Von Neumann equivalence and properly proximal groups.(English)Zbl 1542.22012
Summary: We introduce a new equivalence relation on groups, which we call von Neumann equivalence, that is coarser than both measure equivalence and \(W^\ast \)-equivalence. We introduce a general procedure for inducing actions in this setting and use this to show that many analytic properties, such as amenability, property (T), and the Haagerup property, are preserved under von Neumann equivalence. We also show that proper proximality, which was defined recently in [R. Boutonnet et al., Ann. Sci. Éc. Norm. Supér. (4) 54, No. 2, 445–482 (2021;Zbl 07360850)] using dynamics, is also preserved under von Neumann equivalence. In particular, proper proximality is preserved under both measure equivalence and \(W^\ast \)-equivalence, and from this we obtain examples of non-inner amenable groups that are not properly proximal.
MSC:
| 22D40 | Ergodic theory on groups |
| 22D55 | Kazhdan’s property (T), the Haagerup property, and generalizations |
| 46L36 | Classification of factors |
| 46L55 | Noncommutative dynamical systems |
Citations:
Zbl 07360850Cite
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