A property (T) for C*-algebras

Abstract

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

arXiv comment14 pages

Cited by in corpus (1)

• Kazhdan's Property T and C*-algebras