Uniqueness of trace and C*-simplicity beyond regular representation

Abstract

A discrete group $\Gamma$ is C*-simple if the C*-algebra $C_{\lambda }^{\ast }(\Gamma )$ generated by the range of the left regular representation $\lambda$ on ${\ell }^2(\Gamma )$ is simple. In this case, $\Gamma$ acts faithfully on the Furstenberg boundary ${\partial }_F\Gamma$ and there is a unique trace on $C_{\lambda }^{\ast }(\Gamma )$. In this paper we study the unique trace property for the C*-algebra $C_{\pi }^{\ast }(\Gamma )$ generated by the range of an arbitrary unitary representation $\pi :\Gamma \to B(H_{\pi })$ and relate it to the faithfulness of the action of $\Gamma$ on the Furstenberg-Hamana boundary ${\mathcal{B}}_{\pi }$. Similar relation is obtained between simplicity of $C_{\pi }^{\ast }(\Gamma )$ and (topological) freeness of the action of $\Gamma$ on ${\mathcal{B}}_{\pi }$. Along the way, we extend the Connes-Sullivan and Powers averaging properties for a unitary representation $\pi$ and relate them to simplicity and unique trace property of $C_{\pi }^{\ast }(\Gamma )$.