Tensor Products of Operator Systems
Mark Tomforde
(Joint work with Ali Kavruk, Vern Paulsen, and Ivan Todorov)
East Coast Operator Algebras Symposium
Texas 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).
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
2 / 37
Operator Spaces and Operator Systems
An operator space is a subspace of B(H).
Abstract Characterization [Ruan]: An operator space is a complex vector
space V with a collection of norms {k · kn }∞
n=1 such that
1
(Mn (V ), k · kn ) is a complete normed space for all n,
2
k u0 v0 km+n = max{kukm , kv kn } for u ∈ Mm (V ) and v ∈ Mm (V ),
3
kαuβkn ≤ kαkkukm kβk for u ∈ Mm (V ), α ∈ Mn,m (C), β ∈ Mm,n (C).
Each abstract operator space is completely isometrically isomorphic to a
subspace of B(H).
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
2 / 37
Operator Spaces and Operator Systems
An operator space is a subspace of B(H).
Abstract Characterization [Ruan]: An operator space is a complex vector
space V with a collection of norms {k · kn }∞
n=1 such that
1
(Mn (V ), k · kn ) is a complete normed space for all n,
2
k u0 v0 km+n = max{kukm , kv kn } for u ∈ Mm (V ) and v ∈ Mm (V ),
3
kαuβkn ≤ kαkkukm kβk for u ∈ Mm (V ), α ∈ Mn,m (C), β ∈ Mm,n (C).
Each abstract operator space is completely isometrically isomorphic to a
subspace of B(H).
Category: Operator Spaces and Completely Contractive Maps
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
2 / 37
An operator system is a subspace X ⊆ B(H) with X ∗ = X and 1 ∈ X .
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
3 / 37
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 , with
Cn ⊆ 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 ∗ Cn X ⊆ Cm .
e
..
The element en :=
is an Archimedean order unit for Mn (X ).
.
3
e
(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.)
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
3 / 37
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 , with
Cn ⊆ 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 ∗ Cn X ⊆ Cm .
e
..
The element en :=
is an Archimedean order unit for Mn (X ).
.
3
e
(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.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
3 / 37
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 , with
Cn ⊆ 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 ∗ Cn X ⊆ Cm .
e
..
The element en :=
is an Archimedean order unit for Mn (X ).
.
3
e
(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
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
3 / 37
In an operator system, the positive elements determine the matrix norms:
For A ∈ Mn (X ) we have
ren A
kAkn = inf r :
∈ C2n .
A∗ ren
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
4 / 37
In an operator system, the positive elements determine the matrix norms:
For A ∈ Mn (X ) we have
ren A
kAkn = inf r :
∈ C2n .
A∗ ren
Note: Bigger cones give smaller norms.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
4 / 37
There has been a great deal of work on operator space theory, but not so
much for operator systems.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
5 / 37
There has been a great deal of work on operator space theory, but not so
much for operator systems.
Developing a theory of operator systems is worthwhile — in particular,
theorems for operator systems can tell us something about operator spaces.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
5 / 37
There has been a great deal of work on operator space theory, but not so
much 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. Some
aspects of the theory are expected, and some are surprising — we’ll give a
quick overview of the familiar parts and then focus on the surprising parts
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
5 / 37
Some of the Surprises
1
There exists an operator system S such that S ⊗min A = S ⊗max A for
every C*-algebra A, but S is not nuclear in the category of operator
systems.
2
There exists an operator system S such that whenever A and B are
unital C*-algebras with A ⊆ B, then every completely positive map
φ : A → S has a completely positive extension φ̃ : B → S, but S is
not injective in the category of operator systems.
3
There exists an operator system S such that S ⊗min T = S ⊗c T for
every 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 → S
does not approximately factor through matrices.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
6 / 37
Obtaining Results for Operator Spaces
∗
A TRICK: Given an operator space {X , k · kn }∞
n=1 , let X be the conjugate
vector space of X , and
λ v
∗
∗
SX :=
: λ, µ ∈ C, v ∈ V and w ∈ V
w∗ µ
with matrix addition and multiplication, and
Identify Mn (X ) with
Cn :=
n
P
V∗
V
Q
„
A
W∗
V
B
«
„
λ
w∗
«
„
v ∗
λ
=
µ
v∗
w
µ
: A, B ∈ Mn (C), V ∈ Mn (V ), W ∗ ∈ Mn (V ∗ )
ff
«
.
and define
o
: P, Q ≥ 0 and k(P + I )−1/2 V (Q + I )−1/2 k ≤ 1 ∀ > 0
and e := ( 10 01 ).
Then SX is an operator system and x 7→ ( 00 x0 ) is a complete isometry of X
onto the upper-right-hand corner of SX .
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
7 / 37
Tensor Products of Operator Systems
∞
Given operator systems (S, {Pn }∞
n=1 , e1 ) and (T , {Qn }n=1 , e2 ), we let
S ⊗ 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.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
8 / 37
Tensor Products of Operator Systems
∞
Given operator systems (S, {Pn }∞
n=1 , e1 ) and (T , {Qn }n=1 , e2 ), we let
S ⊗ 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.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
8 / 37
By an operator system tensor product, we mean a mapping
τ : 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 following
property is satisfied:
(T4) For any four operator systems S1 , S2 , T1 , and T2 , we have that if
φ ∈ UCP(S1 , S2 ) and ψ ∈ UCP(T1 , T2 ), then the linear map
φ ⊗ ψ : S1 ⊗ T1 → S2 ⊗ T2 belongs to UCP(S1 ⊗τ T1 , S2 ⊗τ T2 ).
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
9 / 37
By an operator system tensor product, we mean a mapping
τ : 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 following
property is satisfied:
(T4) For any four operator systems S1 , S2 , T1 , and T2 , we have that if
φ ∈ UCP(S1 , S2 ) and ψ ∈ UCP(T1 , T2 ), then the linear map
φ ⊗ 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.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
9 / 37
By an operator system tensor product, we mean a mapping
τ : 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 following
property is satisfied:
(T4) For any four operator systems S1 , S2 , T1 , and T2 , we have that if
φ ∈ UCP(S1 , S2 ) and ψ ∈ UCP(T1 , T2 ), then the linear map
φ ⊗ 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.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
9 / 37
By an operator system tensor product, we mean a mapping
τ : 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 following
property is satisfied:
(T4) For any four operator systems S1 , S2 , T1 , and T2 , we have that if
φ ∈ UCP(S1 , S2 ) and ψ ∈ UCP(T1 , T2 ), then the linear map
φ ⊗ 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 define
τ1 ≥ 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.)
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
9 / 37
Op. System tensor product gives Op. Space tensor product
Recall: If X is an operator space, then
λ x
SX =
:
λ,
µ
∈
C,
x,
y
∈
X
y∗ µ
0 x
is an operator system, and X ⊆ SX via the inclusion x →
.
0 0
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
10 / 37
Op. System tensor product gives Op. Space tensor product
Recall: If X is an operator space, then
λ x
SX =
:
λ,
µ
∈
C,
x,
y
∈
X
y∗ µ
0 x
is an operator system, and X ⊆ SX via the inclusion x →
.
0 0
Definition
Let X and Y be operator spaces and τ be an operator system structure on
SX ⊗ SY . Then the embedding
X ⊗ Y ⊆ SX ⊗τ SY
endows X ⊗ Y with an operator space structure. We call the resulting
operator space the induced operator space tensor product of X and Y and
denote it by X ⊗τ Y .
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
10 / 37
Then X ⊗τ Y is an operator space tensor product in the sense that:
(1) If x ∈ Mn (X ) and y ∈ Mm (Y ), then
kx ⊗ y kMnm (X ⊗τ Y ) ≤ kxkMn (X ) ky kMm (Y ) .
(2) If φ : X → Mn and ψ : Y → Mm are completely bounded, then
φ ⊗ ψ : X ⊗τ Y → Mmn is completely bounded and
kφ ⊗ ψkcb ≤ kφkcb kψkcb .
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
11 / 37
Theorem (Kavruk, Paulsen, Todorov, T)
The collection of all operator system tensor products is a complete lattice
with respect to the order introduced earlier. The collection of all functorial
operator system tensor products is a complete sublattice of this lattice.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
12 / 37
One important concept from the theory of C*-algebras that we shall be
interested in generalizing is nuclearity.
Definition
Let α and β be operator system tensor products. An operator system S
will be called (α, β)-nuclear if the identity map between S ⊗α T and
S ⊗β T is a complete order isomorphism for every operator system T .
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
13 / 37
One important concept from the theory of C*-algebras that we shall be
interested in generalizing is nuclearity.
Definition
Let α and β be operator system tensor products. An operator system S
will be called (α, β)-nuclear if the identity map between S ⊗α T and
S ⊗β T is a complete order isomorphism for every operator system T .
One shortcoming of the theory of operator space tensor products is that
the minimal and maximal operator space tensor products of matrix
algebras do not coincide. For this reason there are essentially no nuclear
spaces in the operator space category. We will see that, unlike the
operator space case, there is a rich theory of nuclear operator systems for
the various tensor products we will introduce subsequently.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
13 / 37
The Minimal Tensor Product
Let S and T be operator systems. For each n ∈ N, we let
+
Cnmin (S, T ) = {(pi,j ) ∈ Mn (S ⊗ T ) : ((φ ⊗ ψ)(pi,j ))i,j ∈ Mnkm
∀ u.c.p. φ : S → Mk and ∀ u.c.p. ψ : T → Mm }.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
14 / 37
The Minimal Tensor Product
Let S and T be operator systems. For each n ∈ N, we let
+
Cnmin (S, T ) = {(pi,j ) ∈ Mn (S ⊗ T ) : ((φ ⊗ ψ)(pi,j ))i,j ∈ Mnkm
∀ u.c.p. φ : S → Mk and ∀ u.c.p. ψ : T → Mm }.
Definition
We call the operator system (S ⊗ T , {Cnmin (S, T )}∞
n=1 , 1 ⊗ 1) the minimal
tensor product of S and T and denote it by S ⊗min T .
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
14 / 37
Theorem (Kavruk, Paulsen, Todorov, T)
The mapping
min : O × O → O
sending (S, T ) to S ⊗min T is an associative, symmetric, functorial
operator system tensor product.
Moreover, if S and T are operator systems and τ is an operator system
structure on S ⊗ T , then min ≤ τ .
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
15 / 37
Theorem (Kavruk, Paulsen, Todorov, T)
The mapping
min : O × O → O
sending (S, T ) to S ⊗min T is an associative, symmetric, functorial
operator system tensor product.
Moreover, if S and T are operator systems and τ is an operator system
structure on S ⊗ T , then min ≤ τ .
Theorem (Kavruk, Paulsen, Todorov, T)
Let S and T be operator systems, and let ιS : S → B(H) and
ιT : T → B(K ) be embeddings that are unital complete order
isomorphisms onto their ranges. The family {Cnmin (S, T )}∞
n=1 is the
operator system structure on S ⊗ T arising from the embedding
ιS ⊗ ιT : S ⊗ T → B(H ⊗ K ).
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
15 / 37
It was shown (Blecher-Paulsen) that the minimal operator space tensor
product, the spatial operator space tensor product, and the injective
operator space tensor product all coincide. For operator spaces X and Y ,
ˇ denote this tensor product.
we will let X ⊗Y
Theorem (Kavruk, Paulsen, Todorov, T)
Let X and Y be operator spaces. Then the induced tensor product
ˇ .
X ⊗min Y coincides with the minimal operator space tensor product X ⊗Y
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
16 / 37
It was shown (Blecher-Paulsen) that the minimal operator space tensor
product, the spatial operator space tensor product, and the injective
operator space tensor product all coincide. For operator spaces X and Y ,
ˇ denote this tensor product.
we will let X ⊗Y
Theorem (Kavruk, Paulsen, Todorov, T)
Let X and Y be operator spaces. Then the induced tensor product
ˇ .
X ⊗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 complete
ˇ .
isometry between the operator spaces S ⊗min T and S ⊗T
Corollary
Let A and B be C*-algebras. Then the minimal operator system tensor
product A ⊗min B is completely order isomorphic to the image of A ⊗ B
inside the minimal C*-algebraic tensor product A ⊗C*min B.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
16 / 37
The Maximal Tensor Product
Let S and T be operator systems. For each n ∈ N, we let
Dnmax (S, T ) = {α(P ⊗ Q)α∗ : P ∈ Mk (S)+ , Q ∈ Mm (T )+ , α ∈ Mn,km ,
k, m ∈ N}.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
17 / 37
The Maximal Tensor Product
Let S and T be operator systems. For each n ∈ N, we let
Dnmax (S, T ) = {α(P ⊗ Q)α∗ : P ∈ Mk (S)+ , Q ∈ Mm (T )+ , α ∈ Mn,km ,
k, m ∈ N}.
• {Dnmax (S, T )}∞
n=1 is a matrix ordering with order unit 1 ⊗ 1.
• However, 1 ⊗ 1 is not an Archimedean matrix order unit.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
17 / 37
Let
Cnmax (S, T ) = {P ∈ Mn (S ⊗ T ) : r (1 ⊗ 1)n + P ∈ Dnmax (S, T ) ∀ r > 0}
Then {Cnmax (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 ⊗ T
for which 1 ⊗ 1 is an Archimedean matrix order unit, then Dnmax ⊆ Cn
implies Cnmax ⊆ Cn .
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
18 / 37
Let
Cnmax (S, T ) = {P ∈ Mn (S ⊗ T ) : r (1 ⊗ 1)n + P ∈ Dnmax (S, T ) ∀ r > 0}
Then {Cnmax (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 ⊗ T
for which 1 ⊗ 1 is an Archimedean matrix order unit, then Dnmax ⊆ Cn
implies Cnmax ⊆ Cn .
This is a special case of a process called “Archimedeanization”, which was
developed in detail in
“Operator system structures on ordered spaces”, by V. Paulsen,
I. Todorov, and M. Tomforde, preprint.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
18 / 37
Definition
We call the operator system (S ⊗ T , {Cnmax (S, T )}∞
n=1 , 1 ⊗ 1) the maximal
tensor product of S and T and denote it by S ⊗max T .
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
19 / 37
Definition
We call the operator system (S ⊗ T , {Cnmax (S, T )}∞
n=1 , 1 ⊗ 1) the maximal
tensor product of S and T and denote it by S ⊗max T .
Theorem (Kavruk, Paulsen, Todorov, T)
The mapping
max : O × O → O
sending (S, T ) to S ⊗max T defines an associative, symmetric, functorial
operator system tensor product. Moreover, if τ is an operator system
structure on S ⊗ T , then τ ≤ max.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
19 / 37
ˆ is the maximal
The operator space projective tensor product X ⊗Y
operator space tensor product.
Theorem (Kavruk, Paulsen, Todorov, T)
Let X and Y be operator spaces. Then the induced tensor product
X ⊗max Y coincides with the operator space projective tensor product
ˆ .
X ⊗Y
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
20 / 37
ˆ is the maximal
The operator space projective tensor product X ⊗Y
operator space tensor product.
Theorem (Kavruk, Paulsen, Todorov, T)
Let X and Y be operator spaces. Then the induced tensor product
X ⊗max Y coincides with the operator space projective tensor product
ˆ .
X ⊗Y
Theorem (Kavruk, Paulsen, Todorov, T)
Let A and B be C*-algebras. Then the operator system A ⊗max B is
completely order isomorphic to the image of A ⊗ B inside the maximal
C*-algebraic tensor product of A ⊗C*max B. Furthermore, we have
Dnmax (A, B) = Cnmax (A, B).
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
20 / 37
The following result shows that the notion of (min, max)-nuclearity of
operator 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 every
operator system S.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
21 / 37
The following result shows that the notion of (min, max)-nuclearity of
operator 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 every
operator system S.
Thus, we see that a C*-algebra is nuclear if and only if
Cnmin (A, S) = Cnmax (A, S) for every n ∈ N and every operator system S.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
21 / 37
The prior result shows every finite-dimensional C*-algebra is
(min-max)-nuclear.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
22 / 37
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 is
also not a C*-algebra.
Theorem (Kavruk, Paulsen, Todorov, T)
Let S = span{E1,1 , E1,2 , E2,1 , E2,2 , E2,3 , E3,2 , E3,3 } ⊆ M3 .


P1,1 P1,2
0
P2,1 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.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
22 / 37
Theorem (Kavruk, Paulsen, Todorov, T)
There exists an operator system S such that whenever A and B are unital
C*-algebras with A ⊆ B, then every completely positive map φ : A → S
has a completely positive extension φ̃ : B → S, but S is not injective in
the category of operator systems.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
23 / 37
There is another operator system tensor product that agrees with the max
tensor product for all pairs of C*-algebras, but does not agree with the
max tensor product on all pairs of operator systems. This gives a different
extension of the maximal C*-algebraic tensor product from the category of
C*-algebras to the category of operator systems.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
24 / 37
There is another operator system tensor product that agrees with the max
tensor product for all pairs of C*-algebras, but does not agree with the
max tensor product on all pairs of operator systems. This gives a different
extension of the maximal C*-algebraic tensor product from the category of
C*-algebras to the category of operator systems.
We call this tensor product the commuting tensor product. In contrast
with the maximal operator system tensor product, but in analogy with the
minimal one, this tensor product is defined by specifying a collection of
completely positive maps rather than specifying the matrix ordering.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
24 / 37
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 ).}
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
25 / 37
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 on
elementary 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 )}.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
25 / 37
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 on
elementary 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.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
25 / 37
Definition
We call the operator system (S ⊗ T , {Pn }∞
n=1 , 1 ⊗ 1) the commuting
tensor product of S and T and denote it by S ⊗c T .
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
26 / 37
Definition
We call the operator system (S ⊗ T , {Pn }∞
n=1 , 1 ⊗ 1) the commuting
tensor product of S and T and denote it by S ⊗c T .
Theorem (Kavruk, Paulsen, Todorov, T)
The mapping
c :O×O →O
sending (S, T ) to S ⊗c T defines a symmetric, functorial operator system
tensor product.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
26 / 37
Theorem (Kavruk, Paulsen, Todorov, T)
The tensor products c and max are different.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
27 / 37
Theorem (Kavruk, Paulsen, Todorov, T)
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.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
27 / 37
Theorem (Kavruk, Paulsen, Todorov, T)
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, then
A ⊗c S = A ⊗max S.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
27 / 37
Theorem (Kavruk, Paulsen, Todorov, T)
There exists an operator system S such that S ⊗min T = S ⊗c T for every
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 → S does not
approximately factor through matrices.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
28 / 37
We recall that for an operator system S, there exists a unital C*-algebra
Cu∗ (S) (called either the universal C*-algebra of S or the maximal
C*-algebra of S) and a unital complete order isomorphism ι : S → Cu∗ (S)
with the properties that ι(S) generates Cu∗ (S) as a C*-algebra, and that
for every unital completely positive map φ : S → B(H) there exists a
unique ∗-homomorphism π : Cu∗ (S) → B(H) such that π ◦ ι = φ.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
29 / 37
We recall that for an operator system S, there exists a unital C*-algebra
Cu∗ (S) (called either the universal C*-algebra of S or the maximal
C*-algebra of S) and a unital complete order isomorphism ι : S → Cu∗ (S)
with the properties that ι(S) generates Cu∗ (S) as a C*-algebra, and that
for every unital completely positive map φ : S → B(H) there exists a
unique ∗-homomorphism π : Cu∗ (S) → B(H) such that π ◦ ι = φ.
Theorem (Kavruk, Paulsen, Todorov, T)
Let S and T be operator systems. The operator system arising from the
inclusion of S ⊗ T into Cu∗ (S) ⊗C*max Cu∗ (T ) coincides with S ⊗c T .
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
29 / 37
Corollary (Kavruk, Paulsen, Todorov, T)
Let S and T be operator systems. A linear map f : S ⊗c T → B(H) is a
unital completely positive map if and only if there exist a Hilbert space K ,
∗-homomorphisms π : Cu∗ (S) → B(K ) and ρ : Cu∗ (T ) → B(K ) with
commuting 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 .
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
30 / 37
Let X and Y be operator spaces. For u ∈ X ⊗ Y , let
kukµ∗ = sup{k(f · g )(u)k : 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 checked
that this gives an operator space structure to X ⊗ Y , and we denote the
resulting operator space X ⊗µ∗ Y .
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
31 / 37
Let X and Y be operator spaces. For u ∈ X ⊗ Y , let
kukµ∗ = sup{k(f · g )(u)k : 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 checked
that this gives an operator space structure to X ⊗ Y , and we denote the
resulting operator space X ⊗µ∗ Y .
Theorem (Kavruk, Paulsen, Todorov, T)
Let X and Y be operator spaces. Then the identity map is a completely
isometric isomorphism betwen X ⊗c Y and X ⊗µ∗ Y .
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
31 / 37
We say that a functorial operator system tensor product τ is injective if
for 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.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
32 / 37
We say that a functorial operator system tensor product τ is injective if
for 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.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
32 / 37
We say that a functorial operator system tensor product τ is injective if
for 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. An
operator system tensor product is injective if it is both left injective and
right injective.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
32 / 37
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 matrix
ordering 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 space
S ⊗ T whose matrix ordering is induced by the inclusion
S ⊗ T ⊆ I (S) ⊗max I (T ).
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
33 / 37
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 matrix
ordering 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 space
S ⊗ T whose matrix ordering is induced by the inclusion
S ⊗ T ⊆ I (S) ⊗max I (T ).
The tensor product e is functorial and injective, and if τ is any injective
operator system tensor product, then τ ≤ e.
The tensor product el (respectively, er) is functorial and left injective, and
if τ is any left injective (respectively, right injective) operator system
tensor product, then τ ≤ e.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
33 / 37
If A is a C ∗ -algebra represented nondegenerately on H with weak closure
A, then φ ∈ B(B(H), A) is a weak expectation if φ is unital completely
positive, kφk = 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 every
faithful representation π of A, it is the case that π(A) has a weak
expectation.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
34 / 37
If A is a C ∗ -algebra represented nondegenerately on H with weak closure
A, then φ ∈ B(B(H), A) is a weak expectation if φ is unital completely
positive, kφk = 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 every
faithful representation π of A, it is the case that π(A) has a weak
expectation.
Theorem (Kavruk, Paulsen, Todorov, T)
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.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
34 / 37
Theorem (Kavruk, Paulsen, Todorov, T)
Let A be a unital C*-algebra. The following are equivalent:
(i) A is nuclear.
(ii) A ⊗er B = A ⊗max B for every unital C*-algebra B.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
35 / 37
Finally, we characterize the norm k · ke induced by the operator system
structure e.
Theorem (Kavruk, Paulsen, Todorov, T)
Let A and B be unital C*-algebras and u ∈ A ⊗ B. Then
kuke = inf{kukA1 ⊗max B1 : A1 and B1 are unital C*-algebras
with 1A1 ∈ A ⊆ A1 and 1B1 ∈ B ⊆ B1 }.
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
36 / 37
Thank you!
Mark Tomforde (Univeristy of Houston)
Tensor Products of Operator Systems
October 19, 2009
37 / 37