The Haagerup property and actions on von Neumann algebras

2021/03/15 by Thiebout Delabie, Delabie, Thiebout, Alexandre Zumbrunnen +1
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2103.08272

openalex publication_date 2021/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In Chapter 2 of "Groups with the Haagerup Property", Jolissaint gives on the one hand a characterization of the Haagerup property in terms of strongly mixing actions on standard probability spaces; on the other hand he gives a noncommutative analogue of this result in terms of actions on factors. In the recent paper "A new characterization of the Haagerup property by actions on infinite measure spaces", the authors give a characterization of the Haagerup property but this time dealing with C₀-actions on infinite measure spaces. Following the spirit of this section, we give a noncommutative analogue in terms of C₀-actions on von Neumann algebras. Next we discuss some natural questions which remained open around C₀-dynamical systems. In particular we give examples of C₀- dynamical systems for groups acting properly on trees. Finally, we give a positive answer to the question of the ergodicity of such systems for non-periodic groups.

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