Property (T ) for non-unital C ∗-algebras ✩

Inspired by the recent work of Bekka, we study two reasonable analogues of property (T ) for not necessarily unital C ∗-algebras. The stronger one of the two is called “property (T )” and the weaker one is called “property (Te).” It is shown that all non-unital C ∗-algebras do not have property (T ) (neither do their unitalizations). Moreover, all non-unital σ-unital C ∗-algebras do not have property (Te). © 2007 Elsevier Inc. All rights reserved