Property (T) for C*-algebras

A notion of Property (T) is defined for an arbitrary unital C*-algebra A admitting a tracial state. This is extended to a notion of Property (T) for a pair (A, B), where B is a C*-subalgebra of A. Let Gamma be a discrete group and C-r* (Gamma) its reduced algebra. We show that C-r* (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* (Gamma), C-r* (Lambda)) has Property (T) if and only if the pair of groups (Gamma, Lambda) has Property (T).