Von Neumann Equivalence in Deformation Rigidity Theory

Abstract

Ishan, Peterson and Ruth introduced the notion of von Neumann equivalence as a non-commutative analogue of measure equivalence. They showed approximation properties like amenability, property (T) and Haagerup property are invariant under von Neumann equivalence in the group case. We show that these results can be extended to the von Neumann algebras by inducing bimodules through von Neumann equivalence. We also prove that proper proximality and $L^2$-rigidity are preserved under von Neumann equivalence.