  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 $\tau:C^*_{\text{red}}(G) \to \mathbb{C}$ vanish on $a$. We show that
all traces $\theta:C^*_{\text{red}}(G) \to \mathbb{C}$ are linear combinations
of traces $\chi_g:C^*_{\text{red}}(G) \to \mathbb{C}$ given by
\[\chi_{g}=\begin{cases} 1 & g \in C(g) \\ 0 & \text{else}\end{cases}\] 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.
