Property $(T)$ and strong Property $(T)$ for unital $C^*$-algebras

Details Property $(T)$ and strong Property $(T)$ for unital $C^*$-algebras Property $(T)$ and strong Property $(T)$ for unital $C^*$-algebras

2009-01-14

arxiv.org/abs/0901.1948v1

Mathematics

Operator Algebras

Scientific paper

In this paper, we will give a thorough study of the notion of Property $(T)$ for $C^*$-algebras (as introduced by M.B. Bekka in \cite{Bek-T}) as well as a slight 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 $\rm II_1$-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.

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