Tensor Products of Operator Systems - Texas A&M …kerr/ecoas09/tomforde.pdf · Operator Spaces and Operator Systems An operator space is a subspace of B(H). Abstract Characterization

Tensor Products of Operator Systems

Mark Tomforde

(Joint work with Ali Kavruk, Vern Paulsen, and Ivan Todorov)

East Coast Operator Algebras SymposiumTexas A&M University

October 19, 2009

Mark Tomforde (Univeristy of Houston) Tensor Products of Operator Systems October 19, 2009 1 / 37

Operator Spaces and Operator Systems

An operator space is a subspace of B(H).

Abstract Characterization [Ruan]: An operator space is a complex vectorspace V with a collection of norms {‖ · ‖n}∞n=1 such that

1 (Mn(V ), ‖ · ‖n) is a complete normed space for all n,

2 ‖ u 00 v ‖m+n = max{‖u‖m, ‖v‖n} for u ∈ Mm(V ) and v ∈ Mm(V ),

3 ‖αuβ‖n ≤ ‖α‖‖u‖m‖β‖ for u ∈ Mm(V ), α ∈ Mn,m(C), β ∈ Mm,n(C).

Each abstract operator space is completely isometrically isomorphic to asubspace of B(H).

Category: Operator Spaces and Completely Contractive Maps




















An operator system is a subspace X ⊆ B(H) with X ∗ = X and 1 ∈ X .

Abstract Characterization [Choi and Effros]: An operator system is a∗-vector space X , a collection of positive elements {Cn}∞n=1, withCn ⊆ Mn(X )h, and an element e ∈ V such that

1 For each n, Cn is a cone with Cn ∩ −Cn = {0}.2 For each n and each m, whenever X ∈ Mn,m(C) we have

X ∗CnX ⊆ Cm.

3 The element en :=

(e. . .

e

)is an Archimedean order unit for Mn(X ).

(Order Unit: For each u ∈ Mn(X )h there exists r > 0 s.t. ren ≥ u.)(Archimedean: For each u ∈ Mn(X ), it is the case that whenever

re + u ≥ 0 for all r > 0, then u ≥ 0.)

Properties (1) and (2) are called a matrix ordering.

Category: Operator Systems and Unital Completely Positive Maps





X ∗CnX ⊆ Cm.

3 The element en :=

(e. . .

e










X ∗CnX ⊆ Cm.

3 The element en :=

(e. . .

e










X ∗CnX ⊆ Cm.

3 The element en :=

(e. . .

e







In an operator system, the positive elements determine the matrix norms:

For A ∈ Mn(X ) we have

‖A‖n = inf

{r :

(ren AA∗ ren

)∈ C2n

}.

Note: Bigger cones give smaller norms.


In an operator system, the positive elements determine the matrix norms:

For A ∈ Mn(X ) we have

‖A‖n = inf

{r :

(ren AA∗ ren

)∈ C2n

}.

Note: Bigger cones give smaller norms.


There has been a great deal of work on operator space theory, but not somuch for operator systems.

Developing a theory of operator systems is worthwhile — in particular,theorems for operator systems can tell us something about operator spaces.

We will discuss a theory of tensor products for operator systems. Someaspects of the theory are expected, and some are surprising — we’ll give aquick overview of the familiar parts and then focus on the surprising parts










Some of the Surprises

1 There exists an operator system S such that S ⊗min A = S ⊗max A forevery C*-algebra A, but S is not nuclear in the category of operatorsystems.

2 There exists an operator system S such that whenever A and B areunital C*-algebras with A ⊆ B, then every completely positive mapφ : A→ S has a completely positive extension φ̃ : B → S, but S isnot injective in the category of operator systems.

3 There exists an operator system S such that S ⊗min T = S ⊗c T forevery operator system T (i.e., S is (min, c)-nuclear), but S is not“classically nuclear” in the sense that the identity map Id : S → Sdoes not approximately factor through matrices.


Obtaining Results for Operator Spaces

A TRICK: Given an operator space {X , ‖ · ‖n}∞n=1, let X ∗ be the conjugatevector space of X , and

SX :=

{(λ v

w∗ µ

): λ, µ ∈ C, v ∈ V and w∗ ∈ V ∗

}with matrix addition and multiplication, and

„λ v

w∗ µ

«∗=

„λ wv∗ µ

«.

Identify Mn(X ) with„

A VW∗ B

«: A, B ∈ Mn(C), V ∈ Mn(V ),W∗ ∈ Mn(V∗)

ffand define

Cn :={(

P VV∗ Q

): P,Q ≥ 0 and ‖(P + εI )−1/2V (Q + εI )−1/2‖ ≤ 1 ∀ε > 0

}and e := ( 1 0

0 1 ).

Then SX is an operator system and x 7→ ( 0 x0 0 ) is a complete isometry of X

onto the upper-right-hand corner of SX .



Given operator systems (S, {Pn}∞n=1, e1) and (T , {Qn}∞n=1, e2), we letS ⊗ T denote the vector space tensor product of S and T .

An operator system structure on S ⊗ T is a family {Cn}∞n=1 of cones,where Cn ⊆ Mn(S ⊗ T ), satisfying:

(T1) (S ⊗ T , {Cn}∞n=1, e1 ⊗ e2) is an operator system,

(T2) Pn ⊗ Qm ⊆ Cnm, for all n,m ∈ N, and

(T3) If φ : S → Mn and ψ : T → Mm are unital completely positive maps,then φ⊗ ψ : S ⊗ T → Mmn is a unital completely positive map.

(T2) is the order analogue of the cross-norm condition(T3) is the analogue of Grothendieck’s “reasonable” property.



Given operator systems (S, {Pn}∞n=1, e1) and (T , {Qn}∞n=1, e2), we letS ⊗ T denote the vector space tensor product of S and T .

An operator system structure on S ⊗ T is a family {Cn}∞n=1 of cones,where Cn ⊆ Mn(S ⊗ T ), satisfying:

(T1) (S ⊗ T , {Cn}∞n=1, e1 ⊗ e2) is an operator system,

(T2) Pn ⊗ Qm ⊆ Cnm, for all n,m ∈ N, and

(T3) If φ : S → Mn and ψ : T → Mm are unital completely positive maps,then φ⊗ ψ : S ⊗ T → Mmn is a unital completely positive map.

(T2) is the order analogue of the cross-norm condition(T3) is the analogue of Grothendieck’s “reasonable” property.


By an operator system tensor product, we mean a mappingτ : O ×O → O, such that for every pair of operator systems S and T ,τ(S, T ) is an operator system structure on S ⊗ T , denoted S ⊗τ T .

We call an operator system tensor product τ functorial if the followingproperty is satisfied:

(T4) For any four operator systems S1,S2, T1, and T2, we have that ifφ ∈ UCP(S1,S2) and ψ ∈ UCP(T1, T2), then the linear mapφ⊗ ψ : S1 ⊗ T1 → S2 ⊗ T2 belongs to UCP(S1 ⊗τ T1,S2 ⊗τ T2).

Symmetric: S ⊗τ T ∼= T ⊗τ S; x ⊗ y 7→ y ⊗ x .

Associative: R⊗τ (S ⊗τ T ) ∼= (R⊗τ S)⊗τ T ; x ⊗ (y ⊗ z) 7→ (x ⊗ y)⊗ z .

Given two operator system structures τ1 and τ2 on S ⊗ T , we defineτ1 ≥ τ2 to mean Id : S ⊗τ1 T → S ⊗τ2 T is completely positive (or,equivalently, Mn(S ⊗τ1 T )+ ⊆ Mn(S ⊗τ2 T )+ for every n ∈ N.)























Op. System tensor product gives Op. Space tensor product

Recall: If X is an operator space, then

SX =

{(λ xy∗ µ

): λ, µ ∈ C, x , y ∈ X

}is an operator system, and X ⊆ SX via the inclusion x →

(0 x0 0

).

Definition

Let X and Y be operator spaces and τ be an operator system structure onSX ⊗ SY . Then the embedding

X ⊗ Y ⊆ SX ⊗τ SY

endows X ⊗ Y with an operator space structure. We call the resultingoperator space the induced operator space tensor product of X and Y anddenote it by X ⊗τ Y .


Op. System tensor product gives Op. Space tensor product

Recall: If X is an operator space, then

SX =

{(λ xy∗ µ

): λ, µ ∈ C, x , y ∈ X

}is an operator system, and X ⊆ SX via the inclusion x →

(0 x0 0

).

Definition

Let X and Y be operator spaces and τ be an operator system structure onSX ⊗ SY . Then the embedding

X ⊗ Y ⊆ SX ⊗τ SY

endows X ⊗ Y with an operator space structure. We call the resultingoperator space the induced operator space tensor product of X and Y anddenote it by X ⊗τ Y .


Then X ⊗τ Y is an operator space tensor product in the sense that:

(1) If x ∈ Mn(X ) and y ∈ Mm(Y ), then

‖x ⊗ y‖Mnm(X⊗τY ) ≤ ‖x‖Mn(X )‖y‖Mm(Y ).

(2) If φ : X → Mn and ψ : Y → Mm are completely bounded, thenφ⊗ ψ : X ⊗τ Y → Mmn is completely bounded and‖φ⊗ ψ‖cb ≤ ‖φ‖cb‖ψ‖cb.


Theorem (Kavruk, Paulsen, Todorov, T)

The collection of all operator system tensor products is a complete latticewith respect to the order introduced earlier. The collection of all functorialoperator system tensor products is a complete sublattice of this lattice.


One important concept from the theory of C*-algebras that we shall beinterested in generalizing is nuclearity.

Definition

Let α and β be operator system tensor products. An operator system Swill be called (α, β)-nuclear if the identity map between S ⊗α T andS ⊗β T is a complete order isomorphism for every operator system T .

One shortcoming of the theory of operator space tensor products is thatthe minimal and maximal operator space tensor products of matrixalgebras do not coincide. For this reason there are essentially no nuclearspaces in the operator space category. We will see that, unlike theoperator space case, there is a rich theory of nuclear operator systems forthe various tensor products we will introduce subsequently.


One important concept from the theory of C*-algebras that we shall beinterested in generalizing is nuclearity.

Definition

Let α and β be operator system tensor products. An operator system Swill be called (α, β)-nuclear if the identity map between S ⊗α T andS ⊗β T is a complete order isomorphism for every operator system T .

One shortcoming of the theory of operator space tensor products is thatthe minimal and maximal operator space tensor products of matrixalgebras do not coincide. For this reason there are essentially no nuclearspaces in the operator space category. We will see that, unlike theoperator space case, there is a rich theory of nuclear operator systems forthe various tensor products we will introduce subsequently.


The Minimal Tensor Product

Let S and T be operator systems. For each n ∈ N, we let

Cminn (S, T ) = {(pi ,j) ∈ Mn(S ⊗ T ) : ((φ⊗ ψ)(pi ,j))i ,j ∈ M+

nkm

∀ u.c.p. φ : S → Mk and ∀ u.c.p. ψ : T → Mm}.

Definition

We call the operator system (S ⊗ T , {Cminn (S, T )}∞n=1, 1⊗ 1) the minimal

tensor product of S and T and denote it by S ⊗min T .


The Minimal Tensor Product


Cminn (S, T ) = {(pi ,j) ∈ Mn(S ⊗ T ) : ((φ⊗ ψ)(pi ,j))i ,j ∈ M+

nkm

∀ u.c.p. φ : S → Mk and ∀ u.c.p. ψ : T → Mm}.

Definition

We call the operator system (S ⊗ T , {Cminn (S, T )}∞n=1, 1⊗ 1) the minimal

tensor product of S and T and denote it by S ⊗min T .



The mappingmin : O ×O → O

sending (S, T ) to S ⊗min T is an associative, symmetric, functorialoperator system tensor product.Moreover, if S and T are operator systems and τ is an operator systemstructure on S ⊗ T , then min ≤ τ .


Let S and T be operator systems, and let ιS : S → B(H) andιT : T → B(K ) be embeddings that are unital complete orderisomorphisms onto their ranges. The family {Cmin

n (S, T )}∞n=1 is theoperator system structure on S ⊗ T arising from the embeddingιS ⊗ ιT : S ⊗ T → B(H ⊗ K ).



The mappingmin : O ×O → O

sending (S, T ) to S ⊗min T is an associative, symmetric, functorialoperator system tensor product.Moreover, if S and T are operator systems and τ is an operator systemstructure on S ⊗ T , then min ≤ τ .


Let S and T be operator systems, and let ιS : S → B(H) andιT : T → B(K ) be embeddings that are unital complete orderisomorphisms onto their ranges. The family {Cmin

n (S, T )}∞n=1 is theoperator system structure on S ⊗ T arising from the embeddingιS ⊗ ιT : S ⊗ T → B(H ⊗ K ).


It was shown (Blecher-Paulsen) that the minimal operator space tensorproduct, the spatial operator space tensor product, and the injectiveoperator space tensor product all coincide. For operator spaces X and Y ,we will let X ⊗̌Y denote this tensor product.


Let X and Y be operator spaces. Then the induced tensor productX ⊗min Y coincides with the minimal operator space tensor product X ⊗̌Y .

Corollary

Let S and T be operator systems. Then the identity map is a completeisometry between the operator spaces S ⊗min T and S⊗̌T .

Corollary

Let A and B be C*-algebras. Then the minimal operator system tensorproduct A⊗min B is completely order isomorphic to the image of A⊗ Binside the minimal C*-algebraic tensor product A⊗C*min B.


It was shown (Blecher-Paulsen) that the minimal operator space tensorproduct, the spatial operator space tensor product, and the injectiveoperator space tensor product all coincide. For operator spaces X and Y ,we will let X ⊗̌Y denote this tensor product.


Let X and Y be operator spaces. Then the induced tensor productX ⊗min Y coincides with the minimal operator space tensor product X ⊗̌Y .

Corollary

Let S and T be operator systems. Then the identity map is a completeisometry between the operator spaces S ⊗min T and S⊗̌T .

Corollary

Let A and B be C*-algebras. Then the minimal operator system tensorproduct A⊗min B is completely order isomorphic to the image of A⊗ Binside the minimal C*-algebraic tensor product A⊗C*min B.


The Maximal Tensor Product


Dmaxn (S, T ) = {α(P ⊗ Q)α∗ : P ∈ Mk(S)+,Q ∈ Mm(T )+, α ∈ Mn,km,

k ,m ∈ N}.

• {Dmaxn (S, T )}∞n=1 is a matrix ordering with order unit 1⊗ 1.

• However, 1⊗ 1 is not an Archimedean matrix order unit.


The Maximal Tensor Product


Dmaxn (S, T ) = {α(P ⊗ Q)α∗ : P ∈ Mk(S)+,Q ∈ Mm(T )+, α ∈ Mn,km,

k ,m ∈ N}.

• {Dmaxn (S, T )}∞n=1 is a matrix ordering with order unit 1⊗ 1.

• However, 1⊗ 1 is not an Archimedean matrix order unit.


Let

Cmaxn (S, T ) = {P ∈ Mn(S ⊗ T ) : r(1⊗ 1)n + P ∈ Dmax

n (S, T ) ∀ r > 0}

Then {Cmaxn (S, T )}∞n=1 is a matrix ordering with Archimedean matrix

order unit 1⊗ 1. Furthermore, if {Cn}∞n=1 is any matrix ordering on S ⊗ Tfor which 1⊗ 1 is an Archimedean matrix order unit, then Dmax

n ⊆ Cn

implies Cmaxn ⊆ Cn.

This is a special case of a process called “Archimedeanization”, which wasdeveloped in detail in

“Operator system structures on ordered spaces”, by V. Paulsen,I. Todorov, and M. Tomforde, preprint.


Let

Cmaxn (S, T ) = {P ∈ Mn(S ⊗ T ) : r(1⊗ 1)n + P ∈ Dmax

n (S, T ) ∀ r > 0}

Then {Cmaxn (S, T )}∞n=1 is a matrix ordering with Archimedean matrix

order unit 1⊗ 1. Furthermore, if {Cn}∞n=1 is any matrix ordering on S ⊗ Tfor which 1⊗ 1 is an Archimedean matrix order unit, then Dmax

n ⊆ Cn

implies Cmaxn ⊆ Cn.

This is a special case of a process called “Archimedeanization”, which wasdeveloped in detail in

“Operator system structures on ordered spaces”, by V. Paulsen,I. Todorov, and M. Tomforde, preprint.


Definition

We call the operator system (S ⊗ T , {Cmaxn (S, T )}∞n=1, 1⊗ 1) the maximal

tensor product of S and T and denote it by S ⊗max T .


The mappingmax : O ×O → O

sending (S, T ) to S ⊗max T defines an associative, symmetric, functorialoperator system tensor product. Moreover, if τ is an operator systemstructure on S ⊗ T , then τ ≤ max.


Definition

We call the operator system (S ⊗ T , {Cmaxn (S, T )}∞n=1, 1⊗ 1) the maximal

tensor product of S and T and denote it by S ⊗max T .


The mappingmax : O ×O → O

sending (S, T ) to S ⊗max T defines an associative, symmetric, functorialoperator system tensor product. Moreover, if τ is an operator systemstructure on S ⊗ T , then τ ≤ max.


The operator space projective tensor product X ⊗̂Y is the maximaloperator space tensor product.


Let X and Y be operator spaces. Then the induced tensor productX ⊗max Y coincides with the operator space projective tensor productX ⊗̂Y .


Let A and B be C*-algebras. Then the operator system A⊗max B iscompletely order isomorphic to the image of A⊗ B inside the maximalC*-algebraic tensor product of A⊗C*max B. Furthermore, we haveDmax

n (A,B) = Cmaxn (A,B).


The operator space projective tensor product X ⊗̂Y is the maximaloperator space tensor product.


Let X and Y be operator spaces. Then the induced tensor productX ⊗max Y coincides with the operator space projective tensor productX ⊗̂Y .


Let A and B be C*-algebras. Then the operator system A⊗max B iscompletely order isomorphic to the image of A⊗ B inside the maximalC*-algebraic tensor product of A⊗C*max B. Furthermore, we haveDmax

n (A,B) = Cmaxn (A,B).


The following result shows that the notion of (min,max)-nuclearity ofoperator systems extends the usual notion of nuclearity of C*-algebras.

Theorem

Let A be a unital C*-algebra. Then A is nuclear if and only if A is(min,max)-nuclear; that is, if and only if A⊗min S = A⊗max S for everyoperator system S.

Thus, we see that a C*-algebra is nuclear if and only ifCmin

n (A,S) = Cmaxn (A,S) for every n ∈ N and every operator system S.


The following result shows that the notion of (min,max)-nuclearity ofoperator systems extends the usual notion of nuclearity of C*-algebras.

Theorem

Let A be a unital C*-algebra. Then A is nuclear if and only if A is(min,max)-nuclear; that is, if and only if A⊗min S = A⊗max S for everyoperator system S.

Thus, we see that a C*-algebra is nuclear if and only ifCmin

n (A,S) = Cmaxn (A,S) for every n ∈ N and every operator system S.


The prior result shows every finite-dimensional C*-algebra is(min-max)-nuclear.

Unlike C*-algebras, finite-dimensional operator systems do not have to be(min-max)-nuclear. We now exhibit an operator system that is “nuclear”when tensored with any C*-algebra, but is not (min, max)-nuclear and isalso not a C*-algebra.


Let S = span{E1,1,E1,2,E2,1,E2,2,E2,3,E3,2,E3,3} ⊆ M3.P1,1 P1,2 0P2,1 P2,2 P2,3

0 P3,2 P3,3

Then• S ⊗min A = S ⊗max A for every C*-algebra A.• S is not completely order isomorphic to a C*-algebra.• S is not (min,max)-nuclear.


The prior result shows every finite-dimensional C*-algebra is(min-max)-nuclear.

Unlike C*-algebras, finite-dimensional operator systems do not have to be(min-max)-nuclear. We now exhibit an operator system that is “nuclear”when tensored with any C*-algebra, but is not (min, max)-nuclear and isalso not a C*-algebra.


Let S = span{E1,1,E1,2,E2,1,E2,2,E2,3,E3,2,E3,3} ⊆ M3.P1,1 P1,2 0P2,1 P2,2 P2,3

0 P3,2 P3,3

Then• S ⊗min A = S ⊗max A for every C*-algebra A.• S is not completely order isomorphic to a C*-algebra.• S is not (min,max)-nuclear.



There exists an operator system S such that whenever A and B are unitalC*-algebras with A ⊆ B, then every completely positive map φ : A→ Shas a completely positive extension φ̃ : B → S, but S is not injective inthe category of operator systems.


There is another operator system tensor product that agrees with the maxtensor product for all pairs of C*-algebras, but does not agree with themax tensor product on all pairs of operator systems. This gives a differentextension of the maximal C*-algebraic tensor product from the category ofC*-algebras to the category of operator systems.

We call this tensor product the commuting tensor product. In contrastwith the maximal operator system tensor product, but in analogy with theminimal one, this tensor product is defined by specifying a collection ofcompletely positive maps rather than specifying the matrix ordering.


There is another operator system tensor product that agrees with the maxtensor product for all pairs of C*-algebras, but does not agree with themax tensor product on all pairs of operator systems. This gives a differentextension of the maximal C*-algebraic tensor product from the category ofC*-algebras to the category of operator systems.

We call this tensor product the commuting tensor product. In contrastwith the maximal operator system tensor product, but in analogy with theminimal one, this tensor product is defined by specifying a collection ofcompletely positive maps rather than specifying the matrix ordering.


Let S and T be operator systems. Set

cp(S, T ) = {(φ,ψ) : H is a Hilbert space, φ ∈ CP(S,B(H)),

ψ ∈ CP(T ,B(H)), and φ(S) commutes with ψ(T ).}

Given (φ, ψ) ∈ cp(S, T ), let φ · ψ : S ⊗ T → B(H) be the map given onelementary tensors by (φ · ψ)(x ⊗ y) = φ(x)ψ(y).

For each n ∈ N, define a cone Pn ⊆ Mn(S ⊗ T ) by letting

Pn = {u ∈ Mn(S ⊗ T ) : (φ · ψ)(n)(u) ≥ 0, for all (φ, ψ) ∈ cp(S, T )}.

Then (S ⊗ T , {Pn}∞n=1, 1⊗ 1) is an operator system.


















Definition

We call the operator system (S ⊗ T , {Pn}∞n=1, 1⊗ 1) the commutingtensor product of S and T and denote it by S ⊗c T .


The mappingc : O ×O → O

sending (S, T ) to S ⊗c T defines a symmetric, functorial operator systemtensor product.


Definition

We call the operator system (S ⊗ T , {Pn}∞n=1, 1⊗ 1) the commutingtensor product of S and T and denote it by S ⊗c T .


The mappingc : O ×O → O

sending (S, T ) to S ⊗c T defines a symmetric, functorial operator systemtensor product.



The tensor products c and max are different.

However . . .

Theorem

If A and B are unital C*-algebras, then A⊗c B = A⊗max B.

and in fact . . .

Theorem

If A is a unital C ∗-algebra and S is an operator system, thenA⊗c S = A⊗max S.




However . . .

Theorem


and in fact . . .

Theorem





However . . .

Theorem


and in fact . . .

Theorem




There exists an operator system S such that S ⊗min T = S ⊗c T for everyoperator system T (i.e., S is (min, c)-nuclear), but S is not “classicallynuclear” in the sense that the identity map Id : S → S does notapproximately factor through matrices.


We recall that for an operator system S, there exists a unital C*-algebraC ∗u (S) (called either the universal C*-algebra of S or the maximalC*-algebra of S) and a unital complete order isomorphism ι : S → C ∗u (S)with the properties that ι(S) generates C ∗u (S) as a C*-algebra, and thatfor every unital completely positive map φ : S → B(H) there exists aunique ∗-homomorphism π : C ∗u (S)→ B(H) such that π ◦ ι = φ.


Let S and T be operator systems. The operator system arising from theinclusion of S ⊗ T into C ∗u (S)⊗C*max C ∗u (T ) coincides with S ⊗c T .


We recall that for an operator system S, there exists a unital C*-algebraC ∗u (S) (called either the universal C*-algebra of S or the maximalC*-algebra of S) and a unital complete order isomorphism ι : S → C ∗u (S)with the properties that ι(S) generates C ∗u (S) as a C*-algebra, and thatfor every unital completely positive map φ : S → B(H) there exists aunique ∗-homomorphism π : C ∗u (S)→ B(H) such that π ◦ ι = φ.


Let S and T be operator systems. The operator system arising from theinclusion of S ⊗ T into C ∗u (S)⊗C*max C ∗u (T ) coincides with S ⊗c T .


Corollary (Kavruk, Paulsen, Todorov, T)

Let S and T be operator systems. A linear map f : S ⊗c T → B(H) is aunital completely positive map if and only if there exist a Hilbert space K ,∗-homomorphisms π : C ∗u (S)→ B(K ) and ρ : C ∗u (T )→ B(K ) withcommuting ranges, and an isometry V : H → K such that

f (x ⊗ y) = V ∗π(x)ρ(y)V

for all x ∈ S and all y ∈ T .


Let X and Y be operator spaces. For u ∈ X ⊗ Y , let

‖u‖µ∗ = sup{‖(f · g)(u)‖ : f : X → B(H) and g : Y → B(H) are

completely contractive maps with the property that f (x)

commutes with {g(y), g(y)∗} for all x ∈ X and y ∈ Y }.

We define norms on Mn(X ⊗ Y ) in a similar fashion. It is easily checkedthat this gives an operator space structure to X ⊗ Y , and we denote theresulting operator space X ⊗µ∗ Y .


Let X and Y be operator spaces. Then the identity map is a completelyisometric isomorphism betwen X ⊗c Y and X ⊗µ∗ Y .


Let X and Y be operator spaces. For u ∈ X ⊗ Y , let

‖u‖µ∗ = sup{‖(f · g)(u)‖ : f : X → B(H) and g : Y → B(H) are

completely contractive maps with the property that f (x)

commutes with {g(y), g(y)∗} for all x ∈ X and y ∈ Y }.

We define norms on Mn(X ⊗ Y ) in a similar fashion. It is easily checkedthat this gives an operator space structure to X ⊗ Y , and we denote theresulting operator space X ⊗µ∗ Y .


Let X and Y be operator spaces. Then the identity map is a completelyisometric isomorphism betwen X ⊗c Y and X ⊗µ∗ Y .


We say that a functorial operator system tensor product τ is injective iffor all operator systems S1 ⊆ S2 and T1 ⊆ T2, the inclusion

S1 ⊗τ T1 ⊆ S2 ⊗τ T2

is a complete order isomorphism onto its range; that is,Mn(S1 ⊗ T1) ∩Mn(S2 ⊗τ T2)+ = Mn(S1 ⊗τ T1)+ for every n ∈ N.

A tensor product τ is called left injective if for all operator systems S1,S2, and T with S1 ⊆ S2, the inclusion of

S1 ⊗τ T ⊆ S2 ⊗τ T

is a complete order isomorphism.

We define a right injective operator system tensor product similarly. Anoperator system tensor product is injective if it is both left injective andright injective.



S1 ⊗τ T1 ⊆ S2 ⊗τ T2



S1 ⊗τ T ⊆ S2 ⊗τ T





S1 ⊗τ T1 ⊆ S2 ⊗τ T2



S1 ⊗τ T ⊆ S2 ⊗τ T




Definition

Let S and T be operator systems. We let S ⊗el T (respectively, S ⊗er T )be the operator system with underlying space S ⊗ T whose matrixordering is induced by the inclusion S ⊗ T ⊆ I (S)⊗max T (respectively,S ⊗ T ⊆ S ⊗max I (T )).

Likewise, we let S ⊗e T be the operator system with underlying spaceS ⊗ T whose matrix ordering is induced by the inclusionS ⊗ T ⊆ I (S)⊗max I (T ).

The tensor product e is functorial and injective, and if τ is any injectiveoperator system tensor product, then τ ≤ e.

The tensor product el (respectively, er) is functorial and left injective, andif τ is any left injective (respectively, right injective) operator systemtensor product, then τ ≤ e.


Definition

Let S and T be operator systems. We let S ⊗el T (respectively, S ⊗er T )be the operator system with underlying space S ⊗ T whose matrixordering is induced by the inclusion S ⊗ T ⊆ I (S)⊗max T (respectively,S ⊗ T ⊆ S ⊗max I (T )).

Likewise, we let S ⊗e T be the operator system with underlying spaceS ⊗ T whose matrix ordering is induced by the inclusionS ⊗ T ⊆ I (S)⊗max I (T ).

The tensor product e is functorial and injective, and if τ is any injectiveoperator system tensor product, then τ ≤ e.

The tensor product el (respectively, er) is functorial and left injective, andif τ is any left injective (respectively, right injective) operator systemtensor product, then τ ≤ e.


If A is a C ∗-algebra represented nondegenerately on H with weak closureA, then φ ∈ B(B(H),A) is a weak expectation if φ is unital completelypositive, ‖φ‖ = 1, and φ(atb) = aφ(t)b for all a, b ∈ A and t ∈ B(H).

A C ∗-algebra has the weak expectation property (WEP) if for everyfaithful representation π of A, it is the case that π(A) has a weakexpectation.


Let A be a unital C*-algebra. The following are equivalent:

(i) A possesses the weak expectation property (WEP).

(ii) A⊗el B = A⊗max B for every C*-algebra B.


If A is a C ∗-algebra represented nondegenerately on H with weak closureA, then φ ∈ B(B(H),A) is a weak expectation if φ is unital completelypositive, ‖φ‖ = 1, and φ(atb) = aφ(t)b for all a, b ∈ A and t ∈ B(H).

A C ∗-algebra has the weak expectation property (WEP) if for everyfaithful representation π of A, it is the case that π(A) has a weakexpectation.



(i) A possesses the weak expectation property (WEP).

(ii) A⊗el B = A⊗max B for every C*-algebra B.




(i) A is nuclear.

(ii) A⊗er B = A⊗max B for every unital C*-algebra B.


Finally, we characterize the norm ‖ · ‖e induced by the operator systemstructure e.


Let A and B be unital C*-algebras and u ∈ A⊗ B. Then

‖u‖e = inf{‖u‖A1⊗maxB1 : A1 and B1 are unital C*-algebras

with 1A1 ∈ A ⊆ A1 and 1B1 ∈ B ⊆ B1 }.


Thank you!


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Tensor Products of Operator Systems - Texas A&M …kerr/ecoas09/tomforde.pdf · Operator Spaces and Operator Systems An operator space is a subspace of B(H). Abstract Characterization