Property (T) for topological groups and C *-algebras

The aim of the present (mostly expository) paper is to show the relationship of a generalization of Kazhdan’s property (T) for C *-algebras introduced in our recent paper to that of B. Bekka. It is shown that our definition coincides with Bekka’s definition for group C *-algebras of locally compact groups, whereas, in general, these definitions are distinct. Criteria for a C *-algebra to possess our property (T) are given. A number of examples of C *-algebras with and without property (T) are considered. Relations to K-theory are studied.