We introduce a new equivalence relation on groups, which we call von Neumann equivalence, that is coarser than both measure equivalence and W*-equivalence. Our general procedure for inducing actions in this setting shows that many properties, such as amenability, property (T), the Haagerup property, and proper proximality, are preserved under von Neumann equivalence. We also introduce a generalization of von Neumann equivalence in the setting of finite von Neumann algebras, which is coarser than the equivalence relation given by virtual isomorphism. This is joint work with Ishan Ishan and Jesse Peterson.
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