Property T** for C*-algebras

Abstract

Extending the notion of property T of finite von Neumann algebras to general von Neumann algebras, we define and study in this paper property T** for (possibly non-unital) C*-algebras. We obtain several results of property T** parallel to those of property T for unital C*-algebras. Moreover, we show that a discrete group Γ has property T if and only if the group C*-algebra C*(Γ) (or equivalently, the reduced group C*-algebra Cr* (Γ)) has property T**. We also show that the compact operators K(l2) has property T** but c0 does not have property T**.