Rigidity for Von Neumann Algebras Given by Locally Compact Groups and Their Crossed Products

© 2018, Springer-Verlag GmbH Germany, part of Springer Nature. We prove the first rigidity and classification theoremsfor crossed product von Neumann algebras given by actions of non-discrete, locally compact groups. We prove that for arbitraryfree probability measure preserving actions of connected simple Lie groups of real rank one, the crossed product has a unique Cartansubalgebra up to unitary conjugacy. We then deduce a W* strong rigidity theorem for irreducible actions of products of such groups. More generally, our results hold for products of locally compact groups that are nonamenable, weakly amenable and that belong to Ozawa’s class S.status: publishe