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Convex Sets Associated to C -Algebras Trace Space Examples Scott - - PowerPoint PPT Presentation

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Convex Sets Associated to C -Algebras S. Atkinson Introduction Classical Situation 2011 Situation H om ( A , M ) Preliminaries Extreme Points Convex Sets Associated to C -Algebras Trace Space Examples Scott Atkinson University of

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Convex Sets Associated to C∗-Algebras

Scott Atkinson

University of Virginia

ECOAS 2014

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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SLIDE 2

Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Classical Situation (1970’s): Ext(A)

Let A be a separable unital C∗-algebra. Ext(A) is given by the set of unital ∗-monomorphisms π : A → B(H)/K(H) modulo B(H)-unitary equivalence.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Classical Situation (1970’s): Ext(A)

Let A be a separable unital C∗-algebra. Ext(A) is given by the set of unital ∗-monomorphisms π : A → B(H)/K(H) modulo B(H)-unitary equivalence. Use a unitarily implemented isomorphism between B(H) and M2(B(H)) to define a semigroup structure on Ext(A).

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Classical Situation (1970’s): Ext(A)

Let A be a separable unital C∗-algebra. Ext(A) is given by the set of unital ∗-monomorphisms π : A → B(H)/K(H) modulo B(H)-unitary equivalence. Use a unitarily implemented isomorphism between B(H) and M2(B(H)) to define a semigroup structure on Ext(A). Here is the picture: [π] + [ρ] = π ρ

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

2011 Situation: Hom(N, RU)

In 2011 Brown introduced the following convex set.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

2011 Situation: Hom(N, RU)

In 2011 Brown introduced the following convex set. For N a separable II1-factor, R the hyperfinite II1-factor, and U a free ultrafilter on N define Hom(N, RU) to be the set of unital ∗-homomorphisms π : N → RU modulo unitary equivalence.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

2011 Situation: Hom(N, RU)

In 2011 Brown introduced the following convex set. For N a separable II1-factor, R the hyperfinite II1-factor, and U a free ultrafilter on N define Hom(N, RU) to be the set of unital ∗-homomorphisms π : N → RU modulo unitary equivalence. We use isomorphisms between RU and pRUp for p a projection in RU to define convex combinations.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

2011 Situation: Hom(N, RU)

Here is a(n incorrect) picture:

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

2011 Situation: Hom(N, RU)

Here is a(n incorrect) picture: t[π] + (1 − t)[ρ] = pπp p⊥ρp⊥

  • where p is a projection in RU and τR(p) = t.
  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

2011 Situation: Hom(N, RU)

Here is a(n incorrect) picture: t[π] + (1 − t)[ρ] = pπp p⊥ρp⊥

  • where p is a projection in RU and τR(p) = t.

With this definition, we may consider Hom(N, RU) as a closed, bounded, convex subset of a Banach space.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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SLIDE 11

Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

2011 Situation: Hom(N, RU)

Here is a(n incorrect) picture: t[π] + (1 − t)[ρ] = pπp p⊥ρp⊥

  • where p is a projection in RU and τR(p) = t.

With this definition, we may consider Hom(N, RU) as a closed, bounded, convex subset of a Banach space. Brown was able to characterize extreme points: Theorem (Brown, 2011) [π] ∈ Hom(N, RU) is extreme if and only if π(N)′ ∩ RU is a factor.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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SLIDE 12

Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

2011 Situation: Hom(N, RU)

Here is a(n incorrect) picture: t[π] + (1 − t)[ρ] = pπp p⊥ρp⊥

  • where p is a projection in RU and τR(p) = t.

With this definition, we may consider Hom(N, RU) as a closed, bounded, convex subset of a Banach space. Brown was able to characterize extreme points: Theorem (Brown, 2011) [π] ∈ Hom(N, RU) is extreme if and only if π(N)′ ∩ RU is a factor.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Preliminaries

Definition For a separable, unital, tracial C∗-algebra A, and a separable McDuff II1-factor M (M ∼ = M ⊗ R), we define Hom(A, M) to be the space of unital ∗-homomorphisms π : A → M modulo the equivalence relation of weak approximate unitary equivalence (w.a.u.e.).

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Preliminaries

Definition For a separable, unital, tracial C∗-algebra A, and a separable McDuff II1-factor M (M ∼ = M ⊗ R), we define Hom(A, M) to be the space of unital ∗-homomorphisms π : A → M modulo the equivalence relation of weak approximate unitary equivalence (w.a.u.e.). That is, [π] = [ρ] if there is a sequence {un} of unitaries in M such that for every a ∈ A we have lim

n ||π(a) − unρ(a)u∗ n||2 = 0.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Preliminaries

Definition For a separable, unital, tracial C∗-algebra A, and a separable McDuff II1-factor M (M ∼ = M ⊗ R), we define Hom(A, M) to be the space of unital ∗-homomorphisms π : A → M modulo the equivalence relation of weak approximate unitary equivalence (w.a.u.e.). That is, [π] = [ρ] if there is a sequence {un} of unitaries in M such that for every a ∈ A we have lim

n ||π(a) − unρ(a)u∗ n||2 = 0.

We endow Hom(A, M) with the topology of pointwise convergence (with appropriate consideration for equivalence classes).

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Convex Structure

Taking advantage of the properties of a McDuff factor (M ∼ = M ⊗ R), we can define convex combinations in Hom(A, M).

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Convex Structure

Taking advantage of the properties of a McDuff factor (M ∼ = M ⊗ R), we can define convex combinations in Hom(A, M). Definition For a McDuff factor M, an isomorphism σM : M ⊗ R → M is a regular isomorphism if σM ◦ (idM ⊗ 1R) ∼ idM.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Convex Structure

Definition For t ∈ [0, 1], [π], [ρ] ∈ Hom(A, M), we define t[π] + (1 − t)[ρ] := [σM(π ⊗ p + ρ ⊗ p⊥)] where σM : M ⊗ R → M is a regular isomorphism and p is a projection in R with τR(p) = t.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Convex Structure

Definition For t ∈ [0, 1], [π], [ρ] ∈ Hom(A, M), we define t[π] + (1 − t)[ρ] := [σM(π ⊗ p + ρ ⊗ p⊥)] where σM : M ⊗ R → M is a regular isomorphism and p is a projection in R with τR(p) = t. (Correct) Picture: (1M ⊗ p)(π ⊗ 1R)(1M ⊗ p) (1M ⊗ p⊥)(ρ ⊗ 1R)(1M ⊗ p⊥)

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Extreme Points

We may also consider Hom(A, M) as a closed, separable, bounded, convex subset of a Banach space.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Extreme Points

We may also consider Hom(A, M) as a closed, separable, bounded, convex subset of a Banach space. We would like to find a nice characterization of extreme points.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Extreme Points

We may also consider Hom(A, M) as a closed, separable, bounded, convex subset of a Banach space. We would like to find a nice characterization of extreme points. The characterization in the ultrapower situation cannot apply here because relative commutants are not well-defined under weak approximate unitary equivalence.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Extreme Points

We may also consider Hom(A, M) as a closed, separable, bounded, convex subset of a Banach space. We would like to find a nice characterization of extreme points. The characterization in the ultrapower situation cannot apply here because relative commutants are not well-defined under weak approximate unitary equivalence. Proposition (A.) Given π : A → M, we have that Hom(A/kerπ, M) is a face of Hom(A, M).

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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SLIDE 24

Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Extreme Points

We may also consider Hom(A, M) as a closed, separable, bounded, convex subset of a Banach space. We would like to find a nice characterization of extreme points. The characterization in the ultrapower situation cannot apply here because relative commutants are not well-defined under weak approximate unitary equivalence. Proposition (A.) Given π : A → M, we have that Hom(A/kerπ, M) is a face of Hom(A, M).

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Connection to Trace Space

Given [π] ∈ Hom(A, M) we get a (unital) trace on A given by τM ◦ π.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Connection to Trace Space

Given [π] ∈ Hom(A, M) we get a (unital) trace on A given by τM ◦ π. The correspondence [π] → τM ◦ π is well-defined, continuous, and affine.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Connection to Trace Space

Given [π] ∈ Hom(A, M) we get a (unital) trace on A given by τM ◦ π. The correspondence [π] → τM ◦ π is well-defined, continuous, and affine. Natural question: For a fixed M, does this give all of the (unital) traces on A?

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Nuclear Case

Theorem (A.) If A is nuclear then for any McDuff M we have Hom(A, M) ∼ = T(A) given by [π] ↔ τM ◦ π.

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Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Nuclear Case

Theorem (A.) If A is nuclear then for any McDuff M we have Hom(A, M) ∼ = T(A) given by [π] ↔ τM ◦ π. English Version: All traces of a separable unital nuclear algebra “lift”through any fixed McDuff factor; and the traces “remember”their homomorphisms up to w.a.u.e.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Nuclear Case

Theorem (A.) If A is nuclear then for any McDuff M we have Hom(A, M) ∼ = T(A) given by [π] ↔ τM ◦ π. English Version: All traces of a separable unital nuclear algebra “lift”through any fixed McDuff factor; and the traces “remember”their homomorphisms up to w.a.u.e. (Recall: A nuclear ⇒ Ext(A) is a group. But the class of algebras A for which Ext(A) is a group is strictly larger than the nuclears. In 1977 Anderson showed that Ext(A) is not always a group.)

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Examples

Similar to the program of Ext(A) we would like to find examples of the following.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Examples

Similar to the program of Ext(A) we would like to find examples of the following.

1 Well-Behaved Non-Nuclear: A non-nuclear A where for

any McDuff M, all traces lift through M and the traces remember their homomorphisms.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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SLIDE 33

Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Examples

Similar to the program of Ext(A) we would like to find examples of the following.

1 Well-Behaved Non-Nuclear: A non-nuclear A where for

any McDuff M, all traces lift through M and the traces remember their homomorphisms.

2 Too Many Traces: A necessarily non-nuclear algebra B

where for some McDuff M, there is a trace on B that does not lift through M.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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SLIDE 34

Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

Examples

Similar to the program of Ext(A) we would like to find examples of the following.

1 Well-Behaved Non-Nuclear: A non-nuclear A where for

any McDuff M, all traces lift through M and the traces remember their homomorphisms.

2 Too Many Traces: A necessarily non-nuclear algebra B

where for some McDuff M, there is a trace on B that does not lift through M.

3 Forgetful Trace: A necessarily non-nuclear algebra C where

for some McDuff M, there is a trace on C lifting through M via two inequivalent homomorphisms.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

(1) Well-Behaved Non-Nuclear

Dadarlat found a tracial non-nuclear A contained in an AF-algebra.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

(1) Well-Behaved Non-Nuclear

Dadarlat found a tracial non-nuclear A contained in an AF-algebra. It follows that any trace on A lifts through R; hence any trace lifts through any McDuff.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

(1) Well-Behaved Non-Nuclear

Dadarlat found a tracial non-nuclear A contained in an AF-algebra. It follows that any trace on A lifts through R; hence any trace lifts through any McDuff. Also, it can be shown that the traces remember their homomorphisms (in any McDuff).

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

(2) Too Many Traces

Brown found an algebra B such that for any McDuff M, there is a trace TM ∈ T(B) so that the von Neumann closure of the GNS representation πTM is isomorphic to M. That is, πTM(B)′′ ∼ = M.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

(2) Too Many Traces

Brown found an algebra B such that for any McDuff M, there is a trace TM ∈ T(B) so that the von Neumann closure of the GNS representation πTM is isomorphic to M. That is, πTM(B)′′ ∼ = M. A result by Ozawa demonstrates that there is no separable universal McDuff factor. So there cannot be an M such that every trace of B lifts through M.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

(3) Forgetful Trace

Finding an example of an algebra C with a forgetful trace would give legitimacy to the expectation that the convex sets Hom(C, M) form an invariant richer than the trace space.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

(3) Forgetful Trace

Finding an example of an algebra C with a forgetful trace would give legitimacy to the expectation that the convex sets Hom(C, M) form an invariant richer than the trace space. A result of Hadwin’s using the concept of dimension ratio (related to free entropy) gives a II1 factor N and two inequivalent homomorphisms π, ρ : C ∗

r (F2) → N such that

τN ◦ π = τN ◦ ρ.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

(3) Forgetful Trace

Finding an example of an algebra C with a forgetful trace would give legitimacy to the expectation that the convex sets Hom(C, M) form an invariant richer than the trace space. A result of Hadwin’s using the concept of dimension ratio (related to free entropy) gives a II1 factor N and two inequivalent homomorphisms π, ρ : C ∗

r (F2) → N such that

τN ◦ π = τN ◦ ρ. Alas, N is not necessarily McDuff.

  • S. Atkinson

Convex Sets Associated to C∗-Algebras

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Convex Sets Associated to C∗-Algebras

  • S. Atkinson

Introduction

Classical Situation 2011 Situation

Hom(A, M)

Preliminaries Extreme Points Trace Space Examples

(3) Forgetful Trace

Finding an example of an algebra C with a forgetful trace would give legitimacy to the expectation that the convex sets Hom(C, M) form an invariant richer than the trace space. A result of Hadwin’s using the concept of dimension ratio (related to free entropy) gives a II1 factor N and two inequivalent homomorphisms π, ρ : C ∗

r (F2) → N such that

τN ◦ π = τN ◦ ρ. Alas, N is not necessarily McDuff. Question: Is the inequivalence of π and ρ preserved when we pass to π ⊗ 1R, ρ ⊗ 1R : C ∗

r (F2) → N ⊗ R?

  • S. Atkinson

Convex Sets Associated to C∗-Algebras