Property (T) for locally compact groups and C*-algebras

Abstract

Let $G$ be a locally compact group and let $C^*(G)$ and $C^*_r(G)$ be the full group $C^*$-algebra and the reduced group $C^*$-algebra of $G$. We investigate the relationship between Property $(T)$ for $G$ and Property $(T)$ as well as its strong version for $C^*(G)$ and $C^*_r(G)$. We show that $G$ has Property $(T)$ if (and only if) $C^*(G)$ has Property $(T)$. In the case where $G$ is a locally compact IN-group, we prove that $G$ has Property $(T)$ if and only if $C^*_r(G)$ has strong Property $(T)$. We also show that $C^*_r(G)$ has strong Property $(T)$ for every non-amenable locally compact group $G$ for which $C^*_r(G)$ is nuclear. Some of these groups (as for instance $G=SL_2(\mathbf{R})$) do not have Property $T$.