Relative weak injectivity of operator system pairs
aa r X i v : . [ m a t h . OA ] M a y RELATIVE WEAK INJECTIVITY OF OPERATOR SYSTEMPAIRS
ANGSHUMAN BHATTACHARYA
Abstract.
The concept of a relatively weakly injective pair of operator sys-tems is introduced and studied in this paper, motivated by relative weak injec-tivity in the C*-algebra category. E. Kirchberg [12] proved that the C ∗ -algebraC ∗ ( F ∞ ) of the free group F ∞ on countably many generators characterises rel-ative weak injectivity for pairs of C ∗ -algebras by means of the maximal tensorproduct. One of the main results of this paper shows that C ∗ ( F ∞ ) also char-acterises relative weak injectivity in the operator system category. A key toolis the theory of operator system tensor products [10, 11]. Introduction
A pair ( A , B ) of unital C ∗ -algebras is a relatively weakly injective pair for everyunital C ∗ -algebra C , A ⊗ max C is a unital C ∗ -subalgebra of B ⊗ max C . (In particular,one has that A is a unital C ∗ -subalgebra of B .) It is common to say that A isrelatively weakly injective in B if the pair ( A , B ) is a relatively weakly injective pair.Relative weak injectivity for pairs of C ∗ -algebras was introduced by E. Kirchberg[12] and was motivated by the work of E.C. Lance [14] on the weak expectationproperty for C ∗ -algebras.The purpose of this paper is to introduce and study a notion of relative weakinjectivity for pairs ( S , T ) of operator systems S and T . To do so, one thereforeneeds to consider operator system tensor products. Although the theory of tensorproducts [10, 11] in the category O , whose objects are operator systems and whosemorphisms are unital completely positive (ucp) linear maps, shares many similar-ities with C ∗ -algebraic tensor products, there some significant differences, particu-larly when considering the operator system analogue of the maximal C ∗ -algebraictensor product, ⊗ max . With the max tensor product, there are two distinct ten-sor products (denoted by ⊗ c and ⊗ max ) in the category O that collapse to themaximal C ∗ -algebraic tensor product on the subcategory of unital C ∗ -algebras andunital ∗ -homomorphisms. In this paper an operator system analogue of relativeweak injectivity will be developed using the commuting tensor product, ⊗ c . Specif-ically, a pair ( S , T ) of operator systems is said to be a relatively weakly injective pair if, for every operator system R , S ⊗ c R is a unital operator subsystem of T ⊗ c R .The C ∗ -algebra C ∗ ( F ∞ ) of the free group F ∞ on countably infinitely many gen-erators is universal in the sense that every unital separable C ∗ -algebra is a quotientof C ∗ ( F ∞ ). Therefore, it is striking that the C ∗ -algebra C ∗ ( F ∞ ) can be used tocharacterise both the weak expectation property and relative weak injectivity, asdemonstrated by two important theorems of Kirchberg. More precisely, A has WEP Mathematics Subject Classification.
Primary 46L07; Secondary 46L06, 47L05.
Key words and phrases. operator system, commuting tensor product, relative weak injectivity. if and only if
A ⊗ min C ∗ ( F ∞ ) = A ⊗ max C ∗ ( F ∞ ) [12, Proposition 1.1], and ( A , B ) isa relatively weakly injective pair if and only if A ⊗ max C ∗ ( F ∞ ) ⊂ B ⊗ max C ∗ ( F ∞ )[12, Proposition 3.1].An operator system analogue of the weak expectation property for C ∗ -algebras–namely the double commutant expectation property–was introduced and studied in[9, 11], and it was shown that C ∗ ( F ∞ ) characterises this property. One of the mainresults of this paper shows that C ∗ ( F ∞ ) also characterises relative weak injectivityof operator system pairs (Theorem 4.1). In addition to establishing some alternatecharacterisations of relative weak injectivity, the existence of relatively weakly in-jective pairs ( S , T ) in the operator system category will be achieved (in Theorem4.2) in a manner similar to Kirchberg’s result [12, Corollary 3.5] that every unitalseparable C ∗ -algebra is a unital C ∗ -subalgebra of a unital separable C ∗ -algebra withthe weak expectation property. The paper concludes with a selection of examples.The theory of operator algebraic tensor products is treated in the books [1,18], while operator system tensors products are developed in the papers [10, 11].Standard references for operator systems and completely positive maps are [16, 17].2. The Commuting Operator System Tensor Product If S and T are operator systems, then the notation S ⊂ T means that S is aunital operator subsystem of T . That is, if 1 S and 1 T denote the distinguishedArchimedean order units for S and T respectively, then 1 S = 1 T . Unless thecontext is not clear, the order unit for an operator system will be denoted simplyby 1.The algebraic tensor product S ⊗ T of operator systems S and T is a ∗ -vectorspace. An operator system tensor product structure on S⊗T is a family τ = {C n } n ∈ N of cones C n ⊂ M n ( S ⊗ T ) such that:(1) (
S ⊗ T , τ, S ⊗ T ) is an operator system, denoted by S ⊗ τ T , in which1 S ⊗ T is an Archimedean order unit,(2) M n ( S ) + ⊗ M m ( T ) + ⊂ C nm , for all n, m ∈ N , and(3) if φ : S → M n and ψ : T → M m are unital completely positive (ucp) maps,then φ ⊗ ψ : S ⊗ τ T → M nm is a ucp map.Recall that a unital completely positive linear (ucp) map φ : S → T of operatorsystems is a complete order isomorphism if it is a linear bijection and if both φ and φ − are completely positive. If the ucp map φ is merely injective, then φ isa complete order injection if φ is a complete order isomorphism of between S andthe operator subsystem φ ( S ) of T .If S ⊂ T and S ⊂ T are inclusions of operator systems, and if ι j : S j → T j are the inclusion maps, then for any operator system structures τ and σ on S ⊗ S and T ⊗ T , respectively, the notation (as used in [6] also) S ⊗ τ S ⊂ + T ⊗ σ T expresses the fact that the linear vector-space embedding ι ⊗ ι : S ⊗ S → T ⊗ T is a ucp map S ⊗ τ S → T ⊗ σ T . That is, S ⊗ τ S ⊂ + T ⊗ σ T if and onlyif M n ( S ⊗ τ S ) + ⊂ M n ( T ⊗ σ T ) + for every n ∈ N . If, in addition, ι ⊗ ι is acomplete order isomorphism onto its range, then this is denoted by S ⊗ τ S ⊂ coi T ⊗ σ T . Thus,
S ⊗ τ T = S ⊗ σ T means S ⊗ τ T ⊂ coi
S ⊗ σ T and S ⊗ σ T ⊂ coi
S ⊗ τ T . ELATIVE WEAK INJECTIVITY OF OPERATOR SYSTEM PAIRS 3
The commuting operator system tensor product ⊗ c was introduced and studiedin [10] and will be defined below. A slight simplification in the definition is affordedby the following lemma, which allows one to restrict to ucp maps rather than useall completely positive maps. Lemma 2.1. [2, Lemma 2.2] , [3, Lemma 5.1.6] Let
S ⊂ B ( K ) be an operatorsystem and φ : S → B ( H ) be a completely positive map. Then there exists a ucpmap ˜ φ : S → B ( H ) such that φ ( · ) = φ (1) ˜ φ ( · ) φ (1) . The proof of the lemma above describes the map ˜ φ as a strong limit of ˜ φ ( n ) in B ( H ), where˜ φ ( n ) ( s ) = (cid:18) φ (1) + 1 n (cid:19) − φ ( s ) (cid:18) φ (1) + 1 n (cid:19) − + h sη, η i (1 − P φ (1) ) , for η ∈ K , and P φ (1) is the projection onto the closure of the range of φ (1). Thus, foroperator systems S ⊂ B ( K S ) and T ⊂ B ( K T ), if φ : S → B ( H ) and ψ : T → B ( H )are completely positive maps with commuting ranges, then the corresponding ucpmaps ˜ φ and ˜ ψ also have commuting ranges.Denote by ucp( S , T ) the set of all pairs ( φ, ψ ) of ucp maps from S and T ,respectively, into B ( H ) for some Hilbert space H , such that φ ( S ) commutes with ψ ( T ). For each ( φ, ψ ) ∈ ucp( S , T ) let φ · ψ : S ⊗ T → B ( H ) be the unique linearmap whose value on elementary tensors is given by φ · ψ ( x ⊗ y ) = φ ( x ) ψ ( y ) . Define cones by C comm n = { η ∈ M n ( S ⊗ T ) : ( φ · ψ ) ( n ) ( η ) ≥ , for all ( φ, ψ ) ∈ ucp( S , T ) } . It was shown in [10] that the collection of cones above is a matrix ordering on
S ⊗ T with Archimedean matrix order unit 1 S ⊗ T . Definition 2.2.
The operator system ( S ⊗ T , {C comm n } n ∈ N , S ⊗ T ) is called the commuting operator system tensor product of S and T and is denoted by S ⊗ c T . The following notation, introduced in [11], will be used.
Notation 2.3. If X and Y are operator systems, then X ˆ ⊗ c Y shall denote thenorm-completion of X ⊗ c Y . For any subspaces X ⊂ X and Y ⊂ Y , X ⊗Y denotes the closure of X ⊗Y in X ˆ ⊗ c Y .The symbol ⊗ max is reserved in this paper (unlike in [10, 11]) for the maximalC ∗ -algebra tensor product. An important fact: if two unital C ∗ -algebras A and B are considered as operator systems, then A ˆ ⊗ c B = A ⊗ max B [10, Theorem 6.6].In principle an abstract operator system S generates many different C ∗ -algebras.The largest such C ∗ -algebra is called the universal C ∗ -algebra generated by S . Thatis, a unital C*-algebra A is universal for S if:(1) there is a unital complete order injection ι u : S → A ,(2) A is generated by ι u ( S ), and(3) if φ : S → B is a ucp map into another C*-algebra B , then there is ahomomorphism π : A → B such that φ = π ◦ ι u . A. BHATTACHARYA
It was shown in [13, Proposition 8] that every operator system has a universalC*-algebra, unique up to isomorphism, and an explicit construction was given.Therefore, C ∗ u ( S ) shall unambiguously denote the universal C*-algebra generatedby S . Theorem 2.4. ([11, Lemma 2.5])
For all operator systems S and T , S ⊗ c T ⊂ coi
S ⊗ c C ∗ u ( T ) ⊂ coi C ∗ u ( S ) ⊗ max C ∗ u ( T ) . Corollary 2.5.
For every unital C ∗ -algebra A , operator system S , and n ∈ N , theoperator systems M n ( S ⊗ c A ) and S ⊗ c M n ( A ) are completely order isomorphic. Preliminary Results
In this section we will use the fact that the matricial order on an operator system S gives rise to a norm k · k M n ( S ) on each matrix space M n ( S ) [16, Chapter 3]. Lemma 3.1.
Let S be an operator system and A be a unital C*-algebra. A linearmap φ : S ⊗ c A → B ( H ) is a ucp map if and only if there is a Hilbert space K ,homomorphisms π : C ∗ u ( S ) → B ( K ) and ρ : A → B ( K ) with commuting ranges, andan isometry V : H → K such that φ ( s ⊗ a ) = V ∗ π ( s ) ρ ( a ) V for all s ∈ S and a ∈ A .Proof. Because
S ⊗ c A ⊂ coi C ∗ u ( S ) ⊗ max A by Proposition 2.4, φ admits a ucp ex-tension Φ : C ∗ u ( S ) ⊗ max A → B ( H ). Therefore, by [10, Corollary 6.5], the restrictionof Φ to S ⊗ c A has the structure indicated in the statement of the lemma. (cid:3) Lemma 3.2.
Let S be a operator system. Let {S i } i ∈I be the set of all separablenontrivial operator subsystems of S (that is, S i ⊂ S ). Then, there is a non-trivialultrafilter U on I such that the map Ψ :
S → Q U C ∗ u ( S i ) given by x ( ψ i ( x )) U , where ψ i ( x ) = x if x ∈ S i or otherwise, is a unital completely positive linear map,where Q U denotes the C*-ultraproduct.Proof. Note that the set I is partially ordered by inclusion of the correspondingoperator subsystems S i and that S = S S i . Consider a cofinal ultrafilter U on thedirected set I . The map Ψ defined in the statement of the lemma is linear becauseof the structure of C ∗ -ultraproducts (see [7]). To show that Ψ is ucp it is sufficientto show that Ψ is a complete isometry (following the discussion after [17, Remark2.8.4]).If x ∈ S , note that the set { i | x ∈ S i } ∈ U . To see this, simply observe that { i | x ∈ S i } = { i | i ≥ i x } , where S i x = span { , x, x ∗ } . Now, for n = 1, k Ψ( x ) k = k ( ψ i ( x )) U k = lim U k ψ i ( x ) k = k x k by the preceding comment.For n >
1, we use a similar argument as follows. Let X = ( x kl ) ∈ M n ( S ). Now,an ultrafilter is closed under finite intersections. So, I X = { i | x kl ∈ S i ∀ k, l } = \ k,l { i | x kl ∈ S i } ELATIVE WEAK INJECTIVITY OF OPERATOR SYSTEM PAIRS 5 is in U . Finally, using the identification M n ( Q U C ∗ u ( S i )) = Q U M n (C ∗ u ( S i )) (seeRemark on Pg-60 of [17]) we obtain k Ψ ( n ) ( X ) k = k (Ψ( x kl )) k,l k = k (( ψ i ( x kl )) U ) k,l k = k (( ψ i ( x kl )) k,l ) U k = lim U k ( ψ i ( x kl )) k,l k = k ( x kl ) k,l k M n ( S i ) ,i ∈ I X = k X k , thereby showing that Ψ is a complete isometry. (cid:3) The following result is of central importance in what follows.
Lemma 3.3.
Assume that A is a C*-algebra and T is an operator system, andfix x ∈ T ⊗ A . If {T i } i ∈I ( x ) is the directed set of all separable unital operatorsubsystems of T for which x ∈ T i ⊗ A , then k x k T ⊗ c A = lim I ( x ) k x k T i ⊗ c A . Proof.
Let us denote by k x k ( · ) the norm k x k ( · ) ⊗ c A . If x ∈ T ⊂ T , then T ⊗ c A ⊂ + T ⊗ c A implies that k x k T ≤ k x k T . Thus, lim I k x k T i exists, since it is a decreasing net, and k x k T ≤ lim I k x k T i . To establish the opposite inequality, following the techniques in the proof of [15,Proposition 3.4],we proceed as follows.Assume that k x k T i ≥ i ∈ I . Thus, k x k T i = k x k C ∗ u ( T i ) ⊗ max A ≥ π i , ρ i of C ∗ u ( T i ) and A respectively, on B ( H i )with commuting ranges such that k π i · ρ i ( x ) k ≥ . Using the map Ψ from Lemma 3.2 above and the injective ∗ -homomorphism ι : A ֒ → Q U A , where U is the same ultrafilter over the same index set I as in Lemma 3.2or above, we have ucp maps φ : T → B ( H T ) and ρ : A → B ( H T ) with commutingranges and such that k φ · ρ ( x ) k ≥ , where H T = Q U H i , φ = ( Q U π i ) ◦ Ψ and ρ = ( Q U ρ i ) ◦ ι .Now, φ · ρ is a ucp map of T ⊗ c A . By Lemma 3.1, there exist representations π and ρ of C ∗ u ( T ) and A with commuting ranges and an isometry V such that φ · ρ ( x ) = V ∗ π · ρ ( x ) V. Since k φ · ρ ( x ) k ≥
1, we have k π · ρ ( x ) k ≥ V is an isometry. But then, k x k T = k x k C ∗ u ( T ) ⊗ max A ≥ , thereby showing that k x k T = lim I k x k T i . (cid:3) Remark 3.4.
Lemma 3.3 is also true if A is only an operator system, as in thatcase, one may simply carry out the argument above with C ∗ u ( A ) and arrive at theconclusion by virtue of Proposition 2.4. A. BHATTACHARYA Main Results
Recall that a pair ( S , T ) of operator systems is a relatively weakly injective pairif, for every operator system R , S ⊗ c R ⊂ coi
T ⊗ c R . It is also convenient to say that T is relatively weakly injective in T if ( S , T ) isrelatively weakly injective pair.The first main result is an operator system version of Kirchberg’s theorem [12,Proposition 3.1]. Theorem 4.1.
The following statements are equivalent for operator systems S and T for which S ⊂ T : (1) ( S , T ) is a relatively weakly injective pair of operator systems; (2) S ⊗ c C ∗ ( F ∞ ) ⊂ coi T ⊗ c C ∗ ( F ∞ ) ; (3) For any ucp map φ : S → B ( H ) , there exist a ucp map Φ :
T → φ ( S ) ′′ suchthat Φ | S = φ ; (4) (C ∗ u ( S ) , C ∗ u ( T )) is a relatively weakly injective pair of C*-algebras.Proof. The order of implications to be proved is (4) ⇒ (2) ⇒ (1) ⇒ (3) ⇒ (4).(4) ⇒ (2). Assume that X ∈ M n ( S ⊗ C ∗ ( F ∞ )) is positive in M n ( T ⊗ c C ∗ ( F ∞ )).We need to show that X ∈ M n ( S ⊗ c C ∗ ( F ∞ )) + . Because X ∈ M n ( T ⊗ c C ∗ ( F ∞ )) + ⊂ M n (C ∗ u ( T ) ⊗ max C ∗ ( F ∞ )) + , hypothesis (4) implies X ∈ M n (C ∗ u ( S ) ⊗ max C ∗ ( F ∞ )) + , and so X is positive in M n ( S ⊗ c C ∗ ( F ∞ )) because S ⊗ c C ∗ ( F ∞ ) ⊂ coi C ∗ u ( S ) ⊗ c C ∗ ( F ∞ ).(2) ⇒ (1). Let R be an arbitrary operator system. By Theorem 2.4, W ⊗ c R ⊂ coi
W ⊗ c C ∗ u ( R ) for every operator system W ; thus, if we can show that S ⊗ c C ∗ u ( R ) ⊂ coi T ⊗ c C ∗ u ( R ), then we deduce immediately that S ⊗ c R ⊂ coi
T ⊗ c R .To begin, assume that R is separable. Hence, there is an ideal K of C ∗ ( F ∞ ) suchthat C ∗ u ( R ) = C ∗ ( F ∞ ) / K . By [11, Corollary 5.17], and using Notation 2.3, S ⊗ c C ∗ u ( R ) ⊂ coi S ˆ ⊗ c C ∗ u ( R ) = S ˆ ⊗ c C ∗ ( F ∞ ) S⊗K . The hypothesis
S ⊗ c C ∗ ( F ∞ ) ⊂ coi T ⊗ c C ∗ ( F ∞ ) implies that S ⊗ c C ∗ ( F ∞ ) ⊂ coi C ∗ u ( T ) ⊗ c C ∗ ( F ∞ ), again by Theorem 2.4. Therefore, [11, Proposition 5.14] yields S ˆ ⊗ c C ∗ ( F ∞ ) S⊗K ⊂ coi C ∗ u ( T ) ˆ ⊗ c C ∗ ( F ∞ )C ∗ u ( T ) ⊗K = C ∗ u ( T ) ⊗ max C ∗ u ( R ) . Thus,
S ⊗ c C ∗ u ( R ) ⊂ coi C ∗ u ( T ) ⊗ c C ∗ u ( R ), which implies S ⊗ c C ∗ u ( R ) ⊂ coi T ⊗ c C ∗ u ( R )and, hence, S ⊗ c R ⊂ coi
T ⊗ c R .Now assume that R is an arbitrary nonseparable operator system. We haveproved above that S ⊗ c R ⊂ coi T ⊗ c R for every separable operator system R .Fix x ∈ S ⊗ R and choose a separable operator subsystem R ⊂ R such that x ∈ S ⊗ R . Thus, S ⊗ c R ⊂ T ⊗ c R . By the beginning of the proof of Lemma3.3 we have the inequality k x k S⊗ c R ≤ k x k S⊗ c R = k x k T ⊗ c R . This inequality above holds for any separable operator subsystem R ⊂ R for which x ∈ S ⊗ R . Lemma 3.3 (or Remark 3.4) thus implies k x k S⊗ c R ≤ k x k T ⊗ c R , which ELATIVE WEAK INJECTIVITY OF OPERATOR SYSTEM PAIRS 7 in turn implies k x k S⊗ c R = k x k T ⊗ c R . Next, for n >
1, fix X ∈ M n ( S ⊗ R ) ⊂ M n ( S ⊗ C ∗ u ( R )) ∼ = S ⊗ M n (C ∗ u ( R )). Onealso has M n ( S ⊗ c C ∗ u ( R )) ∼ = S ⊗ c M n (C ∗ u ( R )). Now, just as in the n = 1 case,there exists a separable operator system R n ⊂ M n (C ∗ u ( R )) such that X ∈ S ⊗ R n and therefore, for any separable operator system R n ⊂ M n (C ∗ u ( R )) for which X ∈S ⊗ R n , we have the inequality k X k M n ( S⊗ c C ∗ u ( R )) = k X k S⊗ c M n (C ∗ u ( R )) ≤ k X k S⊗ c R n = k X k T ⊗ c R n . This implies (as in case of n = 1) that k X k M n ( S⊗ c C ∗ u ( R )) ≤ k X k T ⊗ c M n (C ∗ u ( R )) = k X k M n ( T ⊗ c C ∗ u ( R )) , which in turn implies that k X k M n ( S⊗ c C ∗ u ( R )) = k X k M n ( T ⊗ c C ∗ u ( R )) . That is, theinclusion map S ⊗ R → T ⊗ R is a unital complete isometry
S ⊗ c R → T ⊗ c R and, hence, is a complete order injection.(1) ⇒ (3). Let φ : S → B ( H ) be a ucp map. Since ( S , T ) is a relatively weaklyinjective pair, and because the commutant φ ( S ) ′ ⊂ B ( H ) of φ ( S ) is a C ∗ -algebra, S ⊗ c φ ( S ) ′ ⊂ coi T ⊗ c φ ( S ) ′ ⊂ coi C ∗ u ( T ) ⊗ max φ ( S ) ′ . By the definition of commuting tensor product, φ · id φ ( S ) ′ is a ucp map on S ⊗ c φ ( S ) ′ with values in B ( H ). Take an Arveson extension Ψ of φ · id φ ( S ) ′ to C ∗ u ( T ) ⊗ max φ ( S ) ′ and define a ucp map Φ on T byΦ( t ) = Ψ( t ⊗ , for all t ∈ T . Obviously, Φ | S = φ . Finally, to see that Φ takes values in φ ( S ) ′′ , oneinvokes the usual multiplicative domain argument for completely positive maps.This concludes our claim (1) ⇒ (3).(3) ⇒ (4). Since S ⊂ T , C ∗ u ( S ) is a unital C*-subalgebra of C ∗ u ( T ) [13,Proposition 9]. Let π U : C ∗ u ( S ) → B ( H U ) be the universal representation ofC ∗ u ( S ). Then π U | S : S → B ( H U ) is a ucp map. By hypothesis, π U | S extendsto φ : T → ( π U | S ( S )) ′′ ⊂ ( π U (C ∗ u ( S ))) ′′ . Now, since C ∗ u ( T ) is generated as analgebra by T , the unique homomorphism from C ∗ u ( T ) extending φ takes values in( π U (C ∗ u ( S ))) ′′ . Further, since this homomorphism extends π U | S , it fixes π U , whichcompletes the proof. (cid:3) The second main result shows the abundant existence of pairs of relatively weaklyinjective operator systems and is a generalisation of [12, Lemma 3.4].
Theorem 4.2. If S is a separable operator subsystem of an operator system T ,then there exists a separable operator system R such that S ⊂ coi
R ⊂ coi T and R is relatively weakly injective in T .Proof. Let { s k } k ∈ N be a dense sequence in S ⊗ c C ∗ ( F ∞ ). Using Lemma 3.3, wechoose separable operator subsystems S n of T such that, S ⊂ S ⊂ S ⊂ . . . and k s k k S n ≤ k s k k T + n for 1 ≤ k ≤ n . Let S (1) = S S i . Then S (1) is aseparable operator system containing S , such that, for all x ∈ S ⊗ c C ∗ ( F ∞ ), onehas k x k S (1) = k x k T . By iterating the argument above with S (1) instead of S weobtain a sequence of separable operator systems S ⊂ S (1) ⊂ S (2) ⊂ . . . such that k · k S ( n ) = k · k T on S ( n − ⊗ c C ∗ ( F ∞ ). Define X = S S ( k ) . Thus, X is a separableoperator system containing S such that k · k X = k · k T . A. BHATTACHARYA
Replacing X for S and M (C ∗ ( F ∞ )) for C ∗ ( F ∞ ), repeat the procedure de-scribed above to obtain a separable operator system X such that, for all x ∈X ⊗ c M (C ∗ ( F ∞ )), we have k x k X ⊗ c M (C ∗ ( F ∞ )) = k x k T ⊗ c M (C ∗ ( F ∞ )) . In other words, using the identification
W ⊗ c M (C ∗ ( F ∞ )) = M ( W ⊗ c C ∗ ( F ∞ )) foroperator systems W , we have that the inclusion map X ⊗ c C ∗ ( F ∞ ) → T ⊗ c C ∗ ( F ∞ )is a 2-isometry.Further iterations of the procedure above gives us S ⊂ X ⊂ X ⊂ X ⊂ . . . T such that the inclusion map X k ⊗ c C ∗ ( F ∞ ) → T ⊗ c C ∗ ( F ∞ ) is a k -isometry.Finally, set R = S X k . To show that R is relatively weakly injective in T ,it is enough, by Theorem 4.1, to show that the inclusion map R ⊗ c C ∗ ( F ∞ ) →T ⊗ c C ∗ ( F ∞ ) is a complete isometry.For Y ∈ R ⊗ M n (C ∗ ( F ∞ )) there exists an integer k Y > n such that Y ∈ X k ⊗ M n (C ∗ ( F ∞ )) for all k > k Y . Now recall the fact that the inclusion maps X k ⊗ c C ∗ ( F ∞ ) → T ⊗ c C ∗ ( F ∞ ) are k -isometries. As a consequence, for n < k Y < k the inclusions X k ⊗ c C ∗ ( F ∞ ) → T ⊗ c C ∗ ( F ∞ ) are also n -isometries. Therefore, byLemma 3.3 we have k Y k R⊗ c M n (C ∗ ( F ∞ )) = lim k k Y k X k ⊗ c M n (C ∗ ( F ∞ )) = lim k>k Y k Y k X k ⊗ c M n (C ∗ ( F ∞ )) = k Y k T ⊗ c M n (C ∗ ( F ∞ )) . This shows that R is relatively weakly injective in T , contains S , and is separable,thereby concluding the proof. (cid:3) Remarks on relative weak injectivity with respect to theoperator system maximum tensor product
The maximal C ∗ -tensor product has two distinct generalizations in the O cat-egory, namely the commuting tensor product and the operator system maximaltensor product. See [10, 11] for details. This article focuses on relative weakinjectivity with respect to the former. A natural question would be to seek char-acterisations of relatively weakly injective operator system pairs with respect tothe operator system maximal tensor product. Let us denote the operator systemmaximal tensor product by ⊗ m . Proposition 5.1.
Let
S ⊂ coi T . The following statements are equivalent : (1) For any operator system R , S ⊗ m R ⊂ coi
T ⊗ m R . (2) There exists a ucp map
Φ :
T → S ∗∗ , such that Φ( s ) = s for all s ∈ S .Proof. (1) ⇒ (2). Consider the bidual inclusion S ∗∗ ⊂ coi T ∗∗ ⊂ coi B ( H ), wherethe second inclusion is weak ∗ -WOT homeomorphic exactly as in the proof of [8,Theorem 4.1]. Repeating the proof of [8, Theorem 4.1 (iii) ⇒ (iv)] verbatim givesthe required result.(2) ⇒ (1). For X ∈ M n ( T ⊗ m R ) + ∩ M n ( S ⊗ R ), one has X = (Φ ⊗ id) ( n ) ( X ) ∈ M n ( S ⊗ m R ) ⊂ coi M n ( S ∗∗ ⊗ m R ), where the last inclusion is due to [11, Lemma6.5]. (cid:3) Remark 5.2.
Comparing Proposition 5.1 and Theorem 4.1, it is unlikely that auniversal characterisation of the likes of Theorem 4.1(2) exists in the ⊗ m case. As ELATIVE WEAK INJECTIVITY OF OPERATOR SYSTEM PAIRS 9 a consequence, it cannot be ascertained that an existence result similar to Theorem4.2 holds for the maximal operator system tensor product.6.
Examples
Operator systems generated by free unitaries.
Denote the generators ofthe free group F ∞ by { u j } j ∈ N . In C ∗ ( F ∞ ), each u j is a unitary and so, for each n ∈ N , define S n = span { u − n , . . . , u − , , u , . . . , u n } , which is an operator subsystem of C ∗ ( F n ). Example 6.1.
For n ∈ N , the pair ( S n , C ∗ ( F n )) is a relatively weakly injective pairof operator systems. The proof of this assertion is adapted from the proof of [5, Lemma 4.1] andmakes use of our main result, Theorem 4.1. Let φ : S n → B ( H ) be a ucp map.By Theorem 4.1, it is enough to show that φ extends to C ∗ ( F n ), taking values in φ ( S n ) ′′ . For each contraction φ ( u i ), 1 ≤ i ≤ n , consider its Halmos unitary dilation W i on H ⊕ H given by W i = (cid:20) φ ( u i ) (1 − φ ( u i ) φ ( u − i )) (1 − φ ( u − i ) φ ( u i )) − φ ( u − i ) (cid:21) Let T ∈ φ ( S n ) ′ and consider the operator ˜ T = (cid:20) T T (cid:21) ∈ B ( H ⊕ H ). Now,by functional calculus, ˜ T commutes with W i for all 1 ≤ i ≤ n . Since u , . . . , u n are universal unitaries in C ∗ ( F n ), there is a unique homomorphism π : C ∗ ( F n ) →B ( H ⊕ H ), such that π ( u i ) = W i for 1 ≤ i ≤ n . Let P = (cid:20) I
00 0 (cid:21) . Define ucp map˜ φ : C ∗ ( F n ) → B ( H ) by ˜ φ ( · ) = P π ( · ) | H . Note that, ˜ φ extends φ and ˜ T commuteswith P . Since, ˜ T commutes with every W i , it commutes with π (C ∗ ( F n )). Thus, for x ∈ C ∗ ( F n ) we have˜ φ ( x ) T = P π ( x ) P ˜ T P = P π ( x ) ˜ T P = P ˜ T π ( x ) P = P ˜ T P π ( x ) P = T ˜ φ ( x ) . So, ˜ φ ( x ) ∈ φ ( S n ) ′′ as T was chosen arbitrarily in φ ( S n ) ′ . This concludes our claim.6.2. Operator systems generated from universal relations.
Let G = { h , . . . , h n } and R = { h ∗ j = h j , k h j k ≤ , ≤ j ≤ n } be a set of relations in the set G , and let C ∗ ( G|R ) denote the universal unitalC ∗ -algebra generated by G subject to R . The operator system N C ( n ) = span { , h , ..., h n } ⊂ C ∗ ( G|R ) . is called the operator system of the non-commuting n -cube.It was shown in [6] that the C ∗ -envelope of N C ( n ) is C ∗ ( ∗ n Z ), where ∗ n Z isthe free product of n -copies of Z . The following example is from [6, Lemma 6.2]and can be proved exactly along the lines of the previous example. Example 6.2.
For n ∈ N , the pair ( N C ( n ) , C ∗ ( ∗ n Z )) is a relatively weakly injec-tive pair of operator systems. Inclusion in the double dual.
The dual S ∗ of an operator system is a ma-tricially normed space, but the double dual S ∗∗ is an operator system containing S as an operator subsytem [11]. The following example is established in [11, Corollary6.6]. Example 6.3. ( S , S ∗∗ ) is a relatively weakly injective pair of operator systems, forevery operator system S . Operator systems with DCEP.
An operator system S is said to havethe double commutant expectation property (DCEP) if, for every complete orderembedding S → B ( H ), there exists a completely positive linear map Φ : B ( H ) →S ′′ ⊂ B ( H ), fixing S . Example 6.4. If S has the double commutant expectation property, then ( S , T ) is a relatively weakly injective pair of operator systems, for every operator system T that contains S as an operator subsystem. This assertion above is a consequence of [11, Theorem 7.3, Theorem 7.1], whichstates that if
S ⊂ T and S has the double commutant expectation property, then S ⊗ c R ⊂ coi
T ⊗ c R for every operator system R .7. Acknowledgement
The author would like to thank the referee for his/her valuable suggestions andhis doctoral thesis advisor, Douglas Farenick, for suggesting the topic of this paperand for many helpful discussions during the course of this work. The author’s workat the University of Regina is supported in part by a Saskatchewan Innovation andOpportunity Scholarship and a Faculty of Graduate Studies & Research Dean’sScholarship.
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