦ ) π ( π, π | π₯, π¦ ) = 0 in ξ whenever π β π or π β π . Hence πΈ ( π | π₯ ) and πΉ ( π | π¦ ) are also projections in ξ . Moreover, πΈ ( π | π₯ ) πΉ ( π | π¦ ) = ( β π π ( π, π | π₯, π¦ ))( β π π ( π, π | π₯, π¦ )) = π ( π, π | π₯, π¦ ) since π ( π, π | π₯, π¦ ) π ( π, π | π₯, π¦ ) = 0 whenever π β π or π β π . Finally, by the Arveson Extension Theorem, there existsa state Μπ βΆ ξ β β which extends π , since ξ is a unital self-adjoint subspace of ξ . By part 2 of Proposition 6.1, π ( π, π | π₯, π¦ ) = Μπ ( πΈ ( π | π₯ ) πΉ ( π | π¦ )) is a quantum commuting correlation. But Μπ ( πΈ ( π | π₯ ) πΉ ( π | π¦ )) = π ( π ( π, π | π₯, π¦ )) , so theproof is complete. Acknowledgments
The authors would like to thank Ivan Todorov for pointing us towards the reference [13] and Thomas Sinclair for helpfulcomments on an earlier version of this manuscript. This research was initiated at the conference βQLA meets QIT,"which was funded by the NSF grant DMS-1600857 and the Yue Lin Lawrence Tong Endowment Fund, Purdue Uni-versity. The ο¬rst author was supported by a Purdue Research Foundation Grant from the Department of Mathematics,Purdue University.
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