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C*-simplicity of locally compact Powers groups

Sven Raum

Abstract

In this article we initiate research on locally compact C*-simple groups. We first show that every C*-simple group must be totally disconnected. Then we study C*-algebras and von Neumann algebras associated with certain groups acting on trees. After formulating a locally compact analogue of Powers’ property, we prove that the reduced group C*-algebra of such groups is simple. This is the first simplicity result for C*-algebras of non-discrete groups and answers a question of de la Harpe. We also consider group von Neumann algebras of certain non-discrete groups acting on trees. We prove factoriality, determine their type and show non-amenability. We end the article by giving natural examples of groups satisfying the hypotheses of our work.

Funding statement: The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement No. 622322.

Acknowledgements

We want to thank Alain Valette for his hospitality at the University of Neuchâtel, where part of this work was done. We are grateful to Pierre-Emmanuel Caprace for useful comments on groups acting on trees. We thank Siegfried Echterhoff for asking us whether C*-simple groups are totally disconnected and for a helpful discussion about this question. Further we thank Hiroshi Ando, Pierre de la Harpe, Pierre Julg and Stefaan Vaes for useful comments on the first version this article. Finally, we thank the anonymous referee his comments and for suggesting an easier proof of Theorem 6.1.

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Received: 2015-06-25
Revised: 2016-01-08
Published Online: 2016-07-29
Published in Print: 2019-03-01

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