Uniqueness of trace and C*-simplicity beyond regular representation

TL;DR

This paper explores the conditions under which C*-algebras generated by arbitrary unitary representations of a group have unique traces and are simple, extending known results beyond the regular representation case.

Contribution

It introduces new relations between the faithfulness and freeness of group actions on boundaries and the simplicity and unique trace properties of associated C*-algebras for arbitrary representations.

Findings

Established a link between faithfulness of group action on Furstenberg-Hamana boundary and unique trace property.

Connected topological freeness of the action to the simplicity of the C*-algebra.

Extended Connes-Sullivan and Powers averaging properties to arbitrary unitary representations.

Abstract

A discrete group $\Gamma$ is C*-simple if the C*-algebra $C_\lambda^*(\Gamma)$ 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 $C_\lambda^*(\Gamma)$. In this paper we study the unique trace property for the C*-algebra $C_\pi^*(\Gamma)$ 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 $C_\pi^*(\Gamma)$ 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 $C^*_\pi(\Gamma)$.