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. © 2009 Elsevier Inc. All rights reserved.