Supramenable groups and their actions on locally compact Hausdorff - PowerPoint PPT Presentation

Dynamical systems and crossed product C ∗ -algebras Paradoxical sets - Tarski’s theorem Supramenable groups Supramenable groups and their actions on locally compact Hausdorff spaces Mikael Rørdam rordam@math.ku.dk Department of Mathematics University of Copenhagen Workshop on C ∗ -algebras and Noncommutative Dynamics, Sde Boker, Israel, March 11, 2013 ( Joint with Julian Kellerhals and Nicolas Monod ) nat-logo

Dynamical systems and crossed product C ∗ -algebras Paradoxical sets - Tarski’s theorem Supramenable groups Outline Dynamical systems and crossed product C ∗ -algebras 1 Paradoxical sets - Tarski’s theorem 2 Supramenable groups 3 nat-logo

Dynamical systems and crossed product C ∗ -algebras Paradoxical sets - Tarski’s theorem Supramenable groups X = compact or locally compact Hausdorff space, Γ = (countable) discrete group which acts on X . C 0 ( X ) ⋊ red Γ is nuclear ⇐ ⇒ Γ � X amenable [Anantharaman-Delaroche] Tracial states on C 0 ( X ) ⋊ red Γ ↔ Γ-invariant probability measures on X C 0 ( X ) ⋊ red Γ is simple ⇐ Γ � X is topologically free and minimal . [Archbold–Spielberg] Definition Γ � X is topologically free if ∀ t ∈ Γ \ { e } : { x ∈ X | t . x � = x } is dense in X . Note: C ∗ red ( F n ) is simple, but F n � { pt } is not topologically free (or free). nat-logo

Dynamical systems and crossed product C ∗ -algebras Paradoxical sets - Tarski’s theorem Supramenable groups X = compact or locally compact Hausdorff space, Γ = (countable) discrete group which acts on X . C 0 ( X ) ⋊ red Γ is nuclear ⇐ ⇒ Γ � X amenable [Anantharaman-Delaroche] Tracial states on C 0 ( X ) ⋊ red Γ ↔ Γ-invariant probability measures on X C 0 ( X ) ⋊ red Γ is simple ⇐ Γ � X is topologically free and minimal . [Archbold–Spielberg] Definition Γ � X is topologically free if ∀ t ∈ Γ \ { e } : { x ∈ X | t . x � = x } is dense in X . Note: C ∗ red ( F n ) is simple, but F n � { pt } is not topologically free (or free). nat-logo

Dynamical systems and crossed product C ∗ -algebras Paradoxical sets - Tarski’s theorem Supramenable groups Definition (Anantharaman-Delaroche) An action of a countable discrete group Γ on a locally compact space X is said to be amenable if there is a net of continuous maps m i : X → Prob (Γ) (written x �→ m x i ) such that � t . m x i − m t . x � 1 → 0 i uniformly on all compact subsets of X and for all t ∈ Γ. If Γ is amenable, then we can choose the m i ’s above being constant. Hence any action of an amenable group is amenable. If X = { pt } , then Γ acts amenably on X iff Γ is amenable. Any proper action is amenable. If X has an invariant probability measure, then Γ � X is amenable if and only if Γ is amenable. nat-logo

Dynamical systems and crossed product C ∗ -algebras Paradoxical sets - Tarski’s theorem Supramenable groups Definition (Anantharaman-Delaroche) An action of a countable discrete group Γ on a locally compact space X is said to be amenable if there is a net of continuous maps m i : X → Prob (Γ) (written x �→ m x i ) such that � t . m x i − m t . x � 1 → 0 i uniformly on all compact subsets of X and for all t ∈ Γ. If Γ is amenable, then we can choose the m i ’s above being constant. Hence any action of an amenable group is amenable. If X = { pt } , then Γ acts amenably on X iff Γ is amenable. Any proper action is amenable. If X has an invariant probability measure, then Γ � X is amenable if and only if Γ is amenable. nat-logo

Dynamical systems and crossed product C ∗ -algebras Paradoxical sets - Tarski’s theorem Supramenable groups Definition: Γ � X is regular if C 0 ( X ) ⋊ full Γ = C 0 ( X ) ⋊ red Γ. Anantharaman-Delaroche proved the following: ⇒ Γ � X amenable = Γ � X regular . Γ � X amenable ⇐ ⇒ C 0 ( X ) ⋊ Γ is nuclear. Jean-Louis Tu proved: Γ � X amenable = ⇒ C 0 ( X ) ⋊ Γ is in the UCT class. Example (The Roe algebra) The Roe algebra associated with a discrete group Γ is the crossed product: ℓ ∞ (Γ) ⋊ red Γ = C ( β Γ) ⋊ red Γ where Γ acts on ℓ ∞ (Γ) by left translation. The left multiplication action Γ � Γ extends (uniquely) to an action Γ � β Γ. Γ � β Γ is amenable ⇐ ⇒ Γ is exact. [Ozawa] Γ � β Γ is free for all Γ. nat-logo Γ � β Γ is not minimal (unless Γ is finite).

Dynamical systems and crossed product C ∗ -algebras Paradoxical sets - Tarski’s theorem Supramenable groups Definition: Γ � X is regular if C 0 ( X ) ⋊ full Γ = C 0 ( X ) ⋊ red Γ. Anantharaman-Delaroche proved the following: ⇒ Γ � X amenable = Γ � X regular . Γ � X amenable ⇐ ⇒ C 0 ( X ) ⋊ Γ is nuclear. Jean-Louis Tu proved: Γ � X amenable = ⇒ C 0 ( X ) ⋊ Γ is in the UCT class. Example (The Roe algebra) The Roe algebra associated with a discrete group Γ is the crossed product: ℓ ∞ (Γ) ⋊ red Γ = C ( β Γ) ⋊ red Γ where Γ acts on ℓ ∞ (Γ) by left translation. The left multiplication action Γ � Γ extends (uniquely) to an action Γ � β Γ. Γ � β Γ is amenable ⇐ ⇒ Γ is exact. [Ozawa] Γ � β Γ is free for all Γ. nat-logo Γ � β Γ is not minimal (unless Γ is finite).

Dynamical systems and crossed product C ∗ -algebras Paradoxical sets - Tarski’s theorem Supramenable groups Definition: Γ � X is regular if C 0 ( X ) ⋊ full Γ = C 0 ( X ) ⋊ red Γ. Anantharaman-Delaroche proved the following: ⇒ Γ � X amenable = Γ � X regular . Γ � X amenable ⇐ ⇒ C 0 ( X ) ⋊ Γ is nuclear. Jean-Louis Tu proved: Γ � X amenable = ⇒ C 0 ( X ) ⋊ Γ is in the UCT class. Example (The Roe algebra) The Roe algebra associated with a discrete group Γ is the crossed product: ℓ ∞ (Γ) ⋊ red Γ = C ( β Γ) ⋊ red Γ where Γ acts on ℓ ∞ (Γ) by left translation. The left multiplication action Γ � Γ extends (uniquely) to an action Γ � β Γ. Γ � β Γ is amenable ⇐ ⇒ Γ is exact. [Ozawa] Γ � β Γ is free for all Γ. nat-logo Γ � β Γ is not minimal (unless Γ is finite).

Dynamical systems and crossed product C ∗ -algebras Paradoxical sets - Tarski’s theorem Supramenable groups Recall: Γ � X is regular if C 0 ( X ) ⋊ full Γ = C 0 ( X ) ⋊ red Γ, Γ � X amenable = ⇒ Γ � X regular. Γ � X amenable ⇐ ⇒ C 0 ( X ) ⋊ Γ is nuclear. Proposition (Archbold-Spielberg, 1993) C 0 ( X ) ⋊ full Γ is simple ⇐ ⇒ Γ � X is minimal, topologically free and regular. Corollary (Anantharaman-Delaroche + Archbold-Spielberg) C 0 ( X ) ⋊ red Γ is simple and nuclear ⇐ ⇒ Γ � X is minimal, topologically free and amenable. nat-logo

Dynamical systems and crossed product C ∗ -algebras Paradoxical sets - Tarski’s theorem Supramenable groups Recall: Γ � X is regular if C 0 ( X ) ⋊ full Γ = C 0 ( X ) ⋊ red Γ, Γ � X amenable = ⇒ Γ � X regular. Γ � X amenable ⇐ ⇒ C 0 ( X ) ⋊ Γ is nuclear. Proposition (Archbold-Spielberg, 1993) C 0 ( X ) ⋊ full Γ is simple ⇐ ⇒ Γ � X is minimal, topologically free and regular. Corollary (Anantharaman-Delaroche + Archbold-Spielberg) C 0 ( X ) ⋊ red Γ is simple and nuclear ⇐ ⇒ Γ � X is minimal, topologically free and amenable. nat-logo

Dynamical systems and crossed product C ∗ -algebras Paradoxical sets - Tarski’s theorem Supramenable groups Recall: Γ � X is regular if C 0 ( X ) ⋊ full Γ = C 0 ( X ) ⋊ red Γ, Γ � X amenable = ⇒ Γ � X regular. Γ � X amenable ⇐ ⇒ C 0 ( X ) ⋊ Γ is nuclear. Proposition (Archbold-Spielberg, 1993) C 0 ( X ) ⋊ full Γ is simple ⇐ ⇒ Γ � X is minimal, topologically free and regular. Corollary (Anantharaman-Delaroche + Archbold-Spielberg) C 0 ( X ) ⋊ red Γ is simple and nuclear ⇐ ⇒ Γ � X is minimal, topologically free and amenable. nat-logo

Dynamical systems and crossed product C ∗ -algebras Paradoxical sets - Tarski’s theorem Supramenable groups Consider now the unital case, i.e., X compact. ( C ∗ • C ( X ) ⋊ red Γ nuclear = ⇒ Γ exact red (Γ) ⊆ C ( X ) ⋊ red Γ). Proposition (Anantharaman-Delaroche) Suppose C ( X ) ⋊ red Γ is simple and nuclear. Then: Γ amenable ⇐ ⇒ C ( X ) ⋊ red Γ is stably finite. Γ non-amenable ⇐ ⇒ ∃ n : M n ( C ( X ) ⋊ red Γ) is properly infinite. Question Suppose C ( X ) ⋊ red Γ simple and nuclear, and Γ non-amenable. Is C ( X ) ⋊ red Γ always properly infinite? Is C ( X ) ⋊ red Γ always purely infinite? nat-logo

Dynamical systems and crossed product C ∗ -algebras Paradoxical sets - Tarski’s theorem Supramenable groups Consider now the unital case, i.e., X compact. ( C ∗ • C ( X ) ⋊ red Γ nuclear = ⇒ Γ exact red (Γ) ⊆ C ( X ) ⋊ red Γ). Proposition (Anantharaman-Delaroche) Suppose C ( X ) ⋊ red Γ is simple and nuclear. Then: Γ amenable ⇐ ⇒ C ( X ) ⋊ red Γ is stably finite. Γ non-amenable ⇐ ⇒ ∃ n : M n ( C ( X ) ⋊ red Γ) is properly infinite. Question Suppose C ( X ) ⋊ red Γ simple and nuclear, and Γ non-amenable. Is C ( X ) ⋊ red Γ always properly infinite? Is C ( X ) ⋊ red Γ always purely infinite? nat-logo