Title: Rigidity for von Neumann algebras given by locally compact groups and their crossed products,
Abstract: We prove the first rigidity and classification theorems for crossed product
von Neumann algebras given by actions of non-discrete, locally compact groups.
We prove that for arbitrary free probability measure preserving actions of
connected simple Lie groups of real rank one, the crossed product has a unique
Cartan subalgebra 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.