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arXiv · 1402.0135

Property RD and the Classification of Traces on Reduced Group $C^*$-algebras of Hyperbolic Groups

Sherry Gong
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Abstract

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

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BibTeXRIS

Sherry Gong. 2016-09-29. Property RD and the Classification of Traces on Reduced Group $C^*$-algebras of Hyperbolic Groups. https://doi.org/10.1142/s1793525317500248

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