Property RD and the classification of traces on reduced group C∗-algebras of hyperbolic groups

Abstract

In this paper, we show that if [Formula: see text] is a non-elementary word hyperbolic group, [Formula: see text] is an element, and the conjugacy class of [Formula: see text] is infinite, then all traces [Formula: see text] vanish on [Formula: see text]. Moreover, we completely classify all traces by showing that traces [Formula: see text] are linear combinations of traces [Formula: see text] given by [Formula: see text] where [Formula: see text] is an element with finite conjugacy class, denoted [Formula: see text]. We demonstrate these two statements by introducing a new method to study traces that uses Sobolev norms and the rapid decay property.