W*-superrigiditeit voor groepen en kwantumgroepen

This is a research proposal in rigidity theory for von Neumann algebras. Building upon earlier rigidity and indecomposability results, I will give the first class of II1 factors that are indecomposable in every possible way: even allowing arbitrary amplifications, they cannot be decomposed as II1 factors coming from groups nor group actions, not even twisted by a 2-cocycle. Next, I aim to prove the first superrigidity theorem for group von Neumann algebras up to virtual isomorphism. More precisely, I aim to show that for a certain class of groups, a virtual isomorphism of their group von Neumann algebra and any other group von Neumann algebra will imply that the groups themselves are virtually isomorphic. Finally, I plan to show the first W*-superrigidity theorem for group von Neumann algebras twisted by a 2-cocycle. Namely that for a certain class of groups, any isomorphism between their group von Neumann algebra and an arbitrary amplification of any other twisted group von Neumann algebra implies an isomorphism of the groups, the amplification to be trivial, and the 2-cocycle to be cohomologous with the trivial 2-cocycle. Furthermore, I plan to investigate whether the twisted group von Neumann algebras of this class of superrigid groups remember the cohomology class of the 2-cocycle. Does an isomorphism of two such twisted group von Neumann algebras imply that the 2-cocycles are cohomologous?