Von Neumann Equivalence

Abstract

We introduce a new equivalence relation on groups, which we call von Neumann equivalence, that is coarser than both measure equivalence and W*-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. Furthermore, we introduce a procedure for inducing Herz-Schur multipliers and use this to show that weak amenability, weak Haagerup property, and the approximation property (AP) are also stable under von Neumann equivalence. We also show that proper proximality, which was defined recently by Boutonnet, Ioana, and Peterson using dynamics, is also preserved under von Neumann equivalence. In particular, proper proximality is preserved under both measure equivalence and W*-equivalence, and from this we obtain examples of non-inner amenable groups that are not properly proximal.