Rigidity and structural results in von Neumann algebras and Ergodic Theory

 In their pioneering work, Murray and von Neumann found a natural way to associate a von Neumann algebra to every countable group G and to any of its measure preserving actions. The classification of these von Neumann algebras is in general a hard problem and it is driven by the following fundamental question: what aspects of the group G and of an action of G are remembered by the associated von Neumann algebras? In the amenable case, no information can be recovered excepts the amenability of the group. On the other hand, the non-amenable case revealed a beautiful rigidity theory. Various aspects of the groups and their actions are recovered by their von Neumann algebras. In this proposal we aim to obtain new structural results for von Neumann algebras associated to lattices in higher rank Lie groups and to classify all the tensor product decompositions of von Neumann algebras arising from their actions.
Orbit equivalence (OE) theory has seen an explosion of activity in the last two decades and it was triggered in part by the success of Popa's deformation/rigidity approach in the classification of von Neumann algebras. In this project, we plan to find new classes of actions that are OE superrigid (i.e. the OE relation completely remembers the underlying action) and obtain new structural and rigidity results in the OE theory. In particular, we aim to find new classes of groups G for which any Bernoulli or profinite action of G is cocycle and OE superrigid.