Finite group actions and the UCT problem Workshop on Model Theory - PowerPoint PPT Presentation

Finite group actions and the UCT problem Workshop on Model Theory and Operator Algebras G´ abor Szab´ o (joint work with Sel¸ cuk Barlak) WWU M¨ unster July 2014 1 / 23

Introduction 1 Rokhlin actions on UHF-absorbing C ∗ -algebras 2 Some examples 3 Finite group actions on O 2 and the UCT 4 2 / 23

Introduction Introduction 1 Rokhlin actions on UHF-absorbing C ∗ -algebras 2 Some examples 3 Finite group actions on O 2 and the UCT 4 3 / 23

Introduction Unless specified otherwise, we will stick to the following notation throughout this talk: G is a finite group. A is a separable, unital C ∗ -algebra. α, β or γ are finite group actions on such a C ∗ -algebra. 4 / 23

Introduction Unless specified otherwise, we will stick to the following notation throughout this talk: G is a finite group. A is a separable, unital C ∗ -algebra. α, β or γ are finite group actions on such a C ∗ -algebra. Definition (Izumi) Let α : G � A be given, and let ω ∈ β N \ N be a free ultrafilter. Then α has the Rokhlin property, if there exists a unital, equivariant ∗ -homomorphism → ( A ω ∩ A ′ , α ω ) . ( C ( G ) , G -shift ) ֒ − We also call such α a Rokhlin action. 4 / 23

Introduction Why is this property interesting? 5 / 23

Introduction Why is this property interesting? To give one reason (among many), the crossed products by actions with the Rokhlin property are comparably easy to determine. 5 / 23

Introduction Why is this property interesting? To give one reason (among many), the crossed products by actions with the Rokhlin property are comparably easy to determine. Theorem (Izumi) Let A be simple, G a finite group and α : G � A a Rokhlin action. Then K ∗ ( A ⋊ α G ) is isomorphic to the subgroup � g ∈ G ker(id − K ∗ ( α g )) inside K ∗ ( A ) . 5 / 23

Introduction Why is this property interesting? To give one reason (among many), the crossed products by actions with the Rokhlin property are comparably easy to determine. Theorem (Izumi) Let A be simple, G a finite group and α : G � A a Rokhlin action. Then K ∗ ( A ⋊ α G ) is isomorphic to the subgroup � g ∈ G ker(id − K ∗ ( α g )) inside K ∗ ( A ) . For example, if A belongs to a certain class of C ∗ -algebras classified by K -theory, then (often) so does A ⋊ α G and this helps to determine its isomorphism class. 5 / 23

Introduction Why is this property interesting? To give one reason (among many), the crossed products by actions with the Rokhlin property are comparably easy to determine. Theorem (Izumi) Let A be simple, G a finite group and α : G � A a Rokhlin action. Then K ∗ ( A ⋊ α G ) is isomorphic to the subgroup � g ∈ G ker(id − K ∗ ( α g )) inside K ∗ ( A ) . For example, if A belongs to a certain class of C ∗ -algebras classified by K -theory, then (often) so does A ⋊ α G and this helps to determine its isomorphism class. Theorem (Barlak-S) Let A be given, G a finite group and α : G � A a Rokhlin action. Assume moreover that A ∼ = M | G | ∞ ⊗ A . Then A ⋊ α G decomposes as a direct limit of matrix algebras over A , with connecting maps depending only on α . 5 / 23

Introduction Unfortunately, Rokhlin actions are not always prevalent. Example The Cuntz algebra O ∞ and the Jiang-Su algebra Z admit no finite group actions with the Rokhlin property. 6 / 23

Introduction Unfortunately, Rokhlin actions are not always prevalent. Example The Cuntz algebra O ∞ and the Jiang-Su algebra Z admit no finite group actions with the Rokhlin property. However, there are certain canonical examples. Notation Let G be a finite group. The matrix algebra M | G | is generated by elements { e g,h } g,h ∈ G satisfying the relations e h 1 ,h 2 · e h 3 ,h 4 = δ h 2 ,h 3 e h 1 ,h 4 . One denotes � � � M ⊗ n M | G | ∞ = N M | G | = lim | G | , [ x �→ x ⊗ 1 | G | ] . − → 6 / 23

Introduction Unfortunately, Rokhlin actions are not always prevalent. Example The Cuntz algebra O ∞ and the Jiang-Su algebra Z admit no finite group actions with the Rokhlin property. However, there are certain canonical examples. Notation Let G be a finite group. The matrix algebra M | G | is generated by elements { e g,h } g,h ∈ G satisfying the relations e h 1 ,h 2 · e h 3 ,h 4 = δ h 2 ,h 3 e h 1 ,h 4 . One denotes � � � M ⊗ n M | G | ∞ = N M | G | = lim | G | , [ x �→ x ⊗ 1 | G | ] . − → Example Consider the left-regular representation λ : G → U ( M | G | ) defined by λ ( g ) = � h ∈ G e gh,h . One obtains an induced Rokhlin action β G : G � M | G | ∞ by β G g = � N Ad( λ ( g )) for all g ∈ G . 6 / 23

Rokhlin actions on UHF-absorbing C ∗ -algebras Introduction 1 Rokhlin actions on UHF-absorbing C ∗ -algebras 2 Some examples 3 Finite group actions on O 2 and the UCT 4 7 / 23

Rokhlin actions on UHF-absorbing C ∗ -algebras Fact If A ∼ = M | G | ∞ ⊗ A , then the canonical embedding A ֒ − → M | G | ∞ ⊗ A given by x �→ 1 ⊗ x is approximately unitarily equivalent to an isomorphism. 8 / 23

Rokhlin actions on UHF-absorbing C ∗ -algebras Fact If A ∼ = M | G | ∞ ⊗ A , then the canonical embedding A ֒ − → M | G | ∞ ⊗ A given by x �→ 1 ⊗ x is approximately unitarily equivalent to an isomorphism. Example Let us assume that A ∼ = M | G | ∞ ⊗ A . Let α : G � A be any action. Then β G ⊗ α is an action with the Rokhlin property on M | G | ∞ ⊗ A . Identifying this with A in the above way, this yields a Rokhlin action on A that is pointwise approximately unitarily equivalent to α . 8 / 23

Rokhlin actions on UHF-absorbing C ∗ -algebras Fact If A ∼ = M | G | ∞ ⊗ A , then the canonical embedding A ֒ − → M | G | ∞ ⊗ A given by x �→ 1 ⊗ x is approximately unitarily equivalent to an isomorphism. Example Let us assume that A ∼ = M | G | ∞ ⊗ A . Let α : G � A be any action. Then β G ⊗ α is an action with the Rokhlin property on M | G | ∞ ⊗ A . Identifying this with A in the above way, this yields a Rokhlin action on A that is pointwise approximately unitarily equivalent to α . This seems to suggest that on M | G | ∞ -absorbing C ∗ -algebras, there should be plenty of G -actions with the Rokhlin property, in particular with all kinds of K -theories. 8 / 23

Rokhlin actions on UHF-absorbing C ∗ -algebras Fact If A ∼ = M | G | ∞ ⊗ A , then the canonical embedding A ֒ − → M | G | ∞ ⊗ A given by x �→ 1 ⊗ x is approximately unitarily equivalent to an isomorphism. Example Let us assume that A ∼ = M | G | ∞ ⊗ A . Let α : G � A be any action. Then β G ⊗ α is an action with the Rokhlin property on M | G | ∞ ⊗ A . Identifying this with A in the above way, this yields a Rokhlin action on A that is pointwise approximately unitarily equivalent to α . This seems to suggest that on M | G | ∞ -absorbing C ∗ -algebras, there should be plenty of G -actions with the Rokhlin property, in particular with all kinds of K -theories. However, it is in general not at all clear how many ordinary G -actions exist on a given C ∗ -algebra A , even if one assumes that A is classifiable. 8 / 23

Rokhlin actions on UHF-absorbing C ∗ -algebras Reminder For a finite group action α : G � A , the crossed product A ⋊ α G is defined as the universal C ∗ -algebra generated by a copy of A , and a unitary representation g �→ u g subject to the relations u g au ∗ g = α g ( a ) for all a ∈ A . 9 / 23

Rokhlin actions on UHF-absorbing C ∗ -algebras Reminder For a finite group action α : G � A , the crossed product A ⋊ α G is defined as the universal C ∗ -algebra generated by a copy of A , and a unitary representation g �→ u g subject to the relations u g au ∗ g = α g ( a ) for all a ∈ A . Reminder Let us consider the special case G = Z p for some p ≥ 2 . Set ξ p = exp(2 πi/p ) ∈ C . Then a group action α : Z p � A naturally gives rise to the so-called dual action ˆ α : Z p � A ⋊ α G by setting α ( u ) = ξ p u ˆ and α ( a ) = a ˆ for all a ∈ A. 9 / 23

Rokhlin actions on UHF-absorbing C ∗ -algebras Reminder For a finite group action α : G � A , the crossed product A ⋊ α G is defined as the universal C ∗ -algebra generated by a copy of A , and a unitary representation g �→ u g subject to the relations u g au ∗ g = α g ( a ) for all a ∈ A . Reminder Let us consider the special case G = Z p for some p ≥ 2 . Set ξ p = exp(2 πi/p ) ∈ C . Then a group action α : Z p � A naturally gives rise to the so-called dual action ˆ α : Z p � A ⋊ α G by setting α ( u ) = ξ p u ˆ and α ( a ) = a ˆ for all a ∈ A. Theorem (Takai-duality) α Z p ∼ One always has ( A ⋊ α Z p ) ⋊ ˆ = M p ⊗ A . 9 / 23