The Unique Pseudo-Expectation Property, and a New Proof of ...people.virginia.edu/~des5e/oldsotoa/S12,sotoa/uva.pdf · Operator Systems and Completely Positive Maps De nition (operator

Unique Pseudo-Expectation Property & Mercer’s Extension Theorem

The Unique Pseudo-Expectation Property,and a New Proof of Mercer’s Extension Theorem

Vrej Zarikian (USNA)

joint with Jan Cameron (Vassar) and David Pitts (Nebraska)

UVA Seminar in Operator Theory and Operator Algebras, 17 April 2012

Vrej Zarikian (USNA) Unique Pseudo-Expectation Property & Mercer’s Extension Theorem


The Injective Envelope

Operator Systems and Completely Positive Maps

Definition (operator system)

An operator system is a unital subspace S ⊆ B(H), closed under the adjoint.

Note: Mn(S) ⊆ Mn(B(H)) ∼= B(Hn) is itself an operator system.

Definition (completely positive map)

Let S be an operator system. We say that a linear map ρ : S → B(K) is positive ifρ(S+) ⊆ B(K)+. We say that ρ is completely positive if

ρn : Mn(S)→ Mn(B(K)) = B(Kn) :[xij

]7→[ρ(xij )

]is positive for all n ∈ N.

Example

If A ⊆ B(H) is a unital C?-algebra and π : A → B(K) is a ?-homomorphism, then Ais an operator system and π is a completely positive map.




Operator Systems and Completely Positive Maps II

Theorem (Arveson)

Let S ⊆ S ⊆ B(H) be operator systems and ρ : S → B(K) be a unital completelypositive (ucp) map. Then there exists a ucp map ρ : S → B(K) extending ρ.

Theorem (Stinespring)

Let A be a unital C?-algebra and ρ : A → B(K) be a ucp map. Then there exists a

unital ?-homomorphism π : A → B(K) and an operator V ∈ B(K, K) such that

ρ(a) = V ?π(a)V , a ∈ A .




Operator Spaces and Completely Bounded Maps

Definition (operator space)

An operator space is a subspace E ⊆ B(H).

Note: Mn(E) ⊆ Mn(B(H)) ∼= B(Hn) is itself an operator space.

Definition (completely bounded)

Let E be an operator space and u : E → B(K) be a linear map. We say that u iscompletely bounded if

‖u‖cb := supn‖un : Mn(E)→ B(Kn)‖ <∞.

We say that u is completely contractive if ‖u‖cb ≤ 1, and that u is completelyisometric if un is an isometry for all n ∈ N.

Example

Let A be a unital C?-algebra and π : A → B(K) be a ?-homomorphism. Then A isan operator space and π is completely contractive. If π is injective, then it iscompletely isometric.




Operator Spaces and Completely Bounded Maps II

Theorem (Wittstock)

Let E ⊆ E ⊆ B(H) be operator spaces and u : E → B(K) be a completely boundedmap. Then there exists a completely bounded map u : E → B(K) extending u, with‖u‖cb = ‖u‖cb.

Theorem (Wittstock, Haagerup, Paulsen)

Let A be a unital C?-algebra and u : A → B(K) be a completely bounded map. Then

there exists a unital ?-homomorphism π : A → B(K) and operators V ,W ∈ B(K, K)such that

u(a) = V ?π(a)W , a ∈ A .

Furthermore, we may arrange ‖V ‖‖W ‖ = ‖u‖cb.




Completely Positive vs. Completely Bounded

Proposition

Let S ⊆ B(H) be an operator system and ρ : S → B(K) be a unital linear map. Thenρ is completely positive if and only if ρ is completely contractive.




Injectivity

Definition (injective C?-algebra)

Let A ⊆ B(K) be a unital C?-algebra (or operator system). We say that A isinjective if the following condition holds: Whenever S ⊆ S ⊆ B(H) are operatorsystems and ρ : S → A is a ucp map, there exists a ucp map ρ : S → A extending ρ.

Example

B(K) is injective.

C(X ) is injective iff X is Stonean.





Theorem (Hamana)

Let S ⊆ B(H) be an operator system. Then there exists a minimal injective operatorsystem I (S) ⊆ B(H) containing S. It is unique up to a complete order isomorphismwhich fixes S, and is called the injective envelope of S.

Remark

I (S) is a C?-algebra with respect to the linear and involutive structure it inheritsfrom B(H), but a different product.

If A is a unital C?-algebra, then A ⊆ I (A) is a unital C?-subalgebra.

If D is a unital abelian C?-algebra, then I (D) is a unital abelian C?-algebra.



The Unique Pseudo-Expectation Property

Motivating Result

Theorem (Pitts ’11)

Let (C,D) be a regular MASA inclusion, i.e., D ⊆ C is a MASA andspan(N(C,D)) = C. Then

There exists a unique ucp map Φ : C → I (D) such that Φ|D = id.

Furthermore,LΦ = {x ∈ C : Φ(x?x) = 0}

is a closed 2-sided ideal in C (the largest closed 2-sided ideal in C which intersectsD trivially).

If LΦ = 0, i.e., if Φ is faithful, then D norms C in the sense of Pop, Sinclair, andSmith.





Definition (pseudo-expectation)

Let (C,D) be an inclusion of unital C?-algebras. By a pseudo-expectation for (C,D)we mean a ucp map Φ : C → I (D) such that Φ|D = id. If, in fact, Φ(C) = D, then wesay that Φ is an expectation for (C,D).

Unlike expectations, pseudo-expectations always exist, by injectivity. They maynot be unique, however.

By Choi’s Lemma, pseudo-expectations are D-bimodule maps.

Definition (unique pseudo-expectation property)

We say that (C,D) has the unique pseudo-expectation property if there exists aunique pseudo-expectation Φ : C → I (D). If, in addition, Φ is faithful, we say that(C,D) has the faithful unique pseudo-expectation property.




Elementary Examples

Example

Let (C,D) be a regular MASA inclusion. Then (C,D) has the uniquepseudo-expectation property, by Pitts’ Theorem.

Example

(B(`2), `∞) has the faithful unique pseudo-expectation property. Indeed, (B(`2), `∞)has a unique expectation, which is faithful, and `∞ is injective.

Example

For any unital C?-algebra D, the inclusion (I (D),D) has the faithful uniquepseudo-expectation property. Indeed, the inclusion (I (D),D) is rigid, and soΦ : I (D)→ I (D) ucp and Φ|D = id implies Φ = id.




Non-example

Example (non-example)

Let M be a II1 factor with separable predual, and A ⊆M be a MASA. Then (M,A)does not have the unique pseudo-expectation property. Indeed, since M has separablepredual, A is singly-generated. By a result of Akemann-Sherman, there exist multipleexpectations for (M,A), therefore multiple pseudo-expectations.




Property I: Hereditary

Proposition

The (faithful) unique pseudo-expectation property is hereditary. That is, if (C,D) hasthe (faithful) unique pseudo-expectation property and D ⊆ C0 ⊆ C is a unitalC?-algebra, then (C0,D) has the (faithful) unique pseudo-expectation property.

Proof.

Let Φ : C → I (D) be the unique pseudo-expectation. Let Ψ0 : C0 → I (D) be apseudo-expectation. By injectivity, there exists a pseudo-expectation Ψ : C → I (D)such that Ψ |C0

= Ψ0. By uniqueness, Ψ = Φ, and so Ψ0 = Φ|C0. It follows that (C0,D)

has a unique pseudo-expectation, namely Φ0 = Φ|C0. If Φ is faithful, then so is Φ0.




C ?-Essential Inclusions

Definition (C?-essential inclusion)

Let (C,D) be an inclusion of unital C?-algebras. We say that (C,D) is C?-essential ifevery unital ?-homomorphism π : C → B(H) which is faithful on D is actually faithfulon C. Equivalently, (C,D) is C?-essential if every nontrivial (closed 2-sided) ideal in Cintersects D nontrivially.




Property II: C ?-Essential

Proposition

If (C,D) has the faithful unique pseudo-expectation property, then (C,D) isC?-essential. The converse is false in general.

Proof.

Let Φ : C → I (D) be the unique pseudo-expectation. Let J be a closed 2-sided idealin C such that J ∩D = 0, and let q : C → C /J be the corresponding quotient map,which is faithful on D. By injectivity, there exists a ucp map Ψ : q(C)→ I (D) suchthat Ψ |q(D) = q−1. Then Ψ ◦ q : C → I (D) is a pseudo-expectation, and soΨ ◦ q = Φ. Thus x ∈ J implies

Φ(x?x) = Ψ(q(x?x)) = Ψ(q(x)?q(x)) = 0,

which implies x = 0, since Φ is faithful.




C ?-Envelope

Theorem (Hamana)

Let S be an operator system (or unital operator space). Then there exists a unitalC?-algebra C?env(S) generated by S, and satisfying the following condition: WheneverA is a unital C?-algebra generated by S, there exists a unique ?-homomorphismπ : A → C?env(S) which fixes S. C?env(S) is unique up to a ?-isomorphism which fixesS, and is called the C?-envelope of S.

Proof.

C?env(S) is the C?-subalgebra of I (S) generated by S.




Property III: C ?-Envelope

Corollary

Suppose (C,D) has the faithful unique pseudo-expectation property. If D ⊆ X ⊆ C isan operator space, then C?env(X ) = C?(X ).

Proof.

Since (C,D) has the faithful unique pseudo-expectation property andD ⊆ C?(X ) ⊆ C, (C?(X ),D) has the faithful unique pseudo-expectation property.Then (C?(X ),D) is C?-essential. By the universal property of the C?-envelope, thereexists a unique unital ?-homomorphism π : C?(X )→ C?env(X ) such that π|X = id. Itfollows that π is faithful on D, which implies it is actually faithful on C?(X ).




Abelian Inclusions

Theorem

Let (C(Y ),C(X )) be an inclusion of unital abelian C?-algebras, with correspondingcontinuous surjection j : Y � X . Then the following are equivalent:

1 (C(Y ),C(X )) has the faithful unique pseudo-expectation property.

2 (C(Y ),C(X )) is C?-essential.

3 (Y , j) is an essential cover of X . That is, if K ⊆ Y is closed and j(K) = X , thenK = Y .

When these conditions hold, the unique pseudo-expectation Φ : C(Y )→ I (C(X )) is a?-homomorphism.

Corollary

Suppose D is abelian and (C,D) has the faithful unique pseudo-expectation property.Then the following conditions hold:

Dc = {x ∈ C : xd = dx for all d ∈ D} is abelian, therefore the unique MASAcontaining D.

(Dc ,D) is C?-essential.

Φ|Dc is a ?-homomorphism. In fact, Dc is the multiplicative domain for Φ.




Topological Dynamical Systems

Recall the crossed product construction C(X ) oα Z. It contains two special abelianC?-subalgebras: C(X ) and C(X )c .

Theorem

(C(X ) oα Z,C(X )c ) has the faithful unique pseudo-expectation property.

Corollary

The following are equivalent:

1 (C(X ) oα Z,C(X )) has the faithful unique pseudo-expectation property.

2 (C(X ) oα Z,C(X )) has the unique pseudo-expectation property.

3 C(X )c = C(X ).

4 (X , h) is topologically free.




Norming Subalgebras

Definition (F. Pop , Sinclair, and Smith ’00)

Let A be a unital operator algebra and D ⊆ A be a unital C?-subalgebra. We saythat D norms A if for all X ∈ Mn(A), we have that

‖X‖ = sup{‖RXC‖ : R ∈ Ball(M1,n(D)), C ∈ Ball(Mn,1(D))}.

Theorem (F. Pop, Sinclair, and Smith ’00)

1 Any unital C?-algebra norms itself.

2 Any MASA norms B(H).

3 A unital C?-algebra is normed by the scalars if and only if it is abelian.

4 If N ⊆M is a finite-index inclusion of II1 factors, then N norms M.




Norming and Automatic Complete Boundedness

Theorem (Pitts ’08)

Let A and B be operator algebras and θ : A → B be a bounded isomorphism. If Bcontains a norming C?-subalgebra, then θ is completely bounded and

‖θ‖cb ≤ ‖θ‖‖θ−1‖4.




Faithful Unique Pseudo-Expectation ⇒ Norming?

Recall: If (C,D) is a regular MASA inclusion and if the unique pseudo-expectationΦ : C → I (D) is faithful, then D norms C, by Pitts’ Theorem.

Question

If (C,D) has the faithful unique pseudo-expectation property, does D norm C?

Proposition

For all known examples where (C,D) has the faithful unique pseudo-expectationproperty, D norms C. In more detail:

If (C,D) is a regular MASA inclusion with the faithful unique pseudo-expectationproperty, then D norms C. (Pitts)

`∞ norms B(`2). (Pop, Sinclair, and Smith)

For any unital C?-algebra, D norms I (D).

If (C(Y ),C(X )) has the faithful unique pseudo-expectation property, then C(X )norms C(Y ). (Pop, Sinclair, and Smith)

C(X )c norms C(X ) oα Z. (Pitts)

If (C(X ) oα Z,C(X )) has the faithful unique pseudo-expectation property, thenC(X ) norms C(X ) oα Z. (Pitts)



Mercer’s Extension Theorem


Theorem (Mercer ’91)

Let Ai ⊆ (Mi ,Di ) be Cartan bimodule algebras and θ : A1 → A2 be a Cartanbimodule isomorphism. Then there exists a ?-isomorphism π :M1 →M2 such thatπ|A1

= θ.




Cartan Subalgebras

Definition (Cartan subalgebra)

Let M be a von Neumann algebra. We say that D ⊆M is a Cartan subalgebra if thefollowing conditions hold:

1 D is a MASA in M.

2 spanσ(UN(M,D)) =M.

3 There exists a normal faithful conditional expectation E :M→D.

Example

Dn(C) ⊆ Mn(C) is a Cartan subalgebra. Indeed,

UN(Mn(C),Dn(C)) = {PV : P ∈ Mn(C) permutation matrix, V ∈ Dn(T)}.




Cartan Subalgebras

Definition (Cartan subalgebra)

Let M be a von Neumann algebra. We say that D ⊆M is a Cartan subalgebra if thefollowing conditions hold:

1 D is a MASA in M.

2 spanσ(UN(M,D)) =M.

3 There exists a normal faithful conditional expectation E :M→D.

Example

Dn(C) ⊆ Mn(C) is a Cartan subalgebra. Indeed,

UN(Mn(C),Dn(C)) = {PV : P ∈ Mn(C) permutation matrix, V ∈ Dn(T)}.




The Feldman-Moore Construction (’77)

Inputs:

X , a standard Borel space

R, a countable Borel equivalence relation on X

µ, a probability measure on X which is quasi-invariant for R

s, a normalized 2-cocycle on R

Output:

ν, right counting measure on R relative to µ

L2(R, ν), a separable Hilbert space

M(R, s) ⊆ B(L2(R, ν)), a von Neumann algebra consisting of certain boundedBorel functions T : R → C acting on L2(R, ν) by twisted matrix multiplication:

Tξ(x , y) =∑zRx

T (x , z)ξ(z, y)s(x , z, y), ξ ∈ L2(R, ν), (x , y) ∈ R

D(R, s) = {T ∈ M(R, s) : T (x , y) = 0 if x 6= y}, a Cartan subalgebra of M(R, s)




The Feldman-Moore Construction (’77)

Inputs:

X , a standard Borel space

R, a countable Borel equivalence relation on X

µ, a probability measure on X which is quasi-invariant for R

s, a normalized 2-cocycle on R

Output:

ν, right counting measure on R relative to µ

L2(R, ν), a separable Hilbert space

M(R, s) ⊆ B(L2(R, ν)), a von Neumann algebra consisting of certain boundedBorel functions T : R → C acting on L2(R, ν) by twisted matrix multiplication:

Tξ(x , y) =∑zRx

T (x , z)ξ(z, y)s(x , z, y), ξ ∈ L2(R, ν), (x , y) ∈ R

D(R, s) = {T ∈ M(R, s) : T (x , y) = 0 if x 6= y}, a Cartan subalgebra of M(R, s)




The Feldman-Moore Representation Theorem

Theorem (Feldman, Moore ’77)

Let M be a von Neumann algebra with separable predual and D ⊆M be a Cartansubalgebra. Then there exists X , R, µ, and s such that M∼= M(R, s), withD ∼= D(R, s).




Cartan Bimodule Algebras

Definition (Cartan bimodule algebra)

Let M be a von Neumann algebra with separable predual and D ⊆M be a Cartansubalgebra. We say that D ⊆ A ⊆M is a Cartan bimodule algebra if the followingconditions hold:

1 A is a σ-weakly closed (non-self-adjoint) subalgebra.

2 W ?(A) =M.

Example

D4(C) ⊆

a11 a12 0 a14

0 a22 0 00 a32 a33 00 0 0 a44

: aij ∈ C

⊆ M4(C)

is a Cartan bimodule algebra.




Cartan Bimodule Algebras

Definition (Cartan bimodule algebra)

Let M be a von Neumann algebra with separable predual and D ⊆M be a Cartansubalgebra. We say that D ⊆ A ⊆M is a Cartan bimodule algebra if the followingconditions hold:

1 A is a σ-weakly closed (non-self-adjoint) subalgebra.

2 W ?(A) =M.

Example

D4(C) ⊆

a11 a12 0 a14

0 a22 0 00 a32 a33 00 0 0 a44

: aij ∈ C

⊆ M4(C)

is a Cartan bimodule algebra.




The Spectral Theorem for Bimodules

Theorem (Muhly, Saito, Solel ’88)

Let A ⊆ (M,D) be a Cartan bimodule algebra. Then there exists a unique Borel setΓ(A) ⊆ R such that

A ∼= {T ∈ M(R, s) : T (x , y) = 0 for all (x , y) /∈ Γ(A)}.

In fact, Γ(A) is a reflexive and transitive relation which generates R.

Corollary (abundance of normalizers)

Let A ⊆ (M,D) be a Cartan bimodule algebra. Then

spanσ(GN(A,D)) = A .




The Spectral Theorem for Bimodules

Theorem (Muhly, Saito, Solel ’88)

Let A ⊆ (M,D) be a Cartan bimodule algebra. Then there exists a unique Borel setΓ(A) ⊆ R such that

A ∼= {T ∈ M(R, s) : T (x , y) = 0 for all (x , y) /∈ Γ(A)}.

In fact, Γ(A) is a reflexive and transitive relation which generates R.

Corollary (abundance of normalizers)

Let A ⊆ (M,D) be a Cartan bimodule algebra. Then

spanσ(GN(A,D)) = A .




Cartan Bimodule Isomorphisms

Definition (Cartan bimodule isomorphism)

Let Ai ⊆ (Mi ,Di ), i = 1, 2, be Cartan bimodule algebras. We say that θ : A1 → A2

is a Cartan bimodule isomorphism if the following conditions hold:

1 θ is an isometric isomorphism.

2 θ(D1) = D2.

Example

Let α, β, γ ∈ R. Thena11 a12 0 a14

0 a22 0 00 a32 a33 00 0 0 a44

7→

a11 e iαa12 0 e iβa14

0 a22 0 00 e iγa32 a33 00 0 0 a44

is a Cartan bimodule isomorphism.




Cartan Bimodule Isomorphisms

Definition (Cartan bimodule isomorphism)

Let Ai ⊆ (Mi ,Di ), i = 1, 2, be Cartan bimodule algebras. We say that θ : A1 → A2

is a Cartan bimodule isomorphism if the following conditions hold:

1 θ is an isometric isomorphism.

2 θ(D1) = D2.

Example

Let α, β, γ ∈ R. Thena11 a12 0 a14

0 a22 0 00 a32 a33 00 0 0 a44

7→


0 a22 0 00 e iγa32 a33 00 0 0 a44

is a Cartan bimodule isomorphism.




Mercer’s Representation Theorem


Let Ai ⊆ (Mi ,Di ), i = 1, 2, be a Cartan bimodule algebras and let θ : A1 → A2 bean Cartan bimodule isomorphism. Then there exists a Borel isomorphism τ : X1 → X2

and a Borel function m : Γ(A2)→ T such that the following conditions hold:

1 (τ × τ)(R1) = R2.

2 (τ × τ)(Γ(A1)) = Γ(A2).

3 θ(T )(x , y) = m(x , y)T (τ−1(x), τ−1(y)), T ∈ A1, (x , y) ∈ Γ(A2).

Example


0 a22 0 00 e iγa32 a33 00 0 0 a44

=

1 e iα 0 e iβ

0 1 0 00 e iγ 1 00 0 0 1

•

a11 a12 0 a14

0 a22 0 00 a32 a33 00 0 0 a44




Mercer’s Representation Theorem


Let Ai ⊆ (Mi ,Di ), i = 1, 2, be a Cartan bimodule algebras and let θ : A1 → A2 bean Cartan bimodule isomorphism. Then there exists a Borel isomorphism τ : X1 → X2

and a Borel function m : Γ(A2)→ T such that the following conditions hold:

1 (τ × τ)(R1) = R2.

2 (τ × τ)(Γ(A1)) = Γ(A2).

3 θ(T )(x , y) = m(x , y)T (τ−1(x), τ−1(y)), T ∈ A1, (x , y) ∈ Γ(A2).

Example


0 a22 0 00 e iγa32 a33 00 0 0 a44

=

1 e iα 0 e iβ

0 1 0 00 e iγ 1 00 0 0 1

•

a11 a12 0 a14

0 a22 0 00 a32 a33 00 0 0 a44







= θ.

Proof.

Extend m : Γ(A2)→ T in Mercer’s Representation Theorem to m : R2 → T in anappropriate way and define

π(T )(x , y) = m(x , y)T (τ−1(x), τ−1(y)), T ∈M1, (x , y) ∈ R2.

Example

m =

1 e iα 0 e iβ

0 1 0 00 e iγ 1 00 0 0 1

⇒ m =

1 e iα e i(α−γ) e iβ

e−iα 1 e−iγ e−i(α−β)

e−i(α−γ) e iγ 1 e−i(α−β−γ)

e−iβ e i(α−β) e i(α−β−γ) 1







= θ.

Proof.



Example

m =

1 e iα 0 e iβ

0 1 0 00 e iγ 1 00 0 0 1

⇒ m =











= θ.

Proof.



Example

m =

1 e iα 0 e iβ

0 1 0 00 e iγ 1 00 0 0 1

⇒ m =








On the other hand...


0 a22 0 00 e iγa32 a33 00 0 0 a44

= U

a11 a12 0 a14

0 a22 0 00 a32 a33 00 0 0 a44

U?,

where

U =

e iα 0 0 00 1 0 00 0 e iγ 0

0 0 0 e i(α−β)

.



A New Proof of Mercer’s Extension Theorem

Step 1: Nice properties of θ

1 θ is σ-weakly continuous.

Proof.

Mercer’s Representation Theorem.

2 θ(GN(A1,D1)) = GN(A2,D2).

Proof.







Proof.


2 θ(GN(A1,D1)) = GN(A2,D2).

Proof.







Proof.


2 θ(GN(A1,D1)) = GN(A2,D2).

Proof.







Proof.


2 θ(GN(A1,D1)) = GN(A2,D2).

Proof.







Proof.


2 θ(GN(A1,D1)) = GN(A2,D2).

Proof.





Step 2: Replace weak with norm

Define:

A◦i = span(GN(Ai ,Di )), a unital operator algebra

M◦i = C?(A◦i ), a unital C?-algebra

θ◦ = θ|A◦1

: A◦1 → θ(A◦1 ) ⊆ A2, an isometric isomorphism

Note that:

Di ⊆ A◦i ⊆M◦i

Di ⊆M◦i is a MASA

span(GN(M◦i ,Di )) =M◦i(M◦i ,Di ) has the faithful unique pseudo-expectation property

Di norms M◦i , and therefore A◦iC?env(A◦i ) =M◦iA◦i

σ= Ai

M◦iσ

=Mi

θ◦(A◦1 ) = A◦2




Step 2: Replace weak with norm

Define:

A◦i = span(GN(Ai ,Di )), a unital operator algebra

M◦i = C?(A◦i ), a unital C?-algebra

θ◦ = θ|A◦1

: A◦1 → θ(A◦1 ) ⊆ A2, an isometric isomorphism

Note that:

Di ⊆ A◦i ⊆M◦i

Di ⊆M◦i is a MASA

span(GN(M◦i ,Di )) =M◦i(M◦i ,Di ) has the faithful unique pseudo-expectation property

Di norms M◦i , and therefore A◦iC?env(A◦i ) =M◦iA◦i

σ= Ai

M◦iσ

=Mi

θ◦(A◦1 ) = A◦2




Step 3: Extend θ◦ to π◦

There exists a unique ?-isomorphism π◦ :M◦1 →M◦2 which extends θ◦ : A◦1 → A◦2 .

Proof.

Since θ◦ : A◦1 → A◦2 is an isometric isomorphism and D2 norms A◦2 , θ is completelycontractive, by Pitts’ Automatic Complete Boundedness Theorem. Likewise, since(θ◦)−1 : A◦2 → A◦1 is an isometric isomorphism and D1 norms A◦1 , (θ◦)−1 iscompletely contractive. Since θ◦ : A◦1 → A◦2 is a completely isometric isomorphism,there exists a unique ?-isomorphism π◦ : C?env(A◦1 )→ C?env(A◦2 ) such thatπ◦|A◦

1= θ◦. But C?env(A◦i ) =M◦i .




Step 3: Extend θ◦ to π◦

There exists a unique ?-isomorphism π◦ :M◦1 →M◦2 which extends θ◦ : A◦1 → A◦2 .

Proof.

Since θ◦ : A◦1 → A◦2 is an isometric isomorphism and D2 norms A◦2 , θ is completelycontractive, by Pitts’ Automatic Complete Boundedness Theorem. Likewise, since(θ◦)−1 : A◦2 → A◦1 is an isometric isomorphism and D1 norms A◦1 , (θ◦)−1 iscompletely contractive. Since θ◦ : A◦1 → A◦2 is a completely isometric isomorphism,there exists a unique ?-isomorphism π◦ : C?env(A◦1 )→ C?env(A◦2 ) such thatπ◦|A◦

1= θ◦. But C?env(A◦i ) =M◦i .




Step 4: Define an implementing unitary for π◦

There exists a cyclic and separating vector ξ1 for M1 ⊆ B(H1) and a cyclic vector ξ2

for M2 ⊆ B(H2) such that

M◦1 ξ1 →M◦2 ξ2 : xξ1 7→ π◦(x)ξ2

is isometric. Thus there exists a unitary U : H1 →H2 such that

π◦(x) = UxU?, x ∈M◦1 .

Proof.

Straightforward but a little tedious.




Step 4: Define an implementing unitary for π◦

There exists a cyclic and separating vector ξ1 for M1 ⊆ B(H1) and a cyclic vector ξ2

for M2 ⊆ B(H2) such that

M◦1 ξ1 →M◦2 ξ2 : xξ1 7→ π◦(x)ξ2

is isometric. Thus there exists a unitary U : H1 →H2 such that

π◦(x) = UxU?, x ∈M◦1 .

Proof.

Straightforward but a little tedious.




Step 5: Conclusion

Defineπ(x) = UxU?, x ∈M1 .

Then π :M1 →M2 is a σ-weakly continuous ?-isomorphism such that π|M◦1

= π◦.

Sinceπ|A◦

1= π◦|A◦

1= θ◦ = θ|A◦

1

and θ is σ-weakly continuous,π|A1

= θ.




Future Directions

1 Rely less on Feldman-Moore. In particular, eliminate the use of Mercer’sRepresentation Theorem.

2 Prove Mercer’s Extension Theorem in the norm context. X (Pitts)

3 Study (characterize?) the unique pseudo-expectation property.


Documents

The Unique Pseudo-Expectation Property, and a New Proof of ...people.virginia.edu/~des5e/oldsotoa/S12,sotoa/uva.pdf · Operator Systems and Completely Positive Maps De nition (operator