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

The aim of the present (mostly expository) paper is to show relations of a generalization of Kazhdan's property (T) for $C^*$-algebras introduced in our recent paper, to this 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.