Two Characterizations of the Maximal Tensor Product of Operator Systems
aa r X i v : . [ m a t h . OA ] M a r TWO CHARACTERIZATIONS OF THE MAXIMALTENSOR PRODUCT OF OPERATOR SYSTEMS
WAI HIN NG
Abstract.
In this paper we provide two characterizations of the max-imal tensor product structure for the category of operator systems in-troduced in [4]. The first one is via the schur tensor product given in[9]; the second one employs the idea of the CPAP in [2]. Introduction
An operator system is a self-adjoint and unital subspace of B ( H ) of allbounded operators on a Hilbert space H . In recent years, the theory oftensor products of operator systems has been developed systematically, seee.g. [3, 4, 5]. Given operator systems S and T , their maximal tensor product,denoted by S ⊗ max T , is equipped the smallest family of cones for which thealgebra tensor product S ⊗ T forms an operator system. In the categoryof operator systems, the maximal tensor product is the natural analogue ofthe projective tensor norm of operator spaces, as well as a generalization ofthe maximal tensor norm of C*-algerbas.In fact, given any operator space V , there is an operator system S V containing V completely order isometrically, see [7, Chp 8] . In [4], it isshown that for operator spaces V and W , the projective tensor product V ∧ ⊗ W is completely isometrically included in S V ⊗ max S W . Similary, anyunital C*-algebras A and B are as well operator systems; in the same paperit is proved that their C*-maximal tensor product A ⊗
C*-max B is completelyorder isomorphic to A ⊗ max B .Thus it is natural to characterize the maximal tensor product. In thispaper we give two characterizations of the maximal tensor product fromdifferent approaches. The first approach given in section 3 is to examinethe schur tensor product from [9] in the category of operator systems. Itprovides a different view of the matricial cones of S ⊗ max T . In section 4, weemploy factorization and the idea of the completely positive approximationproperty (CPAP) in [2] to characterize these matricial cones. We showthat this characterization possesses connections to results on (min , max)-nuclearity found in [3]. Date : March 23, 2015.2010
Mathematics Subject Classification.
Primary 46L06; Secondary 47L25.
Key words and phrases. maximal tensor product, operator system, schur tensor prod-uct, factorization. Preliminaries
We outline a few basic facts about the maximal tensor product and referreaders to [1, 3, 4, 5] for the details. Given a pair of operator systems( S , { P n } ∞ n =1 , S ) and ( T , { Q n } ∞ n =1 , T ), by an operator system structure on S ⊗ T , we mean a family τ = { C n } ∞ n =1 of cones, where C n ⊂ M n ( S ⊗ T ),satisfying:(T1) (
S ⊗ T , { C n } ∞ n =1 , S ⊗ T ) is an operator system denoted by S ⊗ τ T .(T2) P n ⊗ Q m ⊂ C nm , for all n, m ∈ N .(T3) If φ : S → M n and ψ : T → M m are unital completely positive maps,then φ ⊗ ψ : S ⊗ τ T → M nm is a unital completely positive map.By an operator system tensor product, we mean a mapping τ taking anypair of operator systems S and T into an operator system structure τ ( S , T ),denoted by S ⊗ τ T . We say τ is fuctorial, provided in addition it satisfiesthe following property:(T4) Given operator systems S i and T i , i = 1 ,
2, if φ i : S i → T i is unitalcompletely positive, then φ ⊗ φ : S ⊗ τ S → T ⊗ τ T is unitalcompletely positive.If for all operator systems S and T , the map θ : x ⊗ y y ⊗ x is a unitalcomplete order isomorphism from S ⊗ τ T onto T ⊗ τ S , then τ is a calledsymmetric.We now recall the construction of the maximal tensor product. Givenoperator systems S and T , we first define the family of cones D max n ( S , T ) = { A ( P ⊗ Q ) A ∗ : P ∈ M k ( S ) + , Q ∈ M m ( T ) + ,A ∈ M n,km , k, m ∈ N } . For short we denote D max n ( S , T ) = D max n . We shall remark the followinguseful representation of D max1 . Lemma 1.
Every u ∈ D max1 can be represented as u = P p ij ⊗ q ij for some [ p ij ] ∈ M n ( S ) + and [ q ij ] ∈ M n ( T ) + .Proof. If u = A ( P ⊗ Q ) A ∗ as above with A ∈ M ,km , note that u is thenthe sum of the entries of the Kronecker tensor product ( A " P ... P ... ...
P ... P A ∗ ) ⊗ Q ,where the operator matrix is in M m ( M k ( S )) + . Since we can replace Q by (cid:2) Q
00 0 (cid:3) of some appropriate size and likewise for the first operator matrix, wededuce such representation as claimed. (cid:3)
This matricial cone structure {D max n } ∞ n =1 is then a compatible family withmatrix order unit 1 S ⊗ T . Yet it is not Archimedean, so we complete thecones through the Archimedeanization process (see [8]) basically by takingthe closure of D max n : C max n ( S , T ) = { U ∈ M n ( S ⊗ T ) : ε (1 S ⊗ T ) + U ∈ D n ( S , T ) , ∀ r > } . WO CHARACTERIZATIONS OF THE MAXIMAL TENSOR PRODUCT 3
Likewise we denote C max n ( S , T ) = C max n . Now S⊗T equipped with this family {C max n } ∞ n =1 satisfies Properties (T1) to (T4) and it defines a symmetric andassociative operator system structure. We call it the maximal tensor productof S and T and denote it S ⊗ max T .The maximal tensor product is projective in the following sense. Let τ bean operator system tensor product. We say that τ is left projective, providedif q : S → R is a complete quotient map ([1, 3]), then for any operator system T , the map q ⊗ id is a complete quotient from S ⊗ τ T onto R ⊗ τ T . It isequivalent to require that for every n ∈ N , every u ∈ M n ( R ⊗ τ T ) + , andevery ε >
0, there is ˜ u ε ∈ M n ( S ⊗ τ T ) + so that q ⊗ id ( ˜ u ε ) = u + ε ( I n ⊗ R ⊗ T ). Right projectivity is defined similarly and we say τ is projective if it isboth left and right projective.This maximal tensor product has the following universal property: Theorem 2. [4, Theorem 5.8]
Let S and T be operator systems. A bilinearmap φ : S × T → B ( H ) is jointly completely positive if and only if its lin-earization L φ : S ⊗ max
T → B ( H ) is a completely positive map. Moreover,if τ is an operator system structure on S ⊗ T satisfying this property, then
S ⊗ τ T = S ⊗ max T . If we take B ( H ) = C , we obtain the following representation of the max-imal tensor product: ( S ⊗ max T ) d, + = CP ( S , T d ) , where the latter set is the cone of all completely positive maps from S to T d . This statement is precisely the operator system analogue of a result byLance in [6].The following lemma is in [4] and will be used in the next section. Weinclude the proof for completion. Lemma 3.
Let S and T be operator systems and { C n } ∞ n =1 be a compatiblefamily of cones of S ⊗ T satisfying Property (T2). Then D max n ⊂ C n .Proof. If P ∈ M n ( S ) + and Q ∈ M m ( T ) + , then Property (T2) implies P ⊗ Q ∈ C nm . By compatibility of { C n } ∞ n =1 , A ( P ⊗ Q ) A ∗ ∈ C k , for all A ∈ M k,nm ; hence D max n ⊂ C n . (cid:3) The Schur Tensor Product
In this section we examine the schur tensor product from [9] in the cate-gory of operator systems. It turns out that in the operator system settings,the matricial cones of the schur tensor product coincide with that of themaximal tensor product, providing a different description of the maximaltensor product.
Definition 4.
Given operator systems S and T , X = [ x ij ] ∈ M n ( S ) + , and Y = [ y ij ] ∈ M n ( T ) + , we define the schur tensor product X ◦ Y to be X ◦ Y := [ x ij ⊗ y ij ] ∈ M n ( S ⊗ T ) . WAI HIN NG
Lemma 5.
Every X ◦ Y ∈ M n ( S ⊗ T ) can be regarded as A ( X ⊗ Y ) A ∗ , forsome A ∈ M n,n .Proof. Let { E ij } ni,j =1 denote the standard matrix units of M n ( C ) and regard X ⊗ Y as the Kronecker tensor product. In the case when n = 2, note that (cid:2) E E (cid:3) X ⊗ Y (cid:2) E E (cid:3) ∗ = (cid:2) E E (cid:3) x ⊗ y x ⊗ y x ⊗ y x ⊗ y x ⊗ y x ⊗ y x ⊗ y x ⊗ y x ⊗ y x ⊗ y x ⊗ y x ⊗ y x ⊗ y x ⊗ y x ⊗ y x ⊗ y (cid:20) E E (cid:21) = (cid:20) x ⊗ y x ⊗ y x ⊗ y x ⊗ y (cid:21) = X ◦ Y. In general, we may view X ◦ Y as a pre-and-post mulitplication of X ⊗ Y by a special n × n matrix E = (cid:2) E E . . . E nn (cid:3) , a similar calculationshows that X ◦ Y = E ( X ⊗ Y ) E ∗ . (cid:3) Lemma 6.
Every P ∈ M n ( S ⊗ T ) can be written as P = A ( X ◦ Y ) B , forsome X ∈ M k ( S ) , Y ∈ M k ( T ) , A ∈ M n,k and B ∈ M k,n . In particular, wemay take B = A ∗ .Proof. Write P as a sum of matrices whose entries are elementary tensors,that is, P = P ml =1 U l , where U l = [ x lij ⊗ y lij ] ∈ M n ( S ⊗ T ). Let U = U ⊕ · · · ⊕ U m , so U = U . . . U . . . . . . U m which is X ◦ Y for some X ∈ M nm ( S ) and Y ∈ M nm ( T ). Now let A = (cid:2) I n I n . . . I n (cid:3) ∈ M n,nm with m copies of I n . Then,( AU ) A ∗ = (cid:2) U U . . . U l (cid:3) n × nm I n I n ... I n nm × n = m X l =1 U l = P. (cid:3) Lemma 5 shows that schur tensor product is in fact of the form of elementsin D max n , except positivity. Motivated by Lemma 6 and the construction ofthe maximal tensor product, we define the following family of cones. Definition 7.
Given operator systems S and T , we define C sn ( S ⊗ T ) := { A ( X ◦ Y ) A ∗ ∈ M n ( S ⊗ T ) : X ∈ M k ( S ) + , Y ∈ M k ( T ) + , A ∈ M n,k , k ∈ N } . For short we denote C sn ( S ⊗ T ) = C sn . WO CHARACTERIZATIONS OF THE MAXIMAL TENSOR PRODUCT 5
Proposition 8.
The family {C sn } ∞ n =1 defines a matrix ordering on S ⊗ T with matrix order unit ⊗ .Proof. We first check that C sn is a cone of M n ( S ⊗ T ). It is obvious fromdefinition that C sn ⊂ M n ( S ⊗ T ) sa . Let A ( X ◦ Y ) A ∗ and B ( X ◦ Y ) B ∗ be in C sn , where X ∈ M k ( S ), Y ∈ M k ( T ), X ∈ M m ( S ), Y ∈ M m ( T ), A ∈ M n,k ( C ), and B ∈ M n,m ( C ). Let X = X ⊕ X = (cid:20) X X (cid:21) ∈ M k + m ( S ) + , and likewise Y = Y ⊕ Y ∈ M k + m ( T ) + . Consider (cid:2) A B (cid:3) ∈ M n,k + m , then (cid:2) A B (cid:3) ( X ◦ Y ) (cid:2) A B (cid:3) ∗ = (cid:2) A B (cid:3) (cid:20) X ◦ Y X ◦ Y (cid:21) (cid:2) A B (cid:3) ∗ = A ( X ◦ Y ) A ∗ + B ( X ◦ Y ) B ∗ is in C sn . If t >
0, then t ( A ( X ◦ Y ) A ∗ ) = ( √ tA )( X ◦ Y )( √ tA ) ∗ ∈ C sn . Also, if B ∈ M r,n then ( BA )( X ◦ Y )( BA ) ∗ ∈ C sr . Therefore, {C sn } ∞ n =1 is a compatiblefamily of cones on S ⊗ T ).Finally, to see that they are proper, we claim that in fact C sn ⊂ D max n .Indeed, let A ( X ◦ Y ) A ∗ ∈ C sn , for some X ∈ M k ( S ) + , Y ∈ M k ( T ) + , and A ∈ M n,k . Then by the previous lemma, A ( X ◦ Y ) A ∗ = A ( E ( X ⊗ Y ) E ∗ ) A ∗ = ( A E )( X ⊗ Y )( A E ) ∗ , which is in D max n ( S , T ) by definition. Since the latter cone is proper, −C sn ∩C sn = { } . The fact that 1 ⊗ {C sn } follows from the inclusion C sn ⊂ D max n and that 1 ⊗ D max n . Consequently, {C sn } ∞ n =1 defines a matrix ordering on S ⊗ T . (cid:3) From the last paragraph of the proof, we see that C sn ⊂ D max n . In fact,one can further deduce that C sn = D max n by Lemma 3, after proving that thisfamily satisfies Property (T2). Lemma 9.
The family {C sn } ∞ n =1 satisfies Property (T2). That is, given X ∈ M n ( S ) + and Y ∈ M m ( T ) + , X ⊗ Y ∈ C snm .Proof. Let X and Y be as above, note that we may view X ⊗ Y = [ x ij ⊗ Y ] ni,j =1 = x ⊗ J m . . . x n ⊗ J m ... . . . ... x n ⊗ J m . . . x nn ⊗ J m nm × nm ◦ Y . . . Y ... . . . ...
Y . . . Y nm × nm , where J k ∈ M k ( C ) is the matrix of entries all 1. It is easy to see that thesecond matrix in the above equation is Y ⊗ J n . A straight-forward calculationshows that for each k ∈ N , J k has eigenvalues 0 and k , so Y ⊗ J n ∈ M nm ( T ) + .On the other hand, after the “canonical shuffle” [7, Chp 3], the first matrix is WAI HIN NG unitarily equivalent to X ⊗ J m , which is also positive in M nm ( S ). Therefore, X ⊗ Y = ( X ⊗ J m ) ◦ ( Y ⊗ J n ) ∈ C snm and the family {C sn } ∞ n =1 satisfies Property(T2). (cid:3) Remark . Now by Lemma 3 we have the reverse inclusion D max n ⊂ C sn , sothe two families of cones are the same. In particular, Lemma 1 follows easily:every u ∈ D max1 = C s can be represented as u = A ( P ◦ Q ) A ∗ = ( A ∗ P A ) ◦ Q ,for some A ∈ M ,n , P ∈ M n ( S ) + , and Q ∈ M n ( T ) + . If we archimedeanizethe cones {C sn } ∞ n =1 , then we obtain the schur tensor product of operatorsystems and denote it S ⊗ s T ; and it is unitally completely order isomorhpicto S ⊗ max T . Theorem 11.
The cones C sn = D max n , for every n ∈ N . Consequently, foroperator systems, the schur tensor product is the maximal tensor product,i.e. S ⊗ s T = S ⊗ max T . Given operator systems S and T , S ⊗ max T when viewed as an operatorspace, possesses a natural operator space matrix norm || · || osy-max ; that is,given U ∈ M n ( S ⊗ max T ), || U || osy-max = inf (cid:26) r : (cid:20) rI UU ∗ rI (cid:21) ∈ M n ( S ⊗ max T ) + (cid:27) . In particular since
A ⊗
C*-max B = A ⊗ max B for C*-algebras, the C*-maximal tensor norm || · || C*-max is precisely || · || osy-max for C*-algebras. Thefollowing proposition is a slightly generalized version of || · ||
C*-max ≤ || · || s in [9]. Proposition 12.
Let S and T be operator systems. Then the identity map φ : S ⊗ s T → S ⊗ max T is a complete contraction.Proof. Let || U || s <
1, then by scaling, there exist scalar contractions
A, B and X ∈ M n ( S ) and Y ∈ M n ( T ), || X || , || Y || ≤ U = A ( X ◦ Y ) B . Hence, the matrices P = (cid:20) I XX ∗ I (cid:21) ∈ M n ( S ) + and Q = (cid:20) I YY ∗ I (cid:21) ∈ M n ( T ) + . Note that (cid:20) A B ∗ (cid:21) P ◦ Q (cid:20) A ∗ B (cid:21) = (cid:20) AA ∗ A ( X ◦ Y ) BB ∗ ( X ∗ ◦ Y ∗ ) A ∗ B ∗ B (cid:21) = (cid:20) AA ∗ UU ∗ B ∗ B (cid:21) , which is in M n ( S ⊗ s T ) + = M n ( S ⊗ max T ) + .On the other hand, since A and B are scalar contractions, I − AA ∗ and I − B ∗ B are positive in M n . Thus, the operator matrix (cid:20) I − AA ∗ I − B ∗ B (cid:21) is positive in M n ( S ⊗ max T ). By adding the two matrices, we obtain (cid:20) I UU ∗ I (cid:21) ∈ M n ( S ⊗ max T ) + which implies that || U || osy-max ≤ . (cid:3) WO CHARACTERIZATIONS OF THE MAXIMAL TENSOR PRODUCT 7 Factorization Through The Matrix Algebras M n We now turn to study the maximal tensor product using factorization.Recall that every u = P ni =1 x i ⊗ y i ∈ S ⊗ T maybe regarded as a linearmap ˆ u : S d → T , ˆ u ( f ) = P ni =1 f ( x i ) y i , where S d is the linear dual of S .The map ˆ u is independent of representation of u and u ˆ u is a one-to-onecorrespondence between S ⊗ T and L ( S d , T ), where the latter is the spaceof linear maps from the linear dual S d to T .In this section, we use the duality results from [1]. Henceforth, to ensure S d is an operator system, we assume S and T to be finite dimensional. Fixa basis { y = 1 T , . . . , y m } for T , where y i = y ∗ i and || y i || = 1, so that every u ∈ S ⊗ T has a unique representation u = P mi =1 x i ⊗ y i , for some x i ∈ S .To obtain the main result in this section, we introduce a temporary normon S ⊗ T by setting ||| u ||| = P mi =1 || x i || . Lemma 13. If u = P mi =1 x i ⊗ y i ∈ S ⊗ T , where x i = x ∗ i , then ||| u ||| (1 S ⊗ T ) + u ∈ D max1 ( S , T ) .Proof. Because (cid:20) || s i || s i s i || s i || (cid:21) ∈ M ( S ) + , (cid:20) t i t i (cid:21) ∈ M ( T ) + , when we form their schur tensor product, we obtain (cid:20) || s i || ⊗ s i ⊗ t i s i ⊗ t i || s i || ⊗ (cid:21) ∈ D max2 ( S , T ) + . Pre-and-post multiply this matrix by [1 ,
1] shows that || s i || (1 ⊗
1) + s i ⊗ t i ∈D ( S , T ) for each i , thus the sum ||| u ||| (1 ⊗
1) + u ∈ D max1 ( S , T ). (cid:3) Lemma 14.
Let u λ be a net in S ⊗ T . Then ||| u λ ||| → in S ⊗ T if andonly if for each f ∈ S d , || ˆ u λ ( f ) || T → .Proof. Since every u λ has a unique representation u λ = P mi =1 x λi ⊗ y i , ||| u λ ||| → λ || x λi || → i ∈ { , . . . m } , which isequivalent to require that x λi → f ∈ S d , || ˆ u λ ( f ) || T ≤ m X i =1 | f ( x λi ) | · || y i || T → . Conversely, it suffices to show that for each i ∈ { , . . . , m } , lim λ || x λi || = 0.Note that for t = P mi =1 c i y i ∈ T , α ( t ) := P mi =1 | c i | defines a norm on T .Since T is finite dimensional, || t || T ≤ α ( t ) ≤ K || t || T for some K >
0. Foreach f ∈ S d , taking c i = f ( x λi ) shows that m X i =1 | f ( x λi ) | ≤ K || ˆ u λ ( f ) || T → . WAI HIN NG
Hence for each f ∈ S d and i ∈ { , . . . , n } , | f ( x λi ) | →
0. The latter conditionis equivalent to ( x λi ) → S is finite dimensional. (cid:3) Definition 15.
A linear map θ : S → T factors through M n approximately,provided there exists nets of completely positive maps φ λ : S → M n λ and ψ λ : M n λ → T such that ψ λ ◦ φ λ converges to θ in the point-norm topol-ogy. An operator system S is said to have complete positive approximationproperty (CPAP) if the identity map factors through M n approximately.In [2] it is shown that S is (min , max)-nuclear if and only if S has CPFP.We now establish the main theorem in the section. Theorem 16.
Let S and T be finite dimensional operator systems and u ∈ ( S ⊗ max T ) + . The following are equivalent: (1) u is positive in S ⊗ max T . (2) The map ˆ u : S d → T factors through M n approximately: S d ˆ u / / ϕ λ ! ! ❉❉❉❉❉❉❉❉❉ T M n λ ψ λ = = ⑤⑤⑤⑤⑤⑤⑤⑤⑤ Proof.
Suppose u ∈ ( S ⊗ max T ) + . Then for each ε > u ε = ε (1 ⊗
1) + u is in D max .By Lemma 1 it can be written as u ε = P p εij ⊗ q εij , where P ε =[ p εij ] ∈ M n ε ( S ) + and Q ε = [ q εij ] ∈ M n ε ( T ) + . Define ϕ ε : S d → M n ε by ϕ ε ( f ) = [ f ( p εij )] and ψ ε : M n ε → T by ψ ε ([ a ij ]) = P a ij q εij . Note that ϕ ε is completely positive by definition of S d . For ψ ε , first consider thecompletely positive map [ a ij ] [ a ij ] ⊗ Q ε . We regard [ a ij ] ⊗ Q ε as thematrix [ q εij [ a kl ]] n ε i,j and pre-and-post multiply it by [ E , E , . . . , E n ε ], thenwe obtain the matrix [ q εij a ij ] n ε i,j =1 ∈ M n ε ( T ) + . Now pre-and-post multiplyit by the row vector of length n ε whose entries are 1; this yields P i,j a ij q εij ,and ψ ε is completely positive. It follows that ˆ u ε = ψ ε ◦ ϕ ε and it convergesto ˆ u as ε → ψ λ ◦ ϕ λ corresponds to a w λ ∈ S ⊗ T so that ˆ w λ = ψ λ ◦ ϕ λ . Identifying ϕ λ to P λ = [ p λij ] ∈ M n λ ( S ) + and ψ λ to Q λ = [ q λij ] ∈ M n λ ( T ) + shows that w λ = P n λ i,j p λij ⊗ q λij ∈ D max1 ( S ⊗ T ). By the point-norm convergence and the last lemma, lim λ ||| u − w λ ||| →
0. Now for each λ , take ε λ = ||| u − w λ ||| , and Lemma 13 asserts that ε λ (1 ⊗
1) + ( u − w λ ) ∈ D max1 ( S , T ). For each ε > λ , so that ε λ < ε so ε (1 ⊗
1) + ( u − w λ ) ∈ D max1 ( S , T ). Hence ε ⊗ u ∈ D max1 ( S , T ) and u ∈ ( S ⊗ max T ) + . (cid:3) By the identification M m ( S ⊗ max T ) ∼ = S ⊗ max M m ( T ), we establish thefollowing characterization of the matricial cone structure of the maximaltensor product. WO CHARACTERIZATIONS OF THE MAXIMAL TENSOR PRODUCT 9
Theorem 17.
An element U ∈ M m ( S ⊗ max T ) is positive if and only if ˆ U : S d → M m ( T ) factors through M n approximately. We would like to remark that this result is rather interesting. In [1] wehave
S ⊗ min T = ( S d ⊗ max T d ) d . Combining with the result after Theorem1, we deduce that ( S ⊗ min T ) + = CP ( S d , T ); whereas by Theorem 16,( S ⊗ max T ) + corresponds to a proper subcone of CP ( S d , T ) whose elementsfactor through M n approximately. Since the minimal and maximal tensorproducts each represents respectively the largest and smallest matricial conestructure one can equip on S ⊗ T , it brings up the natural question aboutthe corresponding subsets with respect to other tensor products in [4].Symmetry and projectivity of the maximal tensor product can also beobtained by this diagram.
Proposition 18.
The maximal tensor product is symmetric and projective.Proof.
Let u = P s i ⊗ t i ∈ S ⊗ max T . By dualizing the diagram in Theorem16, one sees that T d ψ dλ % % ❑❑❑❑❑❑❑❑❑❑ (ˆ u ) d / / S dd = S M dn λ = M n λ ϕ dλ ♣♣♣♣♣♣♣♣♣♣♣ where (ˆ u ) d is the map g P g ( t i ) s i . Consequently, P t i ⊗ s i ∈ ( T ⊗ max S ) + if and only if the above diagram holds, which by duality is equivalent toTheorem 16 (2). This shows that S ⊗ max
T ∼ = max T ⊗ S at the ground level.At each matrix level n , identifying M n ( S ⊗ max T ) = S ⊗ max M n ( T ) andreplacing T by M n ( T ) proves symmetry of the maximal tensor product.For projectivity, first consider a complete quotient map q : S → R . Weclaim that every u ∈ ( R ⊗ max T ) + can be lifted to some w ∈ ( S ⊗ max T ) + .Indeed, by Theorem 16 there are ϕ λ and ψ λ such that ψ λ ◦ ϕ λ convergesto u in the point-norm topology. Since q d : R d → S d is a complete orderinclusion, by the Arveson extension theorem, there is a completely positiveΦ λ : S d → M n λ extending ϕ λ . Hence, the following diagram commutes: R d ˆ u / / q d (cid:15) (cid:15) ϕ λ ' ' ◆◆◆◆◆◆◆◆◆◆◆◆◆ TS d Φ λ / / ❴❴❴❴❴❴ M n λ ψ λ > > ⑤⑤⑤⑤⑤⑤⑤⑤⑤ Let [ s λij ] ∈ M n λ ( S ) + be the corresponding matrix of Φ λ and likewise for[ t λij ] ∈ M n λ ( T ) + of ψ λ . Then w λ = P i,j s λij ⊗ t λij ∈ ( S ⊗ max T ) + by the schurcharacterization and ˆ w = ψ λ ◦ Φ λ . To this end, we claim that there is asubnet w λ α converging to some positive w such that ˆ w ◦ q d = ˆ u . Let δ denote the unit in R d ⊂ coi S d . Then || ˆ w λ ( δ ) || = || ψ λ ◦ ϕ λ ( δ ) || →|| ˆ u ( δ ) || asserts there is λ such that the set {|| ˆ w λ ( δ ) || : λ > λ } is bounded.However, for completely positive maps, || ˆ w λ ( δ ) || = || ˆ w λ || cb = || ˆ w λ || and thelatter norm also defines a norm on S ⊗T . By the equivalence of norm topolo-gies, { w λ : λ > λ } is bounded in ( S ⊗ max T ) + and possesses a convergentsubnet w λ α → w ∈ ( S ⊗ max T ) + . Therefore for each f ∈ R d , || ( ˆ w ◦ q d − ˆ u ) f || = || (lim α ˆ w λ α ◦ q d − ˆ u ) f || = || (lim α ψ λ α ◦ Φ λ α ◦ q d − ˆ u ) f || = || (lim α ψ λ α ◦ ϕ λ α − ˆ u ) f || = lim α || ( ψ λ α ◦ ϕ λ α − ˆ u ) f || = lim λ || ( ψ λ ◦ ϕ λ − ˆ u ) f || → , where the second line follows from Lemma 14. Consequently every positive u ∈ R ⊗ max T can be lifted to a positive w ∈ S ⊗ max T . This implies that forevery such u and for each ε >
0, the element w + ε (1 S ⊗ T ) ∈ ( S ⊗ max T ) + satisfies ( q ⊗ id )( w + ε (1 S ⊗ T )) = u + ε (1 R ⊗ T ).Finally, again by identifying M n ( R ⊗ max T ) to R ⊗ max M n ( T ) and like-wise for S ⊗ max M n ( T ), we prove that the maximal tensor product is leftprojective. By symmetry, it is right projective and hence projective. (cid:3) At last, we remark that this characterization of the maximal tensor prod-uct indeed coincides with the (min , max)-nuclearity result in [2, 3]. Corollary 19.
Let δ i be the dual basis of y i for T d . Then u = P mi =1 δ i ⊗ y i ∈T d ⊗ max T is positive if and only if T is (min, max)-nuclear.Proof. Let S = T d and note that ˆ u is the identity map on T . Moreover, u ∈ ( T d ⊗ max T ) + if and only if ˆ u factors through M n , which by [2, Theorem3.2], if and only if T is (min, max)-nuclear. (cid:3) Acknowledgment
The author would like to thank Vern I. Paulsen for his valuable adviceand inspirations in writing this paper, and thank Prof. Vandana Rajpal forintroducing the schur tensor product to the author.
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