Operator system structures and extensions of Schur multipliers
aa r X i v : . [ m a t h . OA ] D ec OPERATOR SYSTEM STRUCTURES AND EXTENSIONSOF SCHUR MULTIPLIERS
YING-FEN LIN AND IVAN G. TODOROV
Abstract.
For a given C*-algebra A , we establish the existence of max-imal and minimal operator A -system structures on an AOU A -space.In the case A is a W*-algebra, we provide an abstract characterisa-tion of dual operator A -systems, and study the maximal and minimaldual operator A -system structures on a dual AOU A -space. We in-troduce operator-valued Schur multipliers, and provide a Grothendieck-type characterisation. We study the positive extension problem for apartially defined operator-valued Schur multiplier ϕ and, under somerichness conditions, characterise its affirmative solution in terms of theequality between the canonical and the maximal dual operator A -systemstructures on an operator system naturally associated with the domainof ϕ . Introduction
The problem of completing a partially defined matrix to a fully definedpositive matrix has attracted considerable attention in the literature (seee.g. [5] and [8] and the references therein). Given an n by n matrix, only asubset of whose entries are specified, this problem asks whether the remain-ing entries can be determined so as to yield a positive matrix. For blockoperator matrices, this problem was considered in [14], where the authorsshowed that it is closely related to questions about automatic complete posi-tivity of certain positive linear maps. More specifically, one associates to thepattern κ of the partially defined matrix (that is, the set of all given entries)the operator system S ( κ ) of all fully specified matrices supported by κ . Thepositive completion problem is then linked to the question of whether theoperator-valued Schur multiplier with domain S ( κ ) is completely positive.A continuous infinite dimensional version of the scalar-valued completionproblem was considered in [11], where the authors characterised the operatorsystems possessing the positive completion property in terms of an approx-imation of its positive cone via rank one operators. The original motivationbehind the present paper was the study of the operator-valued, infinite di-mensional and continuous, analogue of the positive completion problem. Werelate the question to the automatic complete positivity of operator-valued Date : 2 December 2018.
Schur multipliers; in fact, we characterise the extendability of Schur mul-tipliers in terms of an equality between operator system structures on anassociated Archimedean order unit (AOU) *-vector space.One of the fundamental representation theorems in Operator Space The-ory is Choi-Effros Theorem [13, Theorem 13.1], which characterises operatorsystems (that is, unital selfadjoint linear subspaces S of the space B ( H ) ofall bounded linear operators on a Hilbert space H ) abstractly, in terms ofproperties of the cones of positive elements in the S -valued matrix space M n ( S ). Operator A -systems, that is, the operator systems which admit abimodule action by a unital C*-algebra A , can be characterised similarly ina way that takes into account the extra A -module structure [13, Corollary15.13]. Dual operator systems – that is, operator systems that are also dualoperator spaces – were characterised by D. P. Blecher and B. Magajna in [4].However, no analogous representation of dual operator A -systems, where A is a W*-algebra, has been known.The idea of viewing operator spaces as a quantised version of Banachspaces has been very fruitful in Functional Analysis [6]. Operator systemscan in a similar vein be thought of as a quantised version of Archimedeanorder unit (AOU) *-vector spaces. The possible quantisations, or operatorsystem structures, on a given AOU space, were first studied in [15], whereit was shown that every AOU space possesses two extremal operator systemstructures. However, no similar development has been achieved for dualAOU spaces or for AOU A -spaces.In this paper, we unify all aforementioned strands of questions. We pro-vide a Choi-Effros type representation theorem for dual operator A -systems.We study the operator A -system structures on a given AOU A -space, as wellas the dual operator A -system structures on a given dual AOU A -space. Thelatter results are new even in the case where A coincides with the complexfield. We introduce infinite dimensional measurable operator-valued Schurmultipliers, and provide a characterisation that generalises their well-knowndescription by A. Grothendieck [9] in the scalar case (see also [10] and [17]).Finally, we study the positive extension problem for operator-valued Schurmultipliers, and characterise the possibility of such an extension by equalityof the canonical and the maximal dual operator D -system structures on thedomain of the given Schur multiplier. Our context is that of an arbitrary(albeit standard) measure space ( X, µ ), which includes as a sub-case thediscrete case and thus the finite case considered in [14]. In this context,the algebra D is the maximal abelian selfadjoint algebra corresponding to L ∞ ( X, µ ). Our results are a far reaching generalisation of the results of V.I. Paulsen, S. Power and R. R. Smith [14]; in particular, they provide a dif-ferent view on the positive completion problem for block operator matricesconsidered therein.The paper is organised as follows. After collecting some preliminaries inSection 2, we establish, in Section 3, the existence of the minimal and themaximal operator A -system structures on a AOU A -space V , OMIN A ( V ) PERATOR SYSTEM STRUCTURES 3 and OMAX A ( V ). In case V is a C*-algebra, OMIN A ( V ) was essentiallydefined in [20], in relation with the problem of automatic complete positivityof A -module maps, whose completely bounded version was first consideredby R. R. Smith in [23] (see also the subsequent paper [19]). We show thatOMAX A ( V ) (resp. OMIN A ( V )) is characterised by the automatic completepositivity of A -bimodule positive maps from V into any operator A -system(resp. from any operator A -system into V ).In Section 4, we provide a characterisation theorem for dual operator A -systems and, in Section 5, we define dual AOU A -spaces and undertake adevelopment, analogous to the one in Section 3, for dual operator A -systemstructures.In Section 6, we introduce the operator-valued version of measurableSchur multipliers and provide a Grothendieck-type characterisation, notingthe special case of positive Schur multipliers. In Section 7, we study par-tially defined operator-valued Schur multipliers and their extension proper-ties to a fully defined positive Schur multiplier. Associated with the domain κ ⊆ X × X of the Schur multiplier is an operator system S ( κ ). Our analysisdepends on the presence of sufficiently many operators of finite rank in S ( κ ).We note that, of course, this holds true trivially in the classical matrix case.Under such richness conditions on the domain κ , we show that the positiveextension problem for operator-valued Schur multipliers defined on κ has anaffirmative solution precisely when the canonical operator system structureof S ( κ ) coincides with its maximal dual operator D -system structure.We denote by ( · , · ) the inner product in a Hilbert space, and we use h· , ·i todesignate duality paring. We will assume some basic facts and notions fromOperator Space Theory, for which we refer the reader to the monographs[3, 6, 13, 18]. 2. Preliminaries
In this section we recall basic results and introduce some new notions thatwill be needed subsequently. If W is a real vector space, a cone in W is anon-empty subset C ⊆ W with the following properties:(a) λv ∈ C whenever λ ∈ R + := [0 , ∞ ) and v ∈ C ;(b) v + w ∈ C whenever v, w ∈ C .A *-vector space is a complex vector space V together with a map ∗ : V → V which is involutive (i.e. ( v ∗ ) ∗ = v for all v ∈ V ) and conjugate linear (i.e.( λv + µw ) ∗ = λv ∗ + µw ∗ for all λ, µ ∈ C and all v, w ∈ V ). If V is a *-vectorspace, then we let V h = { x ∈ V : x ∗ = x } and call the elements of V h hermitian . Note that V h is a real vector space.An ordered *-vector space [16] is a pair ( V, V + ) consisting of a *-vectorspace V and a subset V + ⊆ V h satisfying the following properties:(a) V + is a cone in V h ;(b) V + ∩ − V + = { } . Y.-F. LIN AND I. G. TODOROV
Let (
V, V + ) be an ordered *-vector space. We write v ≥ w or w ≤ v if v, w ∈ V h and v − w ∈ V + . Note that v ∈ V + if and only if v ≥
0; for thisreason V + is referred to as the cone of positive elements of V .An element e ∈ V h is called an order unit if for every v ∈ V h there exists r > v ≤ re . The order unit e is called Archimedean if, whenever v ∈ V and re + v ∈ V + for all r >
0, we have that v ∈ V + . In thiscase, we call the triple ( V, V + , e ) an Archimedean order unit *-vector space ( AOU space for short). Note that ( C , R + ,
1) is an AOU space in a canonicalfashion.Let A be a unital C*-algebra. Recall that a (complex) vector space V issaid to be an A -bimodule if it is equipped with bilinear maps A × V → V ,( a, x ) → a · x and V × A → V , ( x, a ) → x · a , such that ( a · x ) · b = a · ( x · b ),( ab ) · x = a · ( b · x ), x · ( ab ) = ( x · a ) · b and 1 · x = x for all x ∈ V and all a, b ∈ A . If V and W are A -bimodules, a linear map φ : V → W is calledan A -bimodule map if φ ( a · x · b ) = a · φ ( x ) · b , for all x ∈ V and all a, b ∈ A . Definition 2.1.
Let A be a unital C*-algebra. An AOU space ( V, V + , e ) will be called an AOU A -space if V is an A -bimodule and the conditions (1) ( a · x ) ∗ = x ∗ · a ∗ , x ∈ V, a ∈ A , (2) a · e = e · a, a ∈ A , and (3) a ∗ · x · a ∈ V + , x ∈ V + , a ∈ A , are satisfied. For a complex vector space V , we let M m,n ( V ) denote the complex vectorspace of all m by n matrices with entries in V , and often use the naturalidentification M m,n ( V ) ≡ M m,n ⊗ V . We write A t for the transpose of amatrix A ∈ M m,n ( V ). We set M n ( V ) = M n,n ( V ), M m,n = M m,n ( C ) and M n = M n ( C ); we write I n for the identity matrix in M n . If V is an AOU A -space, we equip M n ( V ) with an involution by letting ( x i,j ) ∗ = ( x ∗ j,i ) andset(4)( a i,j ) · ( x i,j ) = n X p =1 a i,p · x p,j i,j and ( x i,j ) · ( b i,j ) = n X p =1 x i,p · b p,j i,j , whenever ( x i,j ) ∈ M m,n ( V ), ( a i,j ) ∈ M k,m ( A ) and ( b i,j ) ∈ M n,l ( A ), m, n , k, l ∈ N .Let A be a unital C*-algebra and ( V, V + , e ) be an AOU A -space. Wewrite e n for the element of M n ( V ) whose diagonal entries coincide with e ,while its off-diagonal entries are equal to zero. A family ( P n ) n ∈ N , where P n ⊆ M n ( V ) h is a cone with P n ∩ ( − P n ) = { } , n ∈ N , will be called a matrix ordering of V . A matrix ordering ( P n ) n ∈ N will be called an operator PERATOR SYSTEM STRUCTURES 5 A -system structure on V if P = V + ,(5) A ∗ · X · A ∈ P n , whenever X ∈ P m and A ∈ M m,n ( A ) , and e n ∈ M n ( V ) is an Archimedean order unit for P n for every n ∈ N .Condition (5) will be referred to as the A -compatibility of ( P n ) n ∈ N . Thetriple S = ( V, ( P n ) n ∈ N , e ) is called an operator A -system (see [13]); we write M n ( S ) + = P n . Note that if B ⊆ A is a unital C*-subalgebra, then everyoperator A -system is also an operator B -system in a canonical fashion. Op-erator C -systems are called simply operator systems . We note that everyoperator system has a canonical operator space structure (see [13]). Notethat condition (2) is not a part of the standard definition of an operator A -system; it is however automatically satisfied, as easily follows from Theorem2.2 below.Let H be a Hilbert space and B ( H ) be the space of all bounded linear op-erators on H . We write B ( H ) + for the cone of all positive operators in B ( H ).We identify M n ( B ( H )) with B ( H n ), where H n denotes the direct sum of n copies of H , and write M n ( B ( H )) + = B ( H n ) + , n ∈ N . It is straightforwardto see that B ( H ) is an operator system when equipped with the adjointoperation as an involution, the matrix ordering ( M n ( B ( H )) + ) n ∈ N , and theidentity operator I as an Archimedean matrix order unit.Given AOU spaces ( V, V + , e ) and ( W, W + , f ), a linear map φ : V → W is called unital if φ ( e ) = f , and positive if φ ( V + ) ⊆ W + . A linear map s : V → C is called a state on V if s is unital and positive.Let S and T be operator systems with units e and f , respectively. For alinear map φ : S → T , we let φ ( n,m ) : M n,m ( S ) → M n,m ( T ) be the (linear)map given by φ ( n,m ) (( x i,j ) i,j ) = ( φ ( x i,j )) i,j , and set φ ( n ) = φ ( n,n ) . The map φ is called n -positive if φ ( n ) is positive, and it is called completely positive ifit is n -positive for all n ∈ N . A bijective completely positive map φ : S →T is called a complete order isomorphism if its inverse φ − is completelypositive. In this case, we call S and T are completely order isomorphic;if φ is moreover unital, we say that S and T are unitally completely orderisomorphic. Further, φ is called a complete isometry if φ ( n ) is an isometry foreach n ∈ N . We note that a unital surjective map φ : S → T is a completeisometry if and only if it is a complete order isomorphism [3, 1.3.3].We refer the reader to [13] for the general theory of operator systems andoperator spaces, and in particular for the definition and basic properties ofcompletely bounded maps. The following characterisation, extending thewell-known Choi-Effros representation theorem for operator systems [13,Theorem 13.1], was established in [13, Corollary 15.12].
Theorem 2.2.
Let A be a unital C*-algebra and S be an operator system.The following are equivalent:(i) S is unitally completely order isomorphic to an operator A -system; Y.-F. LIN AND I. G. TODOROV (ii) there exist a Hilbert space H , a unital complete isometry γ : S →B ( H ) and a unital *-homomorphism π : A → B ( H ) such that γ ( a · x ) = π ( a ) γ ( x ) for all x ∈ S and all a ∈ A . We note that, if A is a unital C*-algebra and S is an operator systemthat is also an operator A -bimodule satisfying (1), then S is an operator A -system precisely when the family ( M n ( S ) + ) n ∈ N is A -compatible.3. The extremal operator A -system structures In this section, we show that any AOU A -space can be equipped withtwo extremal operator A -system structures, and establish their universalproperties. We first consider the minimal operator A -system structure. Notethat, in the case where the AOU A -space is a C*-algebra containing A , thisoperator system structure was first defined and studied in [20].Let A be a unital C*-algebra and ( V, V + , e ) be an AOU A -space. For n ∈ N , let C min n ( V ; A ) = { X ∈ M n ( V ) h : C ∗ · X · C ∈ V + , for all C ∈ M n, ( A ) } . Remark 3.1.
Suppose that (
V, V + , e ) is an AOU A -space and that B isa unital C*-subalgebra of A . Then ( V, V + , e ) is also an AOU B -space inthe natural fashion. Clearly, C min n ( V ; A ) ⊆ C min n ( V ; B ). In particular, C min n ( V ; A ) is contained in C min n ( V ; C ); note that the latter set coincideswith the cone C min n ( V ) introduced in [15, Definition 3.1]. Theorem 3.2.
Let A be a unital C*-algebra and ( V, V + , e ) be an AOU A -space. Then ( C min n ( V ; A )) n ∈ N is an operator A -system structure on V .Moreover, if ( P n ) n ∈ N is an operator A -system structure on V then P n ⊆ C min n ( V ; A ) for each n ∈ N .Proof. Since V + is a cone, C min n ( V ; A ) is a cone, too. As a consequence of[15, Theorem 3.2] and Remark 3.1, C min n ( V ; A ) ∩ ( − C min n ( V ; A )) = { } . If X ∈ C min m ( V ; A ), A ∈ M m,n ( A ) and C ∈ M n, ( A ) then AC ∈ M m, ( A ) andhence C ∗ · ( A ∗ · X · A ) · C = ( AC ) ∗ · X · ( AC ) ∈ V + , showing that A ∗ · X · A ∈ C min n ( V ; A ). Thus, the family ( C min n ( V ; A )) n ∈ N is A -compatible.Suppose that ( P n ) n ∈ N is an operator A -system structure on V . If X ∈ P n then, by A -compatibility, C ∗ · X · C ∈ P = V + , and hence X ∈ C min n ( V ; A ).Thus, P n ⊆ C min n ( V ; A ). It will follow from the proof of Theorem 3.7 belowthat e n is an order unit for C min n ( V ; A ). To see that e n is Archimedean,suppose that X + re n ∈ C min n ( V ; A ) for every r >
0. Let C ∈ M n, ( A ).Using (2), we have C ∗ · X · C + rC ∗ C · e = C ∗ · ( X + re n ) · C ∈ V + , for all r > . Let ǫ > T = ( C ∗ C + ǫ − / ∈ A . We have that C ∗ · X · C + rC ∗ C · e + rǫe ∈ V + , for all r > PERATOR SYSTEM STRUCTURES 7 and hence, by (2) and (3), T ( C ∗ · X · C ) T + re ∈ V + , for all r > . Since e is Archimedean for V + , we have that T ( C ∗ · X · C ) T ∈ V + . Applying(3) again, we conclude that C ∗ · X · C = T − ( T ( C ∗ · X · C ) T ) T − ∈ V + ;thus X ∈ C min n ( V ; A ) and the proof is complete. (cid:3) We call ( C min n ( V ; A )) n ∈ N the minimal operator A -system structure on V ,and let OMIN A ( V ) = (cid:0) V, ( C min n ( V ; A )) n ∈ N , e (cid:1) . The following theorem describes its universal property. Part (i) below wasestablished in [20] in the case V is a C*-algebra containing A . Theorem 3.3.
Let A be a unital C*-algebra and ( V, V + , e ) be an AOU A -space.(i) Suppose that S is an operator A -system and φ : S → V is a posi-tive A -bimodule map. Then φ is completely positive as a map from S into OMIN A ( V ) .(ii) If T is an operator A -system with underlying space V and positivecone V + , such that for every operator A -system S , every positive A -bimodulemap φ : S → T is completely positive, then there exists a unital A -bimodulemap ψ : T →
OMIN A ( V ) that is a complete order isomorphism.Proof. (i) Let S be an operator A -system and φ : S → V be a positive A -bimodule map. Suppose that X = ( x i,j ) ∈ M n ( S ) + and C = ( a i ) ni =1 ∈ M n, ( A ). Then C ∗ · X · C ∈ S + ; since φ is a positive A -bimodule map, wehave C ∗ · φ ( n ) ( X ) · C = n X i,j =1 a ∗ i · φ ( x i,j ) · a j = φ n X i,j =1 a ∗ i · x i,j · a j = φ ( C ∗ · X · C ) ∈ V + . Thus, φ ( n ) maps M n ( S ) + into C min n ( V ; A ) and hence φ is completely positive.(ii) Suppose that the operator A -system T satisfies the properties in (ii).Since the identity id : OMIN A ( V ) → V is a positive A -bimodule map,we have that id : OMIN A ( V ) → T is completely positive. On the otherhand, the identity id : T → V is also positive and A -bimodular. By (i),id : T →
OMIN A ( V ) is completely positive, and we can take ψ = id. (cid:3) We next consider the maximal operator A -system structure. For n ∈ N ,set D max n ( V ; A ) = ( k X i =1 A ∗ i · x i · A i : k ∈ N , x i ∈ V + , A i ∈ M ,n ( A ) ) and let D max ( V ; A ) = ( D max n ( V ; A )) n ∈ N . Y.-F. LIN AND I. G. TODOROV
Remark 3.4.
Suppose that (
V, V + , e ) is an AOU A -space and that B is aunital C*-subalgebra of A . Clearly, D max n ( V ; B ) ⊆ D max n ( V ; A ). Given anyAOU space ( V, V + , e ), in [15] the authors defined D max n ( V ) = ( k X i =1 B i ⊗ x i : k ∈ N , x i ∈ V + , B i ∈ M + n ) . Since every matrix B ∈ M + n is the sum of matrices of the form A ∗ A , where A ∈ M ,n , we have that D max n ( V ) = D max n ( V ; C Lemma 3.5.
Let A be a unital C*-algebra and ( V, V + , e ) be an AOU A -space. Let P n ⊆ M n ( V ) h be a cone, n ∈ N , such that the family ( P n ) ∞ n =1 is A -compatible and P = V + . Then D max n ( V ; A ) ⊆ P n , for each n ∈ N .Proof. Let n ∈ N . If A ∈ M ,n ( A ) then A ∗ · V + · A = A ∗ · P · A ⊆ P n . Thus D max n ( V ; A ) ⊆ P n . (cid:3) If x , . . . , x n ∈ V we let diag( x , . . . , x n ) denote the element of M n ( V )with x , . . . , x n on its diagonal (in this order) and zeros elsewhere. Proposition 3.6.
Let A be a unital C*-algebra and ( V, V + , e ) be an AOU A -space. The following hold:(i) D max n ( V ; A ) = { A ∗ · diag( x , . . . , x m ) · A : A ∈ M m,n ( A ) , x i ∈ V + , i =1 , . . . , m, m ∈ N } ;(ii) D max ( V ; A ) is an A -compatible matrix ordering on V and e is a ma-trix order unit for it.Proof. (i) Let D n denote the right hand side of the equality in (i). Wefirst observe that D n is a cone in M n ( V ) h . If x , . . . , x m ∈ V + and A =( a i,k ) i,k ∈ M m,n ( A ) then the ( i, j )-entry of A ∗ · diag( x , . . . , x m ) · A is equalto P mk =1 a ∗ k,i · x k · a k,j and, by (1), m X k =1 a ∗ k,i · x k · a k,j ! ∗ = m X k =1 a ∗ k,j · x k · a k,i ;thus, D n ⊆ M n ( V ) h . It is clear that D n is closed under taking multipleswith non-negative real numbers. Fix elements A ∗ · diag( x , . . . , x m ) · A, and B ∗ · diag( y , . . . , y k ) · B of D n . Letting C = [ A B ] t , we have A ∗ · diag( x , . . . , x m ) · A + B ∗ · diag( y , . . . , y k ) · B = C ∗ · diag( x , . . . , x m , y , . . . , y k ) · C ∈ D n ;in other words, D n is a cone. If B ∈ M n,l ( A ) then B ∗ · ( A ∗ · diag( x , . . . , x m ) · A ) · B = ( AB ) ∗ · diag( x , . . . , x m ) · ( AB ) ∈ D l , and so ( D n ) ∞ n =1 is A -compatible. By (3), D = V + . Lemma 3.5 now impliesthat D max n ( V ; A ) ⊆ D n for n ∈ N . PERATOR SYSTEM STRUCTURES 9
On the other hand, if x , . . . , x m ∈ V + then, letting E i ∈ M ,m ( A ) be therow with 1 at the i th coordinate and zeros elsewhere, we have thatdiag( x , . . . , x m ) = m X i =1 E ∗ i · x i · E i ∈ D max m ( V ; A ) . Since the family D max ( V ; A ) is A -compatible, A ∗ · diag( x , . . . , x m ) · A ∈ D max n ( V ; A ) , A ∈ M m,n ( A ) . Thus, D n ⊆ D max n ( V ; A ) and (i) is established.(ii) By Remark 3.4 and [15, Proposition 3.10], e n is an order unit for D max n ( V ; C e n is an order unit for D max n ( V ; A ). (cid:3) For n ∈ N , let C max n ( V ; A ) = { X ∈ M n ( V ) : X + re n ∈ D max n ( V ; A ) for every r > } . Theorem 3.7.
Let A be a unital C*-algebra and ( V, V + , e ) be an AOU A -space. Then ( C max n ( V ; A )) n ∈ N is an operator A -system structure on V .Moreover, if ( P n ) n ∈ N is an operator A -system structure on V then C max n ( V ; A ) ⊆ P n for each n ∈ N .Proof. Write C n = C max n ( V ; A ), n ∈ N . By Theorem 3.2 and Lemma 3.5, C n ⊆ C min n ( V ; A ); thus, C n ∩ ( − C n ) = { } . Since e n is an order unit for D max n ( V ; A ) and D max n ( V ; A ) ⊆ C n , we have that e n is an order unit for C n .Suppose that X ∈ M n ( V ) h is such that X + re n ∈ C n for every r > ǫ >
0; then X + ǫe n = (cid:16) X + ǫ e n (cid:17) + ǫ e n ∈ D max n ( V ; A )and hence X ∈ C n . Thus, e n is an Archimedean matrix order unit for C n .It remains to show that the family ( C n ) n ∈ N is A -compatible. To this end,let X ∈ C n for some n ∈ N and A ∈ M n,m ( A ). By Proposition 3.6, thereexists R > Re m − A ∗ · e n · A ∈ D max m ( V ; A ) . Let r >
0. Since X + rR e n ∈ D max n ( V ; A ) and the family D max ( V ; A ) is A -compatible (Proposition 3.6), we have A ∗ · X · A + re m = (cid:16) A ∗ · (cid:16) X + rR e n (cid:17) · A (cid:17) + r (cid:18) e m − R A ∗ · e n · A (cid:19) ∈ D max m ( V ; A ) . It follows that A ∗ · X · A ∈ C m . Thus, ( C n ) n ∈ N is an operator A -systemstructure on V .Suppose that ( P n ) n ∈ N is an operator A -system structure on V and X ∈ C n for some n ∈ N . By Lemma 3.5, X + re n ∈ P n for all r > e n is anArchimedean order unit for P n , we conclude that X ∈ P n . Thus, C n ⊆ P n ,and the proof is complete. (cid:3) We call ( C max n ( V ; A )) n ∈ N the maximal operator A -system structure on V and let OMAX A ( V ) = ( V, ( C max n ( V ; A )) n ∈ N , e ) . Remark.
Recall that, given an AOU space (
V, V + , e ), the maximal op-erator system structure ( C max n ( V )) n ∈ N on V was defined in [15] by letting C max n ( V ) be the Archimedeanisation of the cone D max n ( V ) defined in Remark3.4. It follows that the maximal operator system OMAX( V ) defined in [15]coincides with OMAX C ( V ). Theorem 3.8.
Let A be a unital C*-algebra and ( V, V + , e ) be an AOU A -space.(i) Suppose that S is an operator A -system and φ : V → S is a positive A -bimodule map. Then φ is completely positive as a map from OMAX A ( V ) into S .(ii) Suppose that T is an operator A -system with underlying space V andpositive cone V + , such that for every operator A -system S , every positive A -bimodule map φ : T → S is completely positive. Then there exists a unital A -bimodule map ψ : T →
OMAX A ( V ) that is a complete order isomorphism.Proof. (i) Let S is an operator A -system and φ : V → S be a posi-tive A -bimodule map. The modularity property of φ and the definitionof D max n ( V ; A ) imply that φ ( n ) ( D max n ( V ; A )) ⊆ M n ( S ) + . Suppose that X ∈ C max n ( V ; A ). Letting z = φ ( e ), we now have that φ ( n ) ( X ) + r ( z ⊗ I n ) ∈ M n ( S ) + for every r >
0. Since M n ( S ) + is closed, this implies that φ ( n ) ( X ) ∈ M n ( S ) + . Thus, φ is completely positive.(ii) is similar to the proof of Theorem 3.3 (ii). (cid:3) Remark.
Let A be a C*-algebra and A A (resp. S A ) be the category, whoseobjects are AOU A -spaces (resp. operator A -systems) and whose morphismsare unital positive (resp. unital completely positive) maps. It is easy tosee that the correspondences V → OMIN A ( V ) and V → OMAX A ( V ) arecovariant functors from A A into S A .We finish this section with considering the case where V = M k and A coincides with its subalgebra D k of all diagonal matrices. Proposition 3.9.
We have that M k = OMIN D k ( M k ) = OMAX D k ( M k ) .Proof. Suppose that X = ( X i,j ) i,j belongs to M n (OMIN D k ( M k )) + . Let ξ = ( λ i, , . . . , λ i,k ) ni =1 be a vector in C nk . Let D i = diag( λ i, , . . . , λ i,k ), andwrite ξ i for the vector ( λ i, , . . . , λ i,k ) in C k , i = 1 , . . . , n . Letting e be thevector in C k with all entries equal to one, we have( Xξ, ξ ) = n X i,j =1 ( X i,j ξ j , ξ i ) = n X i,j =1 ( D ∗ i X i,j D j e, e ) . It follows by the assumption that (
Xξ, ξ ) ≥
0; thus, X ∈ M + nk and, byTheorem 3.2, M k = OMIN D k ( M k ). PERATOR SYSTEM STRUCTURES 11
Now fix X = ( X i,j ) i,j ∈ M + nk . Since X is the sum of rank one operatorsin M + nk , in order to show that X ∈ M n (OMAX D k ( M k )) + , it suffices toassume that X is itself of rank one. Write X = RR ∗ , where R ∈ M nk, ,and suppose that R = ( R , . . . , R n ) t , where R i ∈ M k, , i = 1 , . . . , n . Wehave that X = ( R i R ∗ j ) ni,j =1 . Let J ∈ M k be the matrix with all its entriesequal to one, and let D i be the diagonal matrix whose entries coincideswith the vector R i , i = 1 , . . . , n . Then X = ( D i J D ∗ j ) ni,j =1 , showing that X ∈ M n (OMAX D k ( M k )) + . By Theorem 3.7, M k = OMAX D k ( M k ). (cid:3) Remark.
We note that the minimal and the maximal operator A -systemstructure are in general distinct. Indeed, this is the case even when V = M k and A = C I [15]. 4. Dual operator A -systems In this section, we establish a representation theorem for dual operator A -systems. An operator system S is called a dual operator system if it is adual operator space, that is, if there exists an operator space S ∗ such that( S ∗ ) ∗ ∼ = S completely isometrically [4]. Here, and in the sequel, we denoteby X ∗ the operator space dual [3] of an operator space X , and we use thesame notation for the dual Banach space of a normed space X ; it will beclear from the context with which category we are working.Let S be an operator system. If H is a Hilbert space and φ : S → B ( H ) isa unital complete isometry such that φ ( S ) is weak* closed, then φ ( S ), andtherefore S , is a dual operator space; thus, in this case, S is a dual operatorsystem. The converse statement was established by Blecher and Magajnain [4]. Theorem 4.1 ([4]) . If S is a dual operator system then there exists a Hilbertspace H , a weak* closed operator system U ⊆ B ( H ) and a unital surjectivecomplete order isomorhism φ : S → U that is also a a weak* homeomor-phism.
Remark 4.2.
Suppose that S is a dual operator system and S ∗ is an op-erator space such that, up to a complete isometry, S = ( S ∗ ) ∗ . Then M n ( S )is an operator system in a canonical fashion; in fact, if S ⊆ B ( H ) for someHilbert space H , then M n ( S ) ⊆ B ( H n ). By [3, 1.6.2], up to a completeisometry, M n ( S ) = ( S ∗ ˆ ⊗ M ∗ n ) ∗ , where ˆ ⊗ is the projective operator spacetensor product. It follows that M n ( S ) is a dual operator system, and itscanonical weak* topology coincides with the topology of entry-wise weak*convergence: for a net (( x αi,j ) i,j ) α ⊆ M n ( S ) and an element ( x i,j ) i,j ∈ M n ( S ),we have (cid:0) ( x αi,j (cid:1) i,j ) α → w ∗ α ( x i,j ) i,j ⇐⇒ (cid:10) x αi,j , φ (cid:11) → α h x i,j , φ i , i, j = 1 , . . . , n, φ ∈ S ∗ . Recall that a
W*-algebra is a C*-algebra that is also a dual Banachspace; by Sakai’s Theorem [21], every W*-algebra possesses a faithful *-representation on a Hilbert space H , whose image is a von Neumann al-gebra (that is, a weak* closed subalgebra of B ( H ) containing the identityoperator), which is also a weak* homeomorphism. Definition 4.3.
Let A be a W*-algebra. An operator system S will be calleda dual operator A -system if (i) S is an operator A -system, (ii) S is a dual operator system, and (iii) the map from A × S into S , sending the pair ( a, x ) to a · x , is sepa-rately weak* continuous. Note that, if S is a dual operator system then the involution is weak*continuous, and thus (1) implies that if S is in addition a dual operator A -system then the map A × S × A → S , ( a, x, b ) → a · x · b, is separately weak* continuous.If S and T are dual operator systems, a linear map φ : S → T will becalled normal if it is weak* continuous. Suppose that H is a Hilbert space, γ : S → B ( H ) is a unital complete order isomorphism such that γ ( S ) isweak* closed and γ : S → γ ( S ) is a weak* homeomorphism, and π : A →B ( H ) is a unital normal *-homomorphism such that γ ( a · x ) = π ( a ) γ ( x ) forall x ∈ S and all a ∈ A . It is clear that, in this case, S is a dual operator A -system. Theorem 4.7 below establishes the converse of this fact. Theresult is both a weak* version of Theorem 2.2 and an A -module version ofTheorem 4.1.We will need two lemmas. Recall that, if A is a W*-algebra and n ∈ N then M n ( A ) is a W*-algebra in a canonical way. Remark 4.4.
Let A be a W*-algebra and S be a dual operator A -system.It is straightforward to verify that M n ( S ) is a dual operator M n ( A )-system,when it is equipped with the action defined in (4). Lemma 4.5.
Let A be a W*-algebra, S be a dual operator A -system and φ : S → C be a normal state. Then the functional ω : A → C given by ω ( a ) = φ ( a · , a ∈ A , is a normal state of A and (6) | φ ( a · x · b ) | ≤ ω ( aa ∗ ) / ω ( b ∗ b ) / , for all a ∈ M ,m ( A ) , b ∈ M m, ( A ) , x ∈ M m ( S ) with k x k ≤ , and m ∈ N .Proof. Let H , γ and π be as in Theorem 2.2, and let φ ′ : γ ( S ) → C be givenby φ ′ ( γ ( x )) = φ ( x ), x ∈ S . If a, b ∈ A then ω ( ab ) = φ (( ab ) ·
1) = φ ′ ( γ (( ab ) · φ ′ ( π ( ab ) γ (1)) = φ ′ ( π ( ab ))= φ ′ ( π ( a ) γ (1) π ( b )) = φ ′ ( γ ( a · · b )) = φ ( a · · b ) . PERATOR SYSTEM STRUCTURES 13
Thus, ω ( a ∗ a ) = φ ( a ∗ · · a ) ≥ a ∈ A , and hence ω is positive.Moreover, ω (1) = φ (1) = 1 and hence ω is a state. By the separate weak*continuity of the A -module action on S , the state ω is normal.Suppose that φ ′ has the form φ ′ ( T ) = ∞ X i =1 ( T ξ i , ξ i ) , T ∈ γ ( S ) , where ( ξ i ) i ∈ N ⊆ H with P ∞ i =1 k ξ i k = 1. If x ∈ M m ( S ), k x k ≤ a ∈ M ,m ( A ) and b ∈ M m, ( A ), then | φ ( a · x · b ) | = (cid:12)(cid:12)(cid:12) φ ′ (cid:16) π (1 ,m ) ( a ) γ ( m ) ( x ) π ( m, ( b ) (cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X i =1 (cid:16) π (1 ,m ) ( a ) γ ( m ) ( x ) π ( m, ( b ) ξ i , ξ i (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X i =1 (cid:12)(cid:12)(cid:12)(cid:16) γ ( m ) ( x ) π ( m, ( b ) ξ i , π ( m, ( a ∗ ) ξ i (cid:17)(cid:12)(cid:12)(cid:12) ≤ ∞ X i =1 (cid:13)(cid:13)(cid:13) π ( m, ( b ) ξ i (cid:13)(cid:13)(cid:13) ! / ∞ X i =1 (cid:13)(cid:13)(cid:13) π ( m, ( a ∗ ) ξ i (cid:13)(cid:13)(cid:13) ! / = φ ′ ( π ( b ∗ b )) / φ ′ ( π ( aa ∗ )) / = ω ( aa ∗ ) / ω ( b ∗ b ) / . (cid:3) We will need the following modification of a result of R. R. Smith [23] onautomatic complete boundedness. Its proof is a straightforward modificationof the proof of [23, Theorem 2.1] and is hence omitted.
Theorem 4.6.
Let A be a unital C*-algebra, S be an operator A -systemand ρ : A → B ( H ) be a cyclic *-representation. Suppose that Φ :
S → B ( H ) is a linear map such that Φ( a · x · b ) = ρ ( a )Φ( x ) ρ ( b ) for all x ∈ S and all a, b ∈ A . If Φ is contractive then Φ is completely contractive. Theorem 4.7.
Let A be a W*-algebra and S be a dual operator A -system.Then there exist a Hilbert space H , a unital complete order embedding γ : S → B ( H ) with the property that γ ( S ) is weak* closed and γ is a weak*homeomorphism, and a unital normal *-homomorphism π : A → B ( H ) ,such that (7) γ ( a · x ) = π ( a ) γ ( x ) , x ∈ S , a ∈ A . Proof.
The proof is motivated by the proof of [4, Theorem 1.1] and relies onideas which go back to the proof of Ruan’s Theorem [6, Theorem 2.3.5]. Fix n ∈ N and let B = M n ( A ). By Remark 4.4, M n ( S ) is a dual operator B -system. Let x ∈ M n ( S ) be a selfadjoint element of norm one and ǫ ∈ (0 , φ on M n ( S ) such that(8) | φ ( x ) | > − ǫ. Let ω : B → C be the normal state given by ω ( b ) = φ ( b · b ∈ B . ByLemma 4.5,(9) | φ ( a · y · b ) | ≤ ω ( aa ∗ ) / ω ( b ∗ b ) / , for all y ∈ M nm ( S ) with k y k ≤ a ∈ M ,m ( B ) and b ∈ M m, ( B ), m ∈ N .Let ρ : B → B ( H ) be the GNS representation arising from ω and ξ be itscorresponding unit cyclic vector. By [24, Proposition III.3.12], ρ is normal.It follows that there exists a normal unital *-representation θ : A → B ( K )such that, up to unitary equivalence, H = K ⊗ C n and ρ = θ ( n ) . Inequality(9) implies | φ ( a ∗ · y · b ) | ≤ k ρ ( b ) ξ kk ρ ( a ) ξ kk y k , a, b ∈ B , y ∈ M n ( S ) . Thus, the sesqui-linear form L y : ( ρ ( B ) ξ ) × ( ρ ( B ) ξ ) → C given by L y ( ρ ( b ) ξ, ρ ( a ) ξ ) = φ ( a ∗ · y · b ) , a, b ∈ B , is bounded and has norm not exceeding k y k . It follows that there exists alinear operator Φ( y ) : ρ ( B ) ξ → ρ ( B ) ξ such that(10) (Φ( y ) ρ ( b ) ξ, ρ ( a ) ξ ) = φ ( a ∗ · y · b ) , a, b ∈ B , and(11) k Φ( y ) k ≤ k y k . Since ρ ( B ) ξ in dense in H , the operator Φ( y ) can be extended to an operatoron H . By (10), the map Φ : M n ( S ) → B ( H ) is linear and hermitian and, by(11), it is contractive.For a, b, c, d ∈ B , by (10), we have(Φ( c ∗ · y · d ) ρ ( b ) ξ, ρ ( a ) ξ ) = ( ρ ( c ∗ )Φ( y ) ρ ( d ) ρ ( b ) ξ, ρ ( a ) ξ ) . The density of ρ ( B ) ξ in H now implies that(12) Φ( c ∗ · y · d ) = ρ ( c ∗ )Φ( y ) ρ ( d ) , c, d ∈ B , y ∈ M n ( S ) . We show that Φ is weak* continuous. Suppose that ( y α ) α ⊆ M n ( S ) isa net of contractions such that y α → α y in the weak* topology, for some y ∈ M n ( S ). Fix δ > η, ζ ∈ H , and choose a, b ∈ B such that k ρ ( b ) ξ − η k < δ and k ρ ( a ) ξ − ζ k < δ. PERATOR SYSTEM STRUCTURES 15
Let α be such that | φ ( a ∗ · y α · b ) − φ ( a ∗ · y · b ) | < δ if α ≥ α . For α ≥ α we have | (Φ( y α ) η, ζ ) − (Φ( y ) η, ζ ) |≤ | (Φ( y α ) η, ζ ) − (Φ( y α ) ρ ( b ) ξ, ρ ( a ) ξ ) | + | (Φ( y α ) ρ ( b ) ξ, ρ ( a ) ξ ) − (Φ( y ) ρ ( b ) ξ, ρ ( a ) ξ ) | + | (Φ( y ) ρ ( b ) ξ, ρ ( a ) ξ ) − (Φ( y ) η, ζ ) | = | (Φ( y α ) η, ζ ) − (Φ( y α ) ρ ( b ) ξ, ρ ( a ) ξ ) | + | φ ( a ∗ · y α · b ) − φ ( a ∗ · y · b ) | + | (Φ( y ) ρ ( b ) ξ, ρ ( a ) ξ ) − (Φ( y ) η, ζ ) |≤ | (Φ( y α ) η, ζ ) − (Φ( y α ) ρ ( b ) ξ, ζ ) | + | (Φ( y α ) ρ ( b ) ξ, ζ ) − (Φ( y α ) ρ ( b ) ξ, ρ ( a ) ξ ) | + | φ ( a ∗ · y α · b ) − φ ( a ∗ · y · b ) | + | (Φ( y ) ρ ( b ) ξ, ρ ( a ) ξ ) − (Φ( y ) η, ρ ( a ) ξ ) | + | (Φ( y ) η, ρ ( a ) ξ ) − (Φ( y ) η, ζ ) |≤ δ ( k ζ k + k η k + k ρ ( a ) ξ k + k ρ ( b ) ξ k + 1) ≤ δ (2 k ζ k + 2 k η k + 2 δ + 1) . We thus showed that Φ( y α ) → α Φ( y ) in the weak operator topology; sincethe net (Φ( y α )) α is bounded, the convergence is in fact in the weak* topology.It follows from Shmulyan’s Theorem that the map Φ is weak* continuous.Identity (12) easily implies that there exists a (normal) map Ψ : S →B ( K ) such that Φ = Ψ ( n ) . Since Φ is hermitian and contractive, so is Ψ. By(12) and Theorem 4.6, the map Φ, and hence Ψ, is completely contractive.Now (12) implies(13) Ψ( a · z · b ) = θ ( a )Ψ( z ) θ ( b ) , z ∈ S , a, b ∈ A . By (10), 1 = φ (1) = (Φ(1) ξ, ξ ) ≤ k Φ(1) kk ξ k ≤ . Thus Φ(1) ξ = ξ ; by (12),Φ(1) ρ ( b ) ξ = ρ ( b )Φ(1) ξ = ρ ( b ) ξ, b ∈ B , and since ξ is cyclic for ρ , we conclude that Φ(1) = 1. It follows thatΨ(1) = 1.The map Ψ, constructed in the previous paragraph, depends on the el-ement x ∈ M n ( S ), and on the chosen ǫ . Note that, by (8) and (10), (cid:13)(cid:13) Ψ ( n ) ( x ) (cid:13)(cid:13) > − ǫ . Let γ (resp. π ) be the direct sum of the maps Ψ(resp. θ ) as above, over all selfadjoint x ∈ M n ( S ) with norm one, all n ∈ N ,and all ǫ ∈ (0 , γ is unital, weak* continuous, hermitian, andhas the property that if x ∈ M n ( S ) is selfadjoint then k x k = 1 implies (cid:13)(cid:13) γ ( n ) ( x ) (cid:13)(cid:13) = 1. This easily yields that γ is completely positive and has acompletely positive inverse. As in the proof of [4, Theorem 1.1], the imageof γ is weak* closed and γ is a weak* homeomorphism onto its range. Inaddition, π is a normal *-representation as a direct sum of such. Condition(7) follows from (13). (cid:3) The dual extremal operator A -system structures In this section, we study dual versions of the extremal operator A -systemstructures considered in Section 3. We start with the definition of a dualAOU space. Note first that, if ( V, V + , e ) is an AOU space then the expression k v k = sup {| f ( v ) | : f a state on V } defines a norm on V , called the order norm [16]; in the sequel we equip V with its order norm. If V is a dual Banach space, the weak* continuousfunctionals on V will be called normal functionals . Definition 5.1. A dual AOU space is an AOU space ( V, V + , e ) , which isalso a dual Banach space, and (i) the involution is weak* continuous; (ii) V + is weak* closed, and (iii) for v ∈ V , k v k = sup {| f ( v ) | : f a normal state on V } , and theweak* topology of V is determined by normal states of V . Suppose that (
V, V + , e ) is a dual AOU space, and let V ∗ be the predualof V . Note that the algebraic tensor product V ∗ ⊗ M ∗ n can be canonicallyembedded into the dual of M n ( V ). By the weak* topology on M n ( V ) we willmean the topology arising from this duality; thus, ( x αi,j ) → α ( x i,j ) if andonly if x αi,j → α x i,j for every i, j . Definition 5.2.
Let A be a W*-algebra. A dual AOU space ( V, V + , e ) willbe called dual AOU A -space if (i) ( V, V + , e ) is an AOU A -space, and (ii) the left (and hence the right) A -module action is separately weak*continuous. Definition 5.3.
Let A be a W*-algebra and ( V, V + , e ) be a dual AOU A -space. A matrix ordering ( C n ) n ∈ N on V will be called a dual operator A -system structure on V if ( V, ( C n ) n ∈ N , e ) is a dual operator A -system whoseweak* topology coincides with that of V , and C = V + . Theorem 5.4.
Let A be a W*-algebra, ( V, V + , e ) be a dual AOU A -spaceand ( C n ) n ∈ N be an operator A -system structure on V . The following areequivalent:(i) ( C n ) n ∈ N is a dual operator A -system structure on V ;(ii) C n is weak* closed for each n ∈ N .Proof. (i) ⇒ (ii) Let S = ( V, ( C n ) n ∈ N , e ). By Theorem 4.7, there exist aHilbert space H and a complete order embedding γ : S → B ( H ) such that γ ( S ) is weak* closed and γ is a weak* homeomorphism. Clearly, M n ( γ ( S )) + is weak* closed in M n ( B ( H )). Note that the weak* topology on M n ( B ( H )) = B ( H n ) is given by entry-wise weak* convergence. On the other hand, since γ is a weak* homeomorphism, we have that if (( x αi,j )) α ⊆ M n ( V ) and ( x i,j ) ∈ M n ( V ) then ( x αi,j ) → α ( x i,j ) weak* if and only if γ ( x αi,j ) → α γ ( x i,j ) for every i, j . It follows that C n is weak* closed. PERATOR SYSTEM STRUCTURES 17 (ii) ⇒ (i) Let S = ( V, ( C n ) n ∈ N , e ). For each n , let P n = { φ : V → M n : weak* continuous unital completely positive map } . Let H = ⊕ n ∈ N ⊕ φ ∈P n C n and let J : V → B ( H ) be the map given by J ( x ) = ⊕ n ∈ N ⊕ φ ∈P n φ ( x ). It is clear that J is a weak* continuous completely positivemap. In addition, by condition (iii) from Definition 5.1, J is isometric.To show that J is a complete order isomorphism, assume that J ( n ) ( X ) ≥ X = ( x i,j ) ∈ M n ( V ) h and that, by way of contradiction, X doesnot belong to C n . The space M n ( V ), equipped with the topology of weak*convergence, is a locally convex topological vector space. By a geometricform of the Hahn-Banach Theorem, there exists a functional s : M n ( V ) → C ,continuous with respect to the topology of entry-wise weak* convergence,such that s ( C n ) ⊆ R + but s ( X ) <
0. By [13, Theorem 6.1], the map φ s : V → M n , given by φ s ( x ) = ( s i,j ( x )) i,j (and where s i,j ( x ) = s ( E i,j ⊗ x )),is completely positive. It is clear that φ s is normal. In addition, φ ( n ) s doesnot map X to a positive matrix. After normalisation, we may assume that φ s is contractive.Let P = φ s ( e ); then P is a positive contraction. Assume that rank( P ) = k and let Q be the projection onto ker( P ) ⊥ . It was shown in the proof of[13, Theorem 13.1] that, if A ∈ M n,k and B ∈ M k,n are matrices suchthat A ∗ P A = I k and AB = Q , and ψ is the mapping given by ψ ( x ) = A ∗ φ s ( x ) A , then ψ is a (unital completely positive) map such that ψ ( n ) ( X )is not positive. Clearly, ψ is normal, and hence an element of P k . Thiscontradicts the fact that J ( n ) ( X ) ≥ J is a weak* homeomorphism, suppose that J ( x α ) → α J ( x )in the weak* topology, for some net ( x α ) ⊆ V and some element x ∈ V .Then φ ( x α ) → φ ( x ) for all normal positive functionals φ . By condition (iii)of Definition 5.1, x α → x in the weak* topology of V .We finally note that J ( V ) is weak* closed in B ( H ). Suppose that J ( x α ) → T , where T ∈ B ( H ) and ( x α ) α ⊆ V is a net such that the net J ( x α ) α isbounded. Since J is an isometry, ( x α ) α is also bounded, and hence has asubnet ( x β ) β , weak* convergent to an element of V , say x . Since J is weak*continuous, we conclude that T = lim β J ( x β ) = J ( x ), and hence T ∈ J ( V ).By the Krein-Smulyan, J ( V ) is weak* closed.By the previous paragraphs, the weak* topology of V coincides with theweak* topology of the operator system S . It now follows that the A -moduleoperations on S are separately weak* continuous; thus, S is a dual operator A -system and the proof is complete. (cid:3) As the next two statements show, if (
V, V + , e ) is a dual AOU A -spacethen the minimal operator A -system structure defined in Section 3 is auto-matically a dual minimal operator A -system structure. Theorem 5.5.
Let A be a W*-algebra and ( V, V + , e ) be a dual AOU A -space. Then ( C min n ( V ; A )) n ∈ N is a dual operator A -system structure. Proof.
Since the A -module actions on V are weak* continuous, C min n ( V ; A )is weak* closed for each n ∈ N . By Theorem 5.4, ( C min n ( V ; A )) n ∈ N is a dualoperator A -system structure. (cid:3) Theorem 5.6.
Let A be a W*-algebra and ( V, V + , e ) be a dual AOU A -space.(i) Suppose that S is a dual operator A -system and φ : S → V is a normalpositive A -bimodule map. Then φ is completely positive as a map from S into OMIN A ( V ) .(ii) If T is a dual operator A -system with underlying space V and posi-tive cone V + , such that for every dual operator A -system S , every normalpositive A -bimodule map φ : S → T is completely positive, then there existsa unital normal A -bimodule map ψ : T →
OMIN A ( V ) that is a completeorder isomorphism and a weak* homeomorphism.Proof. (i) is a direct consequence of Theorem 3.3 (i). The proof of (ii) followsby a standards argument, similar to the one given in the proof of Theorem3.3 (ii). (cid:3) In the remainder of the section, we consider the dual maximal operator A -system structure. For a W*-algebra A and a dual AOU A -space ( V, V + , e ),set W max n ( V ; A ) = C max n ( V ; A ) w ∗ , n ∈ N . Theorem 5.7.
Let A be a W*-algebra and ( V, V + , e ) be a dual AOU A -space. Then ( W max n ( V ; A )) n ∈ N is a dual operator A -system structure on V . Moreover, if ( P n ) n ∈ N is a dual operator A -system structure on V then W max n ( V ; A ) ⊆ P n for each n ∈ N .Proof. By Theorem 3.7, ( C max n ( V ; A )) n ∈ N is an operator system A -structureon V . It follows by the separate weak* continuity of the A -module actionson V and the definition of the M n ( A )-module operations on M n ( V ) (see(4)) that the family ( W max n ( V ; A )) n ∈ N is A -compatible.Since the element e is a matrix order unit for ( D max n ( V ; A )) n ∈ N (see Propo-sition 3.6) and D max n ( V ; A ) ⊆ W max n ( V ; A ) for each n ∈ N , e is a matrixorder unit for ( W max n ( V ; A )) n ∈ N . To show that e is an Archimedean ma-trix order unit for ( W max n ( V ; A )) n ∈ N , suppose that X ∈ M n ( V ) is such that X + re n ∈ W max n ( V ; A ) for all r >
0. Since X + re n → r → X in the weak*topology and W max n ( V ; A ) is weak* closed, X ∈ W max n ( V ; A ).It follows that ( V, ( W max n ( V ; A )) n ∈ N , e ) is an operator A -system; by con-dition (ii) of Definition 5.1, V + = W max1 ( V ; A ). Since its cones are weak*closed, Theorem 5.4 implies that it is a dual operator A -system.Suppose that ( P n ) n ∈ N is a dual operator A -system structure on V . Fix n ∈ N . By Theorem 3.7, C max n ( V ; A ) ⊆ P n . By Theorem 5.4, P n is weak*closed. It follows that W max n ( V ; A ) ⊆ P n . (cid:3) We denote by OMAX w ∗ A ( V ) the operator system ( V, ( W max n ( V ; A )) n ∈ N , e ). PERATOR SYSTEM STRUCTURES 19
Theorem 5.8.
Let A be a W*-algebra and ( V, V + , e ) be a dual AOU A -space.(i) Suppose that S is a dual operator A -system and φ : V → S is anormal positive A -bimodule map. Then φ is completely positive as a mapfrom OMAX w ∗ A ( V ) into S .(ii) If T is a dual operator A -system with underlying space V and posi-tive cone V + , such that for every dual operator A -system S , every normalpositive A -bimodule map φ : T → S is completely positive, then there existsa unital normal A -bimodule map ψ : T →
OMAX w ∗ A ( V ) that is a completeorder isomorphism and a weak* homeomorphism.Proof. (i) By Theorem 3.8 (i), φ ( n ) ( C max n ( V ; A )) ⊆ M n ( S ) + . Since φ isweak* continuous and M n ( S ) + is weak* closed, φ ( n ) ( W max n ( V ; A )) ⊆ M n ( S ) + .(ii) similar to the proof of Theorem 3.3 (ii). (cid:3) Remark.
Let A be a W*-algebra and A w ∗ A (resp. S w ∗ A ) be the category,whose objects are dual AOU A -spaces (resp. dual operator A -systems) andwhose morphisms are weak* continuous unital positive (resp. weak* con-tinuous unital completely positive) maps. It is easy to see that the corre-spondences V → OMIN w ∗ A ( V ) and V → OMAX w ∗ A ( V ) are covariant functorsfrom A w ∗ A into S w ∗ A , here OMIN w ∗ A ( V ) = OMIN A ( V ) as per Theorem 5.5.6. Inflated Schur multipliers
In this section, we introduce an operator-valued version of classical mea-surable Schur multipliers, and characterise them in a fashion, similar to thewell-known descriptions in the scalar-valued case [9, 17].Let (
X, µ ) be a standard measure space. We denote by χ α the characteris-tic function of a measurable set α ⊆ X . If f and g are measurable functionsdefined on X , we write f ∼ g when f ( x ) = g ( x ) for almost all x ∈ X .Throughout the section, let H = L ( X, µ ) and fix a separable Hilbert space K . For a function a ∈ L ∞ ( X, µ ), let M a be the operator on H given by M a f = af , f ∈ H , and set D = { M a : a ∈ L ∞ ( X, µ ) } . We denote by H ⊗ K the Hilbertian tensor product of H and K . Note that H ⊗ K is unitarily equivalent to the space L ( X, K ) of all weakly measurablefunctions g : X → K such that k g k := (cid:0)R X k g ( x ) k dµ ( x ) (cid:1) / < ∞ .If U ⊆ B ( H ) and V ⊆ B ( K ), we denote by U ¯ ⊗V the spacial weak*tensor product of U and V . We write M ( X, B ( K )) for the space of allfunctions F : X → B ( K ) such that, for all ξ ∈ K , the functions x → F ( x ) ξ and x → F ( x ) ∗ ξ are weakly measurable. Note that D ¯ ⊗B ( K ) can becanonically identified with the space L ∞ ( X, B ( K )) of all bounded functions F in M ( X, B ( K )) [24]. Through this identification, a function F gives riseto the operator M F ∈ B ( L ( X, K )), defined by( M F ξ )( x ) = F ( x )( ξ ( x )) , x ∈ X, ξ ∈ L ( X, K ) . It is easy to see that if k ∈ M ( X × X, B ( K )) then the function ( x, y ) →k k ( x, y ) k is measurable as a function from X × X into [0 , + ∞ ]. Let L ( X × X, B ( K )) be the space of all functions k ∈ M ( X × X, B ( K )) for which k k k := (cid:18)Z X × X k k ( x, y ) k dµ ( x ) dµ ( y ) (cid:19) / < ∞ . (Note that the functions from the space L ( X × X, B ( K )) need not be weaklymeasurable.) If k ∈ L ( X × X, B ( K )) and ξ, η ∈ L ( X, K ) then, by [24,Lemma 7.5], the function ( x, y ) → ( k ( x, y )( ξ ( y )) , η ( x )) is measurable. Stan-dard arguments (see [12, p. 391]) show that the formula( T k ξ, η ) = Z X × X ( k ( x, y )( ξ ( y )) , η ( x )) dµ ( y ) dµ ( x ) , x, y ∈ X, ξ, η ∈ L ( X, K ) , defines a bounded operator on L ( X, K ) with k T k k ≤ k k k . If K = C , theoperators of the form T k are precisely the Hilbert-Schmidt operators on H . Remark 6.1.
For an element k ∈ L ( X × X, B ( K )) , we have that T k = 0 if and only if k ( x, y ) = 0 for almost all ( x, y ) ∈ X × X .Proof. Suppose that T k = 0; then, for ξ, η ∈ K and f, g ∈ L ( X ), we have R X × X f ( x ) g ( y )( k ( x, y ) ξ, η ) dµ ( y ) dµ ( x ) = 0. Thus, ( k ( x, y ) ξ, η ) = 0 almosteverywhere. Since K is separable and k ( x, y ) is bounded for all x, y ∈ X ,this implies that k ( x, y ) = 0 almost everywhere. The converse direction istrivial. (cid:3) We equip the linear space { T k : k ∈ L ( X × X, B ( K )) } with the operatorspace structure arising from its inclusion into B ( H ⊗ K ). Similarly, whenever S is an operator system and S ⊆ S is a self-adjoint (not necessarily unital)subspace of S , we equip S with the matrix ordering inherited from S , andthus talk about a linear map from S into an operator system T beingpositive or completely positive.For functions ϕ ∈ L ∞ ( X × X, B ( K )) and k ∈ L ( X × X ), let ϕk : X × X →B ( K ) be the function given by( ϕk )( x, y ) = k ( x, y ) ϕ ( x, y ) , x, y ∈ X. It is straightforward to check that ϕk ∈ L ( X × X, B ( K )). Definition 6.2.
A function ϕ ∈ L ∞ ( X × X, B ( K )) will be called an (in-flated) Schur multiplier if the map T k −→ T ϕk , k ∈ L ( X × X ) , is completely bounded. We will denote by S ( X, K ) the space of all inflated Schur multiplierswith values in B ( K ). If ϕ ∈ S ( X, K ) then the map S ϕ : T k → T ϕk definedon the space S ( H ) of all Hilbert-Schmidt operators on H extends to acompletely bounded map from K ( H ) into B ( H ⊗ K ), which will be denotedin the same way. By taking the second dual of S ϕ , and composing with the PERATOR SYSTEM STRUCTURES 21 weak* continuous projection from B ( H ⊗ K ) ∗∗ onto B ( H ⊗ K ), we obtaina completely bounded weak* continuous map from B ( H ) into B ( H ⊗ K )which for simplicity will still be denoted by S ϕ . Theorem 6.3.
Let ϕ ∈ L ∞ ( X × X, B ( K )) . The following are equivalent:(i) ϕ ∈ S ( X, K ) ;(ii) there exist functions A i ∈ L ∞ ( X, B ( K )) and B i ∈ L ∞ ( X, B ( K )) , i ∈ N , such that the series P ∞ i =1 A i ( x ) A i ( x ) ∗ and P ∞ i =1 B i ( y ) ∗ B i ( y ) convergealmost everywhere in the weak* topology, esssup x ∈ X (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i =1 A i ( x ) A i ( x ) ∗ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ∞ , esssup y ∈ X (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i =1 B i ( y ) ∗ B i ( y ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ∞ , and (14) ϕ ( x, y ) = ∞ X i =1 A i ( x ) B i ( y ) , a.e. on X × X, where the sum is understood in the weak* topology.Proof. (ii) ⇒ (i) Considering A i , B i ∈ D ¯ ⊗B ( K ), i ∈ N , the assumptions implythat A = ( A i ) i ∈ N (resp. B = ( B i ) i ∈ N ) is a bounded row (resp. column)operator. It follows that the map Ψ : B ( H ) → B ( H ⊗ K ), given byΨ( T ) = ∞ X i =1 A i ( T ⊗ I ) B i , T ∈ B ( H ) , is well-defined and completely bounded. Let k ∈ L ( X × X ) ∩ L ∞ ( X × X ), ξ, η ∈ K and f, g ∈ L ( X ) ∩ L ( X ). For almost all ( x, y ) ∈ X × X , we have (cid:12)(cid:12)(cid:12) k ( x, y ) f ( y ) g ( x ) ( ϕ ( x, y ) ξ, η ) (cid:12)(cid:12)(cid:12) ≤ k k k ∞ | f ( y ) || g ( x ) | ∞ X i =1 | ( B i ( y ) ξ, A i ( x ) ∗ η ) |≤ k k k ∞ | f ( y ) || g ( x ) | ∞ X i =1 k B i ( y ) ξ kk A i ( x ) ∗ η k≤ k k k ∞ | f ( y ) || g ( x ) | ∞ X i =1 k B i ( y ) ξ k ! / ∞ X i =1 k A i ( x ) ∗ η k ! / ≤ k k k ∞ | f ( y ) || g ( x ) |k A kk B kk ξ kk η k , while the function ( x, y ) → | f ( y ) || g ( x ) | is integrable with respect to µ × µ .By the Lebesgue Dominated Convergence Theorem, we now have(Ψ( T k )( f ⊗ ξ ) , g ⊗ η )= ∞ X i =1 A i ( T k ⊗ I ) B i ( f ⊗ ξ ) , g ⊗ η ! = ∞ X i =1 Z X × X k ( x, y ) f ( y ) g ( x )( B i ( y ) ξ, A i ( x ) ∗ η ) dµ ( x ) dµ ( y )= Z X × X k ( x, y ) f ( y ) g ( x ) ∞ X i =1 A i ( x ) B i ( y ) ! ξ, η ! dµ ( x ) dµ ( y )= Z X × X k ( x, y ) f ( y ) g ( x ) ( ϕ ( x, y ) ξ, η ) dµ ( x ) dµ ( y )= Z X × X f ( y ) g ( x ) (( ϕk )( x, y ) ξ, η ) dµ ( x ) dµ ( y )= ( T ϕk ( f ⊗ ξ ) , g ⊗ η ) . By linearity and the density of L ( X × X ) ∩ L ∞ ( X × X ) in L ( X × X ) andof L ( X ) ∩ L ( X ) in L ( X ), it follows that ϕ ∈ S ( X, K ) and Ψ = S ϕ .(i) ⇒ (ii) Let ϕ ∈ S ( X, K ). For k ∈ L ( X × X ), a, b ∈ L ∞ ( X ), ξ, η ∈ K and f, g ∈ L ( X ), we have( S ϕ ( M b T k M a )( f ⊗ ξ ) , g ⊗ η )= Z X × X a ( y ) b ( x ) f ( y ) g ( x ) (( ϕk )( x, y ) ξ, η ) dµ ( x ) dµ ( y )= (( M b ⊗ I ) S ϕ ( T k )( M a ⊗ I )( f ⊗ ξ ) , g ⊗ η ) . By continuity, S ϕ ( BT A ) = ( B ⊗ I ) S ϕ ( T )( A ⊗ I ) , T ∈ K ( H ) , A, B ∈ D . Let Φ : K ( H ) ⊗ → B ( H ⊗ K ) be the map given by Φ ( T ⊗ I ) = S ϕ ( T ); thenΦ is a completely bounded D⊗ D ⊗ : B ( H ⊗ K ) → B ( H ⊗ K ) extending Φ . By [10], there exist a boundedrow operator A = ( A i ) ∞ i =1 and a bounded column operator B = ( B i ) i ∈ N ,where A i , B i ∈ D ¯ ⊗B ( K ), i ∈ N , such thatΦ ( T ) = ∞ X i =1 A i T B i , T ∈ B ( H ⊗ K ) . Using the identification D ¯ ⊗B ( K ) ≡ L ∞ ( X, B ( K )), we consider A i (resp. B i ) as a function A i : X → B ( K ) (resp. B i : X → B ( K )). The boundednessof A and B now imply that there exists a null set N ⊆ X such that the PERATOR SYSTEM STRUCTURES 23 series ∞ X i =1 A i ( x ) A i ( x ) ∗ and ∞ X i =1 B i ( y ) ∗ B i ( y )are weak* convergent whenever x, y N . If ( x, y ) N × N then the series P ∞ i =1 A i ( x ) B i ( y ) is weak* convergent. As in the first part of the proof, weconclude that ϕ ( x, y ) coincides with its sum for almost all ( x, y ). (cid:3) An inspection of the proof of Theorem 6.3 shows the following descriptionof inflated Schur multipliers.
Remark 6.4.
The following are equivalent, for a completely bounded map
Φ : K ( H ) → B ( H ⊗ K ) :(i) Φ( BT A ) = ( B ⊗ I )Φ( T )( A ⊗ I ) , for all T ∈ K ( H ) and all A, B ∈ D ;(ii) there exists a Schur multiplier ϕ ∈ S ( X, K ) such that Φ = S ϕ . Definition 6.5.
A Schur multiplier ϕ ∈ S ( X, K ) will be called positive ifthe map S ϕ : B ( H ) → B ( H ⊗ K ) is positive. For the next theorem, note that, if ϕ ∈ L ∞ ( X × X, B ( K )) and α ⊆ X is asubset of finite measure then the function ϕχ α × α belongs to L ( X × X, B ( K ))and hence the operator T ϕχ α × α : H → H ⊗ K is well-defined. Theorem 6.6.
The following are equivalent, for a Schur multiplier ϕ ∈ S ( X, K ) :(i) ϕ is positive;(ii) the map S ϕ : B ( H ) → B ( H ⊗ K ) is completely positive;(iii) for every subset α ⊆ X of finite measure, the operator T ϕχ α × α ispositive;(iv) there exist functions A i ∈ L ∞ ( X, B ( K )) , i ∈ N , such that the series P ∞ i =1 A i ( x ) A i ( x ) ∗ converges almost everywhere in the weak* topology, esssup x ∈ X (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i =1 A i ( x ) A i ( x ) ∗ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ∞ , and ϕ ( x, y ) = ∞ X i =1 A i ( x ) A i ( y ) ∗ , a.e. on X × X. Proof. (i) ⇒ (iii) Let α ⊆ X be a subset of finite measure. Then χ α ∈ H ; let χ α ⊗ χ ∗ α be the corresponding (positive) rank one operator. Then T ϕχ α × α = S ϕ ( χ α ⊗ χ ∗ α ) , and the conclusion follows.(iii) ⇒ (ii) Let n ∈ N , X i = X for i = 1 , . . . , n , Y = X ∪ · · · ∪ X n and ν be the disjoint sum of n copies of the measure µ . Identify C n ⊗ H with L ( Y, ν ), and define ψ : Y × Y → B ( K ) by letting ψ ( x, y ) = ϕ ( x, y ) if ( x, y ) ∈ X i × X j = X × X . Note that S ψ = id M n ⊗ S ϕ and hence ψ ∈ S ( Y, K ). Let α ⊆ X have finite measure and J ∈ M n be the matrix all of whose entries are equal to 1. Let α i ⊆ X i be the set that coincides with α , i = 1 , . . . , n ,and ˜ α = ∪ ni =1 α i ; we have that(15) T ψχ ˜ α × ˜ α ≡ J ⊗ T ϕχ α × α . By assumption, T ϕχ α × α is positive; thus, by (15), T ψχ ˜ α × ˜ α is positive. For g ∈ L ∞ ( Y, ν ) ∩ L ( Y, ν ) and h ∈ L ∞ ( ˜ α ), we have( S ψ ( g ⊗ g ∗ ) h, h ) = (cid:0) T ψχ ˜ α × ˜ α ( gh ) , gh (cid:1) ≥ . Since the set (cid:8) h ∈ L ( Y, ν ) : ∃ a set of finite measure α ⊆ X with h ∈ L ∞ ( ˜ α ) (cid:9) is dense in L ( Y, ν ), we have that S ψ ( g ⊗ g ∗ ) ∈ B ( H ⊗ K ) + . By weak*continuity, S ψ ( T ) ∈ B ( H ⊗ K ) + whenever T ∈ B ( L ( Y, ν )) + . Thus, S ψ ispositive, that is, S ϕ is n -positive.(ii) ⇒ (i) is trivial.(ii) ⇒ (iv) follows from the proof of Theorem 6.3 by noting that in the case S ϕ is completely positive, one can choose B i = A ∗ i , i ∈ N .(iv) ⇒ (i) follows from the proof of Theorem 6.3. (cid:3) Positive extensions
In this section, we apply our results on maximal operator system A -structures to questions about positive extensions of inflated Schur multipli-ers. We first recall some measure theoretic background from [2] and [7],required in the sequel. A subset E ⊆ X × X is called marginally null if E ⊆ ( M × X ) ∪ ( X × M ), where M ⊆ X is null. We call two subsets E, F ⊆ X × X marginally equivalent (resp. equivalent ), and write E ∼ = F (resp. E ∼ F ), if their symmetric difference is marginally null (resp. nullwith respect to product measure). We say that E is marginally contained in F (and write E ⊆ ω F ) if the set difference E \ F is marginally null. Ameasurable subset κ ⊆ X × X is called • a rectangle if κ = α × β where α, β are measurable subsets of X ; • ω -open if it is marginally equivalent to a countable union of rectan-gles, and • ω -closed if its complement κ c is ω -open.Recall that, by [22], if E is any collection of ω -open sets then there existsa smallest, up to marginal equivalence, ω -open set ∪ ω E , called the ω -union of E , such that every set in E is marginally contained in ∪ ω E . Given ameasurable set κ , one defines its ω -interior to beint ω ( κ ) = [ ω { R : R is a rectangle with R ⊆ ω κ } . The ω -closure cl ω ( κ ) of κ is defined to be the complement of int ω ( κ c ). Fora set κ ⊆ X × X , we write ˆ κ = { ( x, y ) ∈ X × X : ( y, x ) ∈ κ } . Thesubset κ ⊆ X × X is said to be generated by rectangles if κ ∼ = cl ω (int ω ( κ ))[7, 11]. PERATOR SYSTEM STRUCTURES 25
For any ω -closed subset κ ⊆ X × X , let S ( κ ) = (cid:8) T k : k ∈ L ( κ ) (cid:9) , S ( κ ) = S ( κ ) k·k and S ( κ ) = S ( κ ) w ∗ , where L ( κ ) is the space of functions in L ( X × X ) which are supportedon κ , up to a set of zero product measure. Note that the spaces S ( κ ), S ( κ ) and S ( κ ) are D -bimodules. We equip them with the operator spacestructures inherited from B ( H ).Partially defined scalar-valued Schur multipliers were defined in [11]. Herewe extend this notion to the operator-valued setting. Definition 7.1.
Let κ ⊆ X × X be a subset generated by rectangles. Afunction ϕ ∈ L ∞ ( κ, B ( K )) will be called a partially defined Schur multiplier if the map S ϕ from S ( κ ) into B ( H ⊗ K ) , given by S ϕ ( T k ) = T ϕk , k ∈ L ( κ ) , is completely bounded. Remark 7.2.
For Schur multipliers ϕ, ψ ∈ L ∞ ( κ, B ( K )) , we have that S ϕ = S ψ if and only if ϕ ∼ ψ .Proof. Suppose ϕ, ψ ∈ L ∞ ( κ, B ( K )) are such that S ϕ = S ψ . Then T ϕk = T ψk for every k ∈ L ( κ ). By Remark 6.1, ϕk ∼ ψk . It now easily followsthat ϕ ∼ ψ . The converse implication follows by reversing the previoussteps. (cid:3) Let κ ⊆ X × X be a subset generated by rectangles. We note that themap S ϕ from Definition 7.1 is D -bimodular. In addition, if ψ ∈ S ( X, K ) isgiven as in Definition 6.2, then its restriction ψ | κ : κ → B ( K ) is an inflatedSchur multiplier. Proposition 7.3.
Let K be a separable Hilbert space, κ ⊆ X × X a subsetgenerated by rectangles and ϕ ∈ L ∞ ( κ, B ( K )) . The following are equivalent:(i) ϕ is a Schur multiplier;(ii) there exists a Schur multiplier ψ : X × X → B ( K ) such that ψ | κ ∼ ϕ ;(iii) there exists a unique completely bounded map Φ : S ( κ ) → B ( H ⊗ K ) such that Φ ( T k ) = T ϕk , for each k ∈ L ( κ ) ;(iv) there exists a unique completely bounded weak* continuous map Φ : S ( κ ) → B ( H ⊗ K ) such that Φ( T k ) = T ϕk , for each k ∈ L ( κ ) .Proof. (i) ⇒ (ii) Since ϕ is a Schur multiplier, the map Φ : S ( κ ) → B ( H ⊗ K ), given by Φ ( T k ) = T ϕk , extends to a completely bounded linear mapΦ : S ( κ ) → B ( H ⊗ K ). By continuity,Φ ( BT A ) = ( B ⊗ I )Φ ( T )( A ⊗ I ) , T ∈ S ( κ ) , A, B ∈ D . Let ˆΦ : S ( κ ) ⊗ → B ( H ⊗ K ) be the map given byˆΦ( T ⊗ I ) = Φ ( T ) , T ∈ S ( κ ) . By [13, Exercise 8.6 (ii)], there exists a completely bounded
D ⊗ : B ( H ⊗ K ) → B ( H ⊗ K ), extending ˆΦ. Let ˆΨ : K ( H ) ⊗ → B ( H ⊗ K ) be the restriction of ˆΦ ; then ˆΨ | S ( κ ) ⊗ = ˆΦ. Let Ψ : K ( H ) → B ( H ⊗ K )be given by Ψ( T ) = ˆΨ( T ⊗ I ). Clearly,Ψ( BT A ) = ( B ⊗ I )Ψ( T )( A ⊗ I ) , T ∈ K ( H ) , A, B ∈ D . By Remark 6.4, there exists ψ ∈ S ( X, K ) such that Ψ = S ψ . For every k ∈ L ( κ ) we have S ψ ( T k ) = S ϕ ( T k ). By Remark 7.2, ψ | κ ∼ ϕ .(ii) ⇒ (iv) Take Φ = S ψ | S ( κ ) . The uniqueness of Φ follows from the factthat the Hilbert-Schmidt operators with integral kernels in L ( κ ) are weak*dense in S ( κ ).(iv) ⇒ (iii) ⇒ (i) are trivial. (cid:3) If ϕ : κ → B ( K ) is a Schur multiplier then we will denote still by S ϕ theweak* continuous map defined on S ( κ ) whose existence was established inProposition 7.3 (iv).We say that a subset κ ⊆ X × X is symmetric if κ ∼ = ˆ κ . We call κ a positivity domain [11] if κ is symmetric, generated by rectangles and thediagonal ∆ := { ( x, x ) : x ∈ X } is marginally contained in κ . The followingwas established in [11]: Proposition 7.4. If κ ⊆ X × X is generated by rectangles, then the followingare equivalent:(i) S ( κ ) is an operator system;(ii) κ is a positivity domain. Let ϕ : κ → B ( K ) be a Schur multiplier. We say that the Schur multiplier ψ : X × X → B ( K ) is a positive extension of ϕ if ψ is positive and ψ | κ ∼ ϕ . Proposition 7.5.
Let κ be a positivity domain and ϕ : κ → B ( K ) be aSchur multiplier. The following are equivalent:(i) ϕ has a positive extension;(ii) the map S ϕ : S ( κ ) → B ( H ⊗ K ) is completely positive.Proof. (i) ⇒ (ii) Suppose that ψ : X × X → B ( K ) is a positive extensionof ϕ . By Theorem 6.6, S ψ is completely positive. On the other hand, S ψ | S ( κ ) = S ψ | κ . Since ψ | κ = ϕ , we conclude that S ϕ is completely positive.(ii) ⇒ (i) Let Φ be the restriction of S ϕ to S ( κ ) + C I ; clearly, Φ is acompletely positive map. By Arveson’s Extension Theorem, there existsa completely positive map Ψ : K ( H ) + C I → B ( H ⊗ K ) extending Φ .The restriction Ψ of Ψ to K ( H ) is then a completely positive extension of S ϕ | S ( κ ) . Let Ψ ∗∗ be the second dual of Ψ, and E : B ( H ⊗ K ) ∗∗ → B ( H ⊗ K )be the canonical projection. We have that the map ˜Ψ = E ◦ Ψ ∗∗ : B ( H ) →B ( H ⊗ K ) is completely positive and weak* continuous extension of S ϕ . LetˆΨ : B ( H ) ⊗ → B ( H ⊗ K ) (resp. ˆΦ : S ( κ ) ⊗ → B ( H ⊗ K )) be themap given by ˆΨ( T ⊗ I ) = ˜Ψ( T ) (resp. ˆΦ( T ⊗ I ) = S ϕ ( T )); then ˆΨ is acompletely positive extension of map ˆΦ. Note that ˆΦ is a D ⊗
D ⊗
PERATOR SYSTEM STRUCTURES 27 there exists ψ ∈ S ( X, K ) such that ˜Ψ = S ψ ; the function ψ is the desiredpositive extension of ϕ . (cid:3) If S is an operator system, we write S ++ for the cone of all positivefinite rank operators in S . If T is an operator system, we call a linear mapΦ : S → T strictly positive if Φ( S ) ∈ T + whenever S ∈ S ++ . We callΦ strictly completely positive if Φ ( n ) is strictly positive for all n ∈ N . ASchur multiplier ϕ : κ → B ( K ) will be called strictly positive (resp. strictlycompletely positive) if the map S ϕ : S ( κ ) → B ( H ⊗ K ) is strictly positive(resp. strictly completely positive). Lemma 7.6.
Let κ be a positivity domain. Every positive finite rank opera-tor in M n ( S ( κ )) has the form ( T k i,j ) ni,j =1 , where k i,j ∈ L ( κ ) , i, j = 1 , . . . , n .Proof. Recall that S ( κ ) = { T k : k ∈ L ( κ ) } and S ( κ ) = S ( κ ) k·k . Itfollows that M n ( S ( κ )) = M n ( S ( κ )) k·k . Suppose that T ∈ M n ( S ( κ )) ++ and let T = ( T i,j ) ni,j =1 , where T i,j ∈ S ( κ ), i, j = 1 , . . . , n . Since T has finiterank, so does T i,j ; in particular, T i,j is a Hilbert-Schmidt operator and, by[7, Lemma 6.1], T i,j ∈ S ( κ ). (cid:3) Recall that the Banach space projective tensor product T ( X ) = L ( X, µ ) ˆ ⊗ L ( X, µ )can be canonically identified with the predual of B ( H ) (and the dual of K ( H )). Indeed, each element h ∈ T ( X ) can be written as a series h = P ∞ i =1 f i ⊗ g i , where P ∞ i =1 k f i k < ∞ and P ∞ i =1 k g i k < ∞ , and the pairingis then given by h T, h i = ∞ X i =1 ( T f i , g i ) , T ∈ B ( H ) . We have [2] that h can be identified with a complex function on X × X ,defined up to a marginally null set, and given by h ( x, y ) = ∞ X i =1 f i ( x ) g i ( y ) . The positive cone T ( X ) + consists, by definition, of all functions h ∈ T ( X )that give rise to positive functionals on B ( H ), that is, functions h of the form h = P ∞ i =1 f i ⊗ f i , where P ∞ i =1 k f i k < ∞ . It is well-known that a function ϕ ∈ L ∞ ( X × X ) is a Schur multiplier if and only if, for every h ∈ T ( X ),there exists h ′ ∈ T ( X ) such that ϕh ∼ h ′ (see [17]). In particular, if themeasure µ is finite then S ( X, C ) can be naturally identified with a subspaceof T ( X ). Theorem 7.7.
Let κ ⊆ X × X be a positivity domain. The following areequivalent:(i) for every separable Hilbert space K , every strictly positive Schur mul-tiplier ϕ : κ → B ( K ) is strictly completely positive; (ii) for every n ∈ N , every positive finite rank operator in M n ( S ( κ )) is thenorm limit of sums of operators of the form ( D i SD ∗ j ) i,j , where ( D i ) ni =1 ⊆ D and S ∈ S ( κ ) ++ .Proof. (i) ⇒ (ii) We first assume that the measure µ is finite. Suppose thatthere exists n ∈ N and a positive finite rank operator T ∈ M n ( S ( κ )) thatis not equal to the limit, in the norm topology, of the operators of theform ( D i SD ∗ j ) ni,j =1 , where ( D i ) ni =1 ⊆ D and S ∈ S ( κ ) ++ . By Lemma 7.6, T = ( T k i,j ) ni,j =1 , for some k i,j ∈ L ( κ ), i, j = 1 , . . . , n . By a geometricform of Hahn-Banach’s Theorem, there exist a norm continuous functional ω : M n ( S ( κ )) → C and γ < ω ( T ) < γ and ω (cid:0) ( D i SD ∗ j ) ni,j =1 (cid:1) ≥ , S ∈ S ( κ ) ++ , ( D i ) ni =1 ⊆ D . Let ω i,j : S ( κ ) → C be the norm continuous functionals such that ω (( S i,j ) ni,j =1 ) = n X i,j =1 ω i,j ( S i,j ) , S i,j ∈ S ( κ ) , i, j = 1 , . . . , n. After extending ω i,j to K ( H ), we may assume that ω i,j ∈ T ( X ) for i, j =1 , . . . , n .Suppose first that ω i,j ∈ S ( X, C ), i, j = 1 , . . . , n . Identify ω with thefunction (denoted by the same symbol) ω : X × X → M n , given by ω ( x, y ) =( ω i,j ( x, y )) ni,j =1 . Since S ω : S ( H ) → B ( H ) ⊗ M n is given by S ω ( T k ) =( S ω i,j ( T k )), k ∈ L ( X × X ), and the maps S ω i,j are completely bounded, wehave that the map S ω is completely bounded, that is, ω ∈ S ( X, M n ).We claim that S ( n ) ω is not strictly positive. Note that S ( n ) ω ( T ) = (cid:0) S ω i,j ( T k p,q ) (cid:1) i,j,p,q . Writing e for the vector in H n with all its entries equal to the constantfunction 1, we have that γ > ω ( T ) = n X i,j =1 Z κ ω i,j ( x, y ) k i,j ( x, y ) d ( µ × µ )( x, y )= (cid:16)(cid:0) S ω i,j ( T k i,j ) (cid:1) i,j e, e (cid:17) . (17)Suppose that S ( n ) ω ( T ) is positive. Then its submatrix ( S ω i,j ( T k i,j )) i,j is pos-itive, which contradicts (17).We now show that S ω is strictly positive. Let S ∈ S ( κ ) ++ . Using Lemma7.6, write S = T k for some k ∈ L ( κ ). We have that S ω ( S ) = ( T ω i,j k ) ni,j =1 .For i = 1 , . . . , n , let ξ i ∈ L ∞ ( X, µ ) and note that, since µ is finite, ξ i ∈ H . PERATOR SYSTEM STRUCTURES 29
Let D i = M ξ i , i = 1 , . . . , n , and set ξ = ( ξ i ) ni =1 . We have that( S ω ( S ) ξ, ξ ) = n X i,j =1 ( T ω i,j k ξ j , ξ i )= n X i,j =1 Z κ ω i,j ( x, y ) k ( x, y ) ξ j ( x ) ξ i ( y ) d ( µ × µ )( x, y )= ω (cid:0) ( D ∗ i SD j ) ni,j =1 (cid:1) ≥ . Since L ∞ ( X, µ ) is dense in H , we have that S ω ( S ) ∈ M n ( B ( H )) + .Now relax the assumption that ω i,j ∈ S ( X, C ). By standard arguments(see e.g. the proof of [1, Lemma 3.13]), there exist measurable sets X m ⊆ X with X m ⊆ X m +1 , m ∈ N , such that µ ( X \ X m ) → m →∞ ω ( m ) i,j of ω i,j to X m × X m belongs to S ( X m , C ) for all m ∈ N . Let ω ( m ) : X × X → M n be the function given by ω ( m ) ( x, y ) = ( ω ( m ) i,j ( x, y )) i,j if( x, y ) ∈ X m × X m and ω ( m ) ( x, y ) = 0 otherwise, and note that ω ( m ) definesa functional on M n ( K ( H )) in the natural way (which will be denoted by thesame symbol). Let P m be the projection from H onto L ( X m ). We havethat ω ( m ) ( R ) = ω (( P m ⊗ I n ) R ( P m ⊗ I n )) , R ∈ M n ( K ( H )) . Since ( P m ⊗ I n ) R ( P m ⊗ I n ) → m →∞ R in norm, for every R ∈ M n ( K ( H )),we have that (16) eventually holds true for ω ( m ) in the place of ω . By theprevious paragraph, ω ( m ) is a Schur multiplier for which S ω ( m ) is strictlypositive, but not strictly completely positive.Finally, relax the assumption that µ be finite. Let ( X m ) m ∈ N be an in-creasing sequence of sets of finite measure such that ∪ ∞ m =1 X m = X , and let Q m be the projection from H onto L ( X m ), m ∈ N . Let T ∈ M n ( S ( κ )) ++ .Since T is a positive operator of finite rank, ( Q m T Q m ) m ∈ N is a sequence ofpositive finite rank operators, converging to T in norm. By the first partof the proof, Q m T Q m is a norm limit of operators of the form ( D i SD ∗ j ) i,j ,where ( D i ) ni =1 ⊆ D and S ∈ S ( κ ) ++ . The conclusion follows.(ii) ⇒ (i) Let ϕ : κ → B ( K ) be a Schur multiplier such that S ϕ : S ( κ ) →B ( H ⊗ K ) is strictly positive. It follows from the assumption and fact that S ϕ is a D -bimodule map that S ( n ) ϕ ( T ) is positive whenever T ∈ M n ( S ( κ )) ++ . (cid:3) Definition 7.8.
Let κ be a positivity domain. We call κ rich if M n ( S ( κ )) + = M n ( S ( κ )) ++ w ∗ for every n ∈ N . Suppose that X is a countable set equipped with counting measure. Inthis case, positivity domains can be identified with undirected graphs withvertex set X in the natural way. This identification will be made in thesubsequent remark and in Theorem 7.12. Remark 7.9.
Let X be a countable set. Then any graph κ ⊆ X × X is rich. Proof.
For X = N , write Q m for the projection onto the span of { e i } mi =1 , m ∈ N , where { e i } i ∈ N is the standard basis of ℓ . If T ∈ M n ( S ( κ )) + then(( Q m ⊗ I n ) T ( Q m ⊗ I n )) m ∈ N is a sequence in M n ( S ( κ )) ++ , converging inthe weak* topology to T . (cid:3) By Proposition 7.5, if a Schur multiplier ϕ : κ → B ( K ) has a positiveextension then the map S ϕ : S ( κ ) → B ( H ⊗ K ) is necessarily positive. Wecall ϕ admissible if S ϕ is a positive map. The main result of this sectionis a characterisation of when an admissible Schur multiplier has a positiveextension, in terms of the maximal operator D -system structure defined inSection 5. Note that S ( κ ) is a dual AOU D -space in the natural fashion. Theorem 7.10.
Let κ ⊆ X × X be a rich positivity domain. The followingare equivalent:(i) for every separable Hilbert space K , every admissible Schur multiplier ϕ : κ → B ( K ) has a positive extension;(ii) S ( κ ) = OMAX w ∗ D ( S ( κ )) .Proof. (i) ⇒ (ii) Let ϕ : κ → B ( K ) be a strictly positive Schur multiplier.Since S ( κ ) + = S ( κ ) ++ w ∗ and S ϕ is weak* continuous, S ϕ is positive. By theassumption and Proposition 7.5, S ϕ is completely positive. In particular, S ϕ is strictly completely positive. By Theorem 7.7 and the fact that thematricial cones of any operator system are norm closed, we have that(18) M n ( S ( κ )) ++ ⊆ M n (OMAX D ( S ( κ ))) + . Since κ is rich, by taking weak* closures on both sides in (18) we obtainthat(19) M n ( S ( κ )) + ⊆ M n (OMAX w ∗ D ( S ( κ ))) + . Since the converse inclusion in (19) always holds, we conclude that S ( κ ) =OMAX w ∗ D ( S ( κ )).(ii) ⇒ (i) follows from Theorem 5.8 and Proposition 7.5. (cid:3) Theorem 7.10 and Remark 7.9 have the following immediate corollary. Inthe case where X is finite, it is a reformulation, in terms of operator systemstructures, of [14, Theorem 4.6]. Corollary 7.11.
Let X be a countable set, equipped with counting measureand κ ⊆ X × X be a symmetric set containing the diagonal. The followingare equivalent:(i) for every Hilbert space K , every admissible Schur multiplier ϕ : κ →B ( K ) has a positive extension;(ii) S ( κ ) = OMAX w ∗ D ( S ( κ )) . Let X be a countable set. Recall that a graph κ ⊆ X × X is called chordalif every 4-cycle in κ has an edge connecting two non-consecutive vertices ofthe cycle (see e.g. [14]). PERATOR SYSTEM STRUCTURES 31
Theorem 7.12.
Let X be a countable set and κ ⊆ X × X be a chordalgraph. Then S ( κ ) = OMAX w ∗ D ( S ( κ )) .Proof. Fix n ∈ N and let [ n ] = { , . . . , n } . Suppose that κ ⊆ X × X is achordal graph. Let κ ( n ) = { (( x, i ) , ( y, j )) ∈ ( X × [ n ]) × ( X × [ n ]) : ( x, y ) ∈ κ } . Then κ ( n ) is a chordal graph on X × [ n ]. By [11, Theorem 2.5], every positiveoperator in M n ( S ( κ )) is a weak* limit of rank one positive operators in M n ( S ( κ )).Suppose that K is a Hilbert space and ϕ : κ → B ( K ) is a Schur multipliersuch that S ϕ : S ( κ ) → B ( H ⊗ K ) is a positive map. Let R ∈ M n ( S ( κ )) be apositive rank one operator. After identifying M n ( S ( κ )) with S ( κ ( n ) ), we seethat there exists a subset α ⊆ X × [ n ] such that R is supported on α × α .Let β = { x ∈ X : ∃ i ∈ [ n ] with ( x, i ) ∈ α } . Since α × α ⊆ κ ( n ) , we have that β × β ⊆ κ . Setting ˜ β = β × [ n ], we have that α ⊆ ˜ β , and hence R is supported on ˜ β × ˜ β . The restriction ψ of ϕ to β × β is a positive Schur multiplier. By Theorem 6.6, the map S ψ : S ( β × β ) →B ( H ⊗ K ) is completely positive. Thus, S ( n ) ϕ ( R ) = S ( n ) ψ ( R ) ∈ B ( H ⊗ K ) + .Since S ϕ is weak* continuous, the previous paragraph implies that S ϕ iscompletely positive. By Proposition 7.5, ϕ has a positive extension and, byCorollary 7.11, S ( κ ) = OMAX w ∗ D ( S ( κ )). (cid:3) References [1]
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Mathematical Sciences Research Centre, Queen’s University Belfast, BelfastBT7 1NN, United Kingdom
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