Property (T) and strong property (T) for unital C∗-algebras

In this paper we will give a thorough study of the notion of property (T) for C∗-algebras (as introduced by M.B. Bekka) as well as a slightly stronger version of it, called “strong property (T)” (which is also an analogue of the corresponding concept in the case of discrete groups and type II1-factors). More precisely, we will give some interesting equivalent formulations as well as some permanence properties for both property (T) and strong property (T). We will also relate them to certain (T)-type properties of the unitary group of the underlying C∗-algebra.