A property (T) for C*-algebras

Abstract

We define a notion of Property (T) for an arbitrary $C^*$-algebra $A$ admitting a tracial state. We extend this to a notion of Property (T) for the pair $(A,B),$ where $B$ is a $C^*$-subalgebra of $A.$ Let $Γ$ be a discrete group and $C^*_r(Γ)$ its reduced algebra. We show that $C^*_r(Γ)$ has Property (T) if and only if the group $Γ$ has Property (T) . More generally, given a subgroup $Λ$ of $Γ$, the pair $(C^*_r(Γ),C^*_r(Λ))$ has Property (T) if and only if the pair of groups $(Γ, Λ)$ has Property (T).