Property RD and the Classification of Traces on Reduced Group $C^{\ast }$-algebras of Hyperbolic Groups

In this paper, we show that if $G$ by a non-elementary word hyperbolic group and $a\in G$ an element, if the conjugacy class of $a$ is infinite, then all traces $\tau :C_{\text{red}}^{\ast }(G)\to \mathbb{C}$ vanish on $a$. We show that all traces $\theta :C_{\text{red}}^{\ast }(G)\to \mathbb{C}$ are linear combinations of traces ${\chi }_g:C_{\text{red}}^{\ast }(G)\to \mathbb{C}$ given by $${\chi }_g=\{\begin{array}{ll}1& g\in C(g)\\ 0& \text{else}\end{array}$$ where $C(g)$ is the conjugacy class of $g$. To do this, we introduce a new method to study traces, using Sobolev norms and property RD.