Boundaries of reduced C*-algebras of discrete groups

Abstract

For a discrete group G, we consider the minimal C*-subalgebra of ${\ell }^{\infty }(G)$ that arises as the image of a unital positive G-equivariant projection. This algebra always exists and is unique up to isomorphism. It is trivial if and only if G is amenable. We prove that, more generally, it can be identified with the algebra $C({\partial }_{F}G)$ of continuous functions on Furstenberg's universal G-boundary ${\partial }_{F}G$. This operator-algebraic construction of the Furstenberg boundary has a number of interesting consequences. We prove that G is exact precisely when the G-action on ${\partial }_{F}G$ is amenable, and use this fact to prove Ozawa's conjecture that if G is exact, then there is an embedding of the reduced C*-algebra ${\mathrm{C}}_r^{\ast }(G)$ of G into a nuclear C*-algebra which is contained in the injective envelope of ${\mathrm{C}}_r^{\ast }(G)$. It is a longstanding open problem to determine which groups are C*-simple, in the sense that the algebra ${\mathrm{C}}_r^{\ast }(G)$ is simple. We prove that this problem can be reformulated as a problem about the structure of the G-action on the Furstenberg boundary. Specifically, we prove that a discrete group G is C*-simple if and only if the G-action on the Furstenberg boundary is topologically free. We apply this result to prove that Tarski monster groups are C*-simple. This provides another solution to a problem of de la Harpe (recently answered by Olshanskii and Osin) about the existence of C*-simple groups with no free subgroups.

arXiv comment26 pages; establishes equivalence of C*-simplicity and the existence of a topologically free boundary action

Cited by in corpus (18)

• Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems
• The Dixmier property and tracial states for C*-algebras
• Bernoulli disjointness
• Noncommutative Furstenberg boundary
• Some remarks about the weak containment property for groupoids and semigroups
• On C*-algebras of irreversible algebraic dynamical systems
• C*-simplicity of free products with amalgamation and radical classes of groups
• Minimal ambient nuclear C*-algebras
• $C^{\ast }$-simplicity and representations of topological full groups of groupoids
• Subgroup dynamics and $C^{\ast }$-simplicity of groups of homeomorphisms
• Non-amenable tight squeezes by Kirchberg algebras
• A generalized Powers averaging property for commutative crossed products
• Relative C*-simplicity and characterizations for normal subgroups
• C*-simplicity has no local obstruction
• Arboreal structures on groups and the associated boundaries
• $C^{\ast }$-simplicity and Ozawa conjecture for groupoid $C^{\ast }$-algebras, part I: injective envelopes
• Continuity of the stabilizer map and irreducible extensions
• Boundary rigidity for free product C*-algebras