Property (T) for non-unital C*-algebras

TL;DR

This paper explores two variants of property (T) for non-unital C*-algebras, showing that such algebras generally lack these properties, extending Bekka's work on unital cases.

Contribution

It introduces and analyzes two analogues of property (T) for non-unital C*-algebras, establishing their absence in broad classes of these algebras.

Findings

Non-unital C*-algebras do not have property (T).

Unitalizations of non-unital C*-algebras also lack property (T).

Non-unital σ-unital C*-algebras do not have property (T_e).

Abstract

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 (T_{e})''. It is shown that all non-unital C*-algebras do not have property (T) (neither do their unitalizations). Moreover, all non-unital $\sigma$-unital C*-algebras do not have property (T_e).