Examples of non-amenable, boundary-amenable dynamical systems

TL;DR

This paper proves that certain boundary actions of free groups are topologically amenable, leading to unique ideal structures in associated C*-algebras, using advanced dynamical boundary analysis.

Contribution

It establishes topological amenability of boundary actions for free groups with specific parameters, extending understanding of their C*-algebraic properties.

Findings

Boundary actions of free groups are topologically amenable.

The associated C*-algebras have a unique non-trivial ideal.

The study advances dynamical boundary analysis techniques.

Abstract

It is proved that for every $r,s \in \mathbb{N}\backslash \{0,1\}$ the action of $\mathbb{F}_{r+s}$ on $\partial_\beta (\mathbb{F}_{r+s}/\mathbb{F}_r)$ is topologically amenable. In particular the $C^*$-algebra associated to the corresponding quasi-regular representation has a unique non-trivial ideal. The techniques involved rely on a study of dynamical properties for actions on non-standard boundaries studied by the author and F. R{\u a}dulescu in previous works.