Boundaries of reduced C*-algebras of discrete groups Matthew - PowerPoint PPT Presentation

Boundaries of reduced C*-algebras of discrete groups Matthew Kennedy (joint work with Mehrdad Kalantar) Carleton University, Ottawa, Canada June 23, 2014 1

Definition A discrete group G is amenable if there is a left-invariant mean i.e. a unital positive G -invariant linear map. In this case, is a unital positive G -equivariant projection. λ : ℓ ∞ ( G ) → C ,

Definition A discrete group G is amenable if there is a left-invariant mean i.e. a unital positive G -invariant linear map. λ : ℓ ∞ ( G ) → C , In this case, λ is a unital positive G -equivariant projection.

Reframed Definition A discrete group G is amenable if there is a unital positive G -equivariant projection Therefore, G is non-amenable if is “too small” to be the range of a unital positive G -equivariant projection on G . λ : ℓ ∞ ( G ) → C .

Reframed Definition A discrete group G is amenable if there is a unital positive G -equivariant projection λ : ℓ ∞ ( G ) → C . Therefore, G is non-amenable if C is “too small” to be the range of a unital positive G -equivariant projection on ℓ ∞ ( G ) .

G should somehow “measure” the non-amenability of G . Idea unital positive G -equivariant projection The size of Consider the minimal C*-subalgebra A G of ℓ ∞ ( G ) such that there is a P : ℓ ∞ ( G ) → A G .

Idea unital positive G -equivariant projection Consider the minimal C*-subalgebra A G of ℓ ∞ ( G ) such that there is a P : ℓ ∞ ( G ) → A G . The size of A G should somehow “measure” the non-amenability of G .

Theorem (Kalantar-K 2014) unital positive G-equivariant projection There is a unique minimal C*-algebra A G arising as the range of a P : ℓ ∞ ( G ) → A G . The algebra A G is isomorphic to the algebra C ( ∂ F G ) of continuous functions on the Furstenberg boundary ∂ F G of G .

Motivation

Kirchberg proved that every exact C*-algebra can be embedded into a nuclear C*-algebra. In the separable case, Kirchberg and Phillips proved the nuclear C*-algebra can be taken to be the Cuntz algebra on two generators.

Kirchberg proved that every exact C*-algebra can be embedded into a nuclear C*-algebra. In the separable case, Kirchberg and Phillips proved the nuclear C*-algebra can be taken to be the Cuntz algebra on two generators.

Ozawa conjectured the existence of what he calls a “tight” nuclear embedding. Conjecture (Ozawa 2007) The algebra will inherit many properties from , for example simplicity and primality. Let A be an exact C*-algebra. There is a canonical nuclear C*-algebra N ( A ) such that A ⊂ N ( A ) ⊂ I ( A ) , where I ( A ) denotes the injective envelope of A .

Ozawa conjectured the existence of what he calls a “tight” nuclear embedding. Conjecture (Ozawa 2007) simplicity and primality. Let A be an exact C*-algebra. There is a canonical nuclear C*-algebra N ( A ) such that A ⊂ N ( A ) ⊂ I ( A ) , where I ( A ) denotes the injective envelope of A . The algebra N ( A ) will inherit many properties from A , for example

n denote the reduced C*-algebra of n for n n . n is exact since n is an exact group. n Note that C r denotes the injective envelope of C r n where I C r n I C r Ozawa proved this conjecture for the reduced C*-algebra of the free N C r C r such that n canonical nuclear C*-algebra N C r . There is a Let C r Theorem (Ozawa 2007) n group F n on n ≥ 2 generators.

n is exact since n is an exact group. Note that C r Ozawa proved this conjecture for the reduced C*-algebra of the free Theorem (Ozawa 2007) group F n on n ≥ 2 generators. Let C ∗ r ( F n ) denote the reduced C*-algebra of F n for n ≥ 2 . There is a canonical nuclear C*-algebra N ( C ∗ r ( F n )) such that C ∗ r ( F n ) ⊂ N ( C ∗ r ( F n )) ⊂ I ( C ∗ r ( F n )) , where I ( C ∗ r ( F n )) denotes the injective envelope of C ∗ r ( F n ) .

Ozawa proved this conjecture for the reduced C*-algebra of the free Theorem (Ozawa 2007) group F n on n ≥ 2 generators. Let C ∗ r ( F n ) denote the reduced C*-algebra of F n for n ≥ 2 . There is a canonical nuclear C*-algebra N ( C ∗ r ( F n )) such that C ∗ r ( F n ) ⊂ N ( C ∗ r ( F n )) ⊂ I ( C ∗ r ( F n )) , where I ( C ∗ r ( F n )) denotes the injective envelope of C ∗ r ( F n ) . Note that C ∗ r ( F n ) is exact since F n is an exact group.

Key Proposition (Ozawa 2007) Let be a quasi-invariant doubly ergodic measure on G . If C n L G is a unital positive n -equivariant map, then id. Ozawa takes N ( C ∗ r ( F n )) = C ( ∂ F n ) ⋊ r F n , where ∂ F n denotes the hyperbolic boundary of F n .

Key Proposition (Ozawa 2007) Ozawa takes N ( C ∗ r ( F n )) = C ( ∂ F n ) ⋊ r F n , where ∂ F n denotes the hyperbolic boundary of F n . Let µ be a quasi-invariant doubly ergodic measure on ∂ G . If φ : C ( ∂ F n ) → L ∞ ( ∂ G , µ ) is a unital positive F n -equivariant map, then φ = id.

Equivariant Injective Envelopes

An operator system is a unital self-adjoint subspace of a C*-algebra. A G-operator system is an operator system equipped with the action of a group G , i.e. a unital homomorphism from G into the group of order isomorphisms on .

An operator system is a unital self-adjoint subspace of a C*-algebra. A G-operator system is an operator system equipped with the action of a group G , i.e. a unital homomorphism from G into the group of order isomorphisms on S .

When the objects are operator systems and the morphisms are unital completely positive maps, we get injectivity . When the objects are G -operator systems and the morphisms are G -equivariant unital completely positive maps, we get G-injectivity . Let C be a category consisting of objects and morphisms. An object I is injective in C if, for every pair of objects E ⊂ F and and every morphism φ : E → I , there is an extension ˜ φ : F → I .

When the objects are operator systems and the morphisms are unital completely positive maps, we get injectivity . When the objects are G -operator systems and the morphisms are G -equivariant unital completely positive maps, we get G-injectivity . Let C be a category consisting of objects and morphisms. An object I is injective in C if, for every pair of objects E ⊂ F and and every morphism φ : E → I , there is an extension ˜ φ : F → I .

When the objects are operator systems and the morphisms are unital completely positive maps, we get injectivity . When the objects are G -operator systems and the morphisms are G -equivariant unital completely positive maps, we get G-injectivity . Let C be a category consisting of objects and morphisms. An object I is injective in C if, for every pair of objects E ⊂ F and and every morphism φ : E → I , there is an extension ˜ φ : F → I .

The G-injective envelope of a G -operator system is the minimal G -injective operator system containing . The injective envelope of an operator system S is the minimal injective operator system containing S .

The injective envelope of an operator system S is the minimal injective operator system containing S . The G-injective envelope of a G -operator system S is the minimal G -injective operator system containing S .

Theorem (Hamana 1985) extends to a unital completely isometric G-equivariant embedding Since there is a unital completely isometric G -equivariant embedding of into G there are unital completely isometric G -equivariant embeddings I G G If S is a G-operator system, then S has a unique G-injective envelope I G ( S ) . Every unital completely isometric G-equivariant embedding φ : S → T , φ : I G ( S ) → T . ˜

Theorem (Hamana 1985) extends to a unital completely isometric G-equivariant embedding Since there is a unital completely isometric G -equivariant embedding embeddings If S is a G-operator system, then S has a unique G-injective envelope I G ( S ) . Every unital completely isometric G-equivariant embedding φ : S → T , φ : I G ( S ) → T . ˜ of S into ℓ ∞ ( G , S ) there are unital completely isometric G -equivariant S ⊂ I G ( S ) ⊂ ℓ ∞ ( G , S ) .

Upshot unital completely isometric G -equivariant embeddings The G -injective envelope I G has a natural C*-algebra structure (induced by the Choi-Effros product). If S is an operator system equipped with a G -action, then there are S ⊂ I G ( S ) ⊂ ℓ ∞ ( G , S ) , and a unital positive G -equivariant projection P : ℓ ∞ ( G , S ) → I G ( S ) .

Upshot unital completely isometric G -equivariant embeddings (induced by the Choi-Effros product). If S is an operator system equipped with a G -action, then there are S ⊂ I G ( S ) ⊂ ℓ ∞ ( G , S ) , and a unital positive G -equivariant projection P : ℓ ∞ ( G , S ) → I G ( S ) . The G -injective envelope I G ( S ) has a natural C*-algebra structure

Corollary and there is a unital positive G-equivariant projection The G -injective envelope I G is a commutative C*-algebra equipped with a G -action, so there is a compact G -space space H G such that I G C H G . We call H G the Hamana boundary of G . Let G be a discrete group acting trivially on C and let I G ( C ) denote the G-injective envelope of C . Then C ⊂ I G ( C ) ⊂ ℓ ∞ ( G ) , P : ℓ ∞ ( G ) → I G ( C ) .