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

TL;DR

This paper thoroughly investigates Property (T) and strong Property (T) for unital C*-algebras, providing equivalent formulations, permanence properties, and connections to unitary groups, extending concepts from group theory.

Contribution

It introduces and analyzes strong Property (T) for C*-algebras, establishing new equivalences and permanence results, and relating these properties to unitary group characteristics.

Findings

Equivalent formulations of Property (T) and strong Property (T) for C*-algebras

Permanence properties under algebraic operations

Connections between properties and unitary group behaviors

Abstract

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.