On cocycle superrigidity for Gaussian actions (Q2884086)

A central motivating problem in the theory of measure preserving actions of countable groups on probability spaces is to classify certain actions up to orbit equivalence, i.e. isomorphisms of the underlying probability spaces such that the orbits of one group are carried onto the orbits of another. For non-amenable groups investigations are difficult. S. Popa made a breakthrough using deformation/rigidity techniques in von Neumann algebras.NEWLINENEWLINEIn this paper a general setting is presented to investigate \({\mathcal U}_{fin}\)-cocycle superrigidity for Gaussian actions in terms of closable derivations on von Neumann algebras. In this setting new proofs to some \({\mathcal U}_{fin}\)-cocycle superrigidity results of S. Popa are given and new examples of this phenomenon are produced. A result of K. R. Parthasarathy and K. Schmidt to give a necessary cohomological condition on a group representation in order for the resulting Gaussian actions to be \({\mathcal U}_{fin}\)-cocycle superrigid, is used.