Boundaries of reduced C*-algebras of discrete groups

TL;DR

This paper explores the boundaries of reduced C*-algebras of discrete groups, linking algebraic properties to group actions on the Furstenberg boundary, and proves several conjectures and characterizations related to exactness and C*-simplicity.

Contribution

It introduces a new operator-algebraic perspective on the Furstenberg boundary, characterizes C*-simplicity via boundary actions, and proves the existence of C*-simple groups with no free subgroups.

Findings

The minimal G-invariant C*-subalgebra is isomorphic to C(∂_F G).

G is exact iff the action on ∂_F G is amenable.

Tarski monster groups are C*-simple.

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 $\mathrm{C}_r^*(G)$ of G into a nuclear C*-algebra which is contained in the injective envelope of $\mathrm{C}_r^*(G)$. It is a longstanding open problem to determine which groups are C*-simple, in the sense that the algebra $\mathrm{C}_r^*(G)$ 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.