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Ideal structure of the C*-algebra of Thompson group T

Research output: Chapter in Book/Report/Conference proceedingChapter (peer-reviewed)

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Abstract

In a recent paper Uffe Haagerup and Kristian Knudsen Olesen show that for Richard Thompson’s group T, if there exists a finite set H which can be decomposed as disjoint union of sets H1 and H2 with \sum_{g∈H1} π(g) = \sum_{h∈H2} π(h) and such that the closed ideal generated by \sum_{g∈H1} λ(g) − \sum_{h∈H2} λ(h) coincides with C*_λ(T ), then Richard Thompson's group F is not amenable. In particular, if C*_λ(T) is simple then F is not amenable. Here we prove the converse, namely, if F is not amenable then we can find two sets H1 and H2 with the above properties. The only currently available tools for proving the simplicity of group C*-algebras is Power’s condition. We show that it fails for C*_λ(T) and present an apparent weakening of that condition which could potentially be used for various new groups H to show the simplicity of C*_λ(H). While we use our weakening in the proof of the first result, we also show that the new condition is still too strong to be used to show the simplicity of C*_λ(T ). Along the way, we give a new application of the Ping-Pong Lemma to find free groups as subgroups in groups of homeomorphisms of the circle generated by elements with rational rotation number.
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Details

Original languageEnglish
Title of host publicationTopological Methods in Geometric Group Theory
PublisherCambridge University Press
Number of pages10
Publication statusAccepted/In press - 25 Apr 2016

Publication series

NameLondon Mathematical Society Lecture Note Series
ISSN (Print)0076-0552

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