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Raum, Sven

\(C^*\)-simplicity of locally compact Powers groups. (English) Zbl 1419.46035

J. Reine Angew. Math. 748, 173-205 (2019); erratum ibid. 772, 223-225 (2021).
Summary: 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 P. de la Harpe [Bull. Lond. Math. Soc. 39, No. 1, 1–26 (2007; Zbl 1123.22004), Question 5]. 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.

Cited in 2 Reviews
Cited in 10 Documents

MSC:

46L05 General theory of \(C^*\)-algebras
46L10 General theory of von Neumann algebras
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
20F99 Special aspects of infinite or finite groups

Keywords:

\({C}^\ast\)-simple group; reduced group \({C}^\ast\)-algebra; group von Neumann algebras; groups acting on trees

Citations:

Zbl 1123.22004
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Full Text: DOI arXiv

References:

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