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

Abstract

Let $G$ be a locally compact group and let $C^{\ast }(G)$ and $C_r^{\ast }(G)$ be the full group $C^{\ast }$-algebra and the reduced group $C^{\ast }$-algebra of $G$. We investigate the relationship between Property $(T)$ for $G$ and Property $(T)$ as well as its strong version for $C^{\ast }(G)$ and $C_r^{\ast }(G)$. We show that $G$ has Property $(T)$ if (and only if) $C^{\ast }(G)$ 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 $C_r^{\ast }(G)$ has strong Property $(T)$. We also show that $C_r^{\ast }(G)$ has strong Property $(T)$ for every non-amenable locally compact group $G$ for which $C_r^{\ast }(G)$ is nuclear. Some of these groups (as for instance $G=S{L}_2(\mathbf{R})$) do not have Property $T$.

9 pages