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).