Property RD and the Classification of Traces on Reduced Group \(C^\)-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 \(\tau:C^*_{\text{red}}(G) \to \mathbb{C}\) vanish on \(a\). We show that all traces \(\theta:C^*_{\text{red}}(G) \to \mathbb{C}\) a...

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Bibliographic Details
Published in:arXiv.org 2016-09
Main Author: Gong, Sherry
Format: Article
Language:English
Subjects:
Group theory
Norms
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1402.0135