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

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 Virginia ECOAS 2014 S. Atkinson Convex Sets Associated to C ∗ -Algebras

Classical Situation (1970’s): Ext( A ) Convex Sets Associated to Let A be a separable unital C ∗ -algebra. Ext( A ) is given by the C ∗ -Algebras set of unital ∗ -monomorphisms π : A → B ( H ) / K ( H ) modulo S. Atkinson B ( H )-unitary equivalence. Introduction Classical Situation 2011 Situation H om ( A , M ) Preliminaries Extreme Points Trace Space Examples S. Atkinson Convex Sets Associated to C ∗ -Algebras

Classical Situation (1970’s): Ext( A ) Convex Sets Associated to Let A be a separable unital C ∗ -algebra. Ext( A ) is given by the C ∗ -Algebras set of unital ∗ -monomorphisms π : A → B ( H ) / K ( H ) modulo S. Atkinson B ( H )-unitary equivalence. Introduction Classical Situation 2011 Situation Use a unitarily implemented isomorphism between B ( H ) and H om ( A , M ) M 2 ( B ( H )) to define a semigroup structure on Ext( A ). Preliminaries Extreme Points Trace Space Examples S. Atkinson Convex Sets Associated to C ∗ -Algebras

Classical Situation (1970’s): Ext( A ) Convex Sets Associated to Let A be a separable unital C ∗ -algebra. Ext( A ) is given by the C ∗ -Algebras set of unital ∗ -monomorphisms π : A → B ( H ) / K ( H ) modulo S. Atkinson B ( H )-unitary equivalence. Introduction Classical Situation 2011 Situation Use a unitarily implemented isomorphism between B ( H ) and H om ( A , M ) M 2 ( B ( H )) to define a semigroup structure on Ext( A ). Preliminaries Extreme Points Trace Space Examples Here is the picture: �� π �� 0 [ π ] + [ ρ ] = 0 ρ S. Atkinson Convex Sets Associated to C ∗ -Algebras

2011 Situation: H om( N , R U ) Convex Sets Associated to C ∗ -Algebras In 2011 Brown introduced the following convex set. S. Atkinson Introduction Classical Situation 2011 Situation H om ( A , M ) Preliminaries Extreme Points Trace Space Examples S. Atkinson Convex Sets Associated to C ∗ -Algebras

2011 Situation: H om( N , R U ) Convex Sets Associated to C ∗ -Algebras In 2011 Brown introduced the following convex set. S. Atkinson Introduction Classical Situation For N a separable II 1 -factor, R the hyperfinite II 1 -factor, and U 2011 Situation a free ultrafilter on N define H om( N , R U ) to be the set of unital H om ( A , M ) ∗ -homomorphisms π : N → R U modulo unitary equivalence. Preliminaries Extreme Points Trace Space Examples S. Atkinson Convex Sets Associated to C ∗ -Algebras

2011 Situation: H om( N , R U ) Convex Sets Associated to C ∗ -Algebras In 2011 Brown introduced the following convex set. S. Atkinson Introduction Classical Situation For N a separable II 1 -factor, R the hyperfinite II 1 -factor, and U 2011 Situation a free ultrafilter on N define H om( N , R U ) to be the set of unital H om ( A , M ) ∗ -homomorphisms π : N → R U modulo unitary equivalence. Preliminaries Extreme Points Trace Space Examples We use isomorphisms between R U and pR U p for p a projection in R U to define convex combinations. S. Atkinson Convex Sets Associated to C ∗ -Algebras

2011 Situation: H om( N , R U ) Convex Sets Here is a(n incorrect) picture: Associated to C ∗ -Algebras S. Atkinson Introduction Classical Situation 2011 Situation H om ( A , M ) Preliminaries Extreme Points Trace Space Examples S. Atkinson Convex Sets Associated to C ∗ -Algebras

2011 Situation: H om( N , R U ) Convex Sets Here is a(n incorrect) picture: Associated to C ∗ -Algebras �� p π p S. Atkinson 0 �� t [ π ] + (1 − t )[ ρ ] = Introduction p ⊥ ρ p ⊥ 0 Classical Situation 2011 Situation where p is a projection in R U and τ R ( p ) = t . H om ( A , M ) Preliminaries Extreme Points Trace Space Examples S. Atkinson Convex Sets Associated to C ∗ -Algebras

2011 Situation: H om( N , R U ) Convex Sets Here is a(n incorrect) picture: Associated to C ∗ -Algebras �� p π p S. Atkinson 0 �� t [ π ] + (1 − t )[ ρ ] = Introduction p ⊥ ρ p ⊥ 0 Classical Situation 2011 Situation where p is a projection in R U and τ R ( p ) = t . H om ( A , M ) Preliminaries With this definition, we may consider H om( N , R U ) as a closed, Extreme Points Trace Space Examples bounded, convex subset of a Banach space. S. Atkinson Convex Sets Associated to C ∗ -Algebras

2011 Situation: H om( N , R U ) Convex Sets Here is a(n incorrect) picture: Associated to C ∗ -Algebras �� p π p S. Atkinson 0 �� t [ π ] + (1 − t )[ ρ ] = Introduction p ⊥ ρ p ⊥ 0 Classical Situation 2011 Situation where p is a projection in R U and τ R ( p ) = t . H om ( A , M ) Preliminaries With this definition, we may consider H om( N , R U ) as a closed, Extreme Points Trace Space Examples bounded, convex subset of a Banach space. Brown was able to characterize extreme points: Theorem (Brown, 2011) [ π ] ∈ H om ( N , R U ) is extreme if and only if π ( N ) ′ ∩ R U is a factor. S. Atkinson Convex Sets Associated to C ∗ -Algebras

2011 Situation: H om( N , R U ) Convex Sets Here is a(n incorrect) picture: Associated to C ∗ -Algebras �� p π p S. Atkinson 0 �� t [ π ] + (1 − t )[ ρ ] = Introduction p ⊥ ρ p ⊥ 0 Classical Situation 2011 Situation where p is a projection in R U and τ R ( p ) = t . H om ( A , M ) Preliminaries With this definition, we may consider H om( N , R U ) as a closed, Extreme Points Trace Space Examples bounded, convex subset of a Banach space. Brown was able to characterize extreme points: Theorem (Brown, 2011) [ π ] ∈ H om ( N , R U ) is extreme if and only if π ( N ) ′ ∩ R U is a factor. S. Atkinson Convex Sets Associated to C ∗ -Algebras

Preliminaries Convex Sets Definition Associated to C ∗ -Algebras For a separable, unital, tracial C ∗ -algebra A , and a separable S. Atkinson McDuff II 1 -factor M ( M ∼ = M ⊗ R ), we define H om( A , M ) to Introduction be the space of unital ∗ -homomorphisms π : A → M modulo Classical Situation the equivalence relation of weak approximate unitary 2011 Situation equivalence (w.a.u.e.). H om ( A , M ) Preliminaries Extreme Points Trace Space Examples S. Atkinson Convex Sets Associated to C ∗ -Algebras

Preliminaries Convex Sets Definition Associated to C ∗ -Algebras For a separable, unital, tracial C ∗ -algebra A , and a separable S. Atkinson McDuff II 1 -factor M ( M ∼ = M ⊗ R ), we define H om( A , M ) to Introduction be the space of unital ∗ -homomorphisms π : A → M modulo Classical Situation the equivalence relation of weak approximate unitary 2011 Situation equivalence (w.a.u.e.). H om ( A , M ) Preliminaries Extreme Points Trace Space That is, [ π ] = [ ρ ] if there is a sequence { u n } of unitaries in M Examples such that for every a ∈ A we have n || π ( a ) − u n ρ ( a ) u ∗ lim n || 2 = 0 . S. Atkinson Convex Sets Associated to C ∗ -Algebras

Preliminaries Convex Sets Definition Associated to C ∗ -Algebras For a separable, unital, tracial C ∗ -algebra A , and a separable S. Atkinson McDuff II 1 -factor M ( M ∼ = M ⊗ R ), we define H om( A , M ) to Introduction be the space of unital ∗ -homomorphisms π : A → M modulo Classical Situation the equivalence relation of weak approximate unitary 2011 Situation equivalence (w.a.u.e.). H om ( A , M ) Preliminaries Extreme Points Trace Space That is, [ π ] = [ ρ ] if there is a sequence { u n } of unitaries in M Examples such that for every a ∈ A we have n || π ( a ) − u n ρ ( a ) u ∗ lim n || 2 = 0 . We endow H om( A , M ) with the topology of pointwise convergence (with appropriate consideration for equivalence classes). S. Atkinson Convex Sets Associated to C ∗ -Algebras

Convex Structure Convex Sets Associated to C ∗ -Algebras Taking advantage of the properties of a McDuff factor S. Atkinson ( M ∼ = M ⊗ R ), we can define convex combinations in Introduction H om( A , M ). Classical Situation 2011 Situation H om ( A , M ) Preliminaries Extreme Points Trace Space Examples S. Atkinson Convex Sets Associated to C ∗ -Algebras

Convex Structure Convex Sets Associated to C ∗ -Algebras Taking advantage of the properties of a McDuff factor S. Atkinson ( M ∼ = M ⊗ R ), we can define convex combinations in Introduction H om( A , M ). Classical Situation 2011 Situation H om ( A , M ) Preliminaries Extreme Points Definition Trace Space Examples For a McDuff factor M , an isomorphism σ M : M ⊗ R → M is a regular isomorphism if σ M ◦ (id M ⊗ 1 R ) ∼ id M . S. Atkinson Convex Sets Associated to C ∗ -Algebras

Convex Structure Convex Sets Associated to C ∗ -Algebras Definition S. Atkinson For t ∈ [0 , 1] , [ π ] , [ ρ ] ∈ H om( A , M ), we define Introduction t [ π ] + (1 − t )[ ρ ] := [ σ M ( π ⊗ p + ρ ⊗ p ⊥ )] Classical Situation 2011 Situation H om ( A , M ) where σ M : M ⊗ R → M is a regular isomorphism and p is a Preliminaries Extreme Points projection in R with τ R ( p ) = t . Trace Space Examples S. Atkinson Convex Sets Associated to C ∗ -Algebras