Property T of reduced C∗-crossed products by discrete groups

Abstract

We generalize a result of Kamalov and we show that if $G$ is an amenable discrete group with an action $\alpha$ on a finite nuclear unital $C^{\ast }$-algebra $A$ such that the reduced crossed product $A{⋊}_{\alpha ,r}G$ has property $T$, then $G$ is finite and $A$ is finite-dimensional. As an application, an infinite discrete group $H$ is nonamenable if and only if the corresponding uniform Roe algebra $C_u^{\ast }\left(H\right)$ has property $T$.