Existence of tracial states on reduced group C*-algebras

Abstract

Let $G$ be a locally compact group. It is not always the case that its reduced C*-algebra $C_r^{\ast }(G)$ admits a tracial state. We exhibit closely related necessary and sufficient conditions for the existence of such. We gain a complete answer when $G$ compactly generated. In particular for $G$ almost connected, or more generally when $C_r^{\ast }(G)$ is nuclear, the existence of a trace is equivalent to amenability. We exhibit two examples of classes of totally disconnected groups for which $C_r^{\ast }(G)$ does not admit a tracial state.

The proof of Remark 2.6 (ii) has been corrected, and a typo has been fixed