Uniqueness of trace and C*-simplicity beyond regular representation

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