One-cohomology and the uniqueness of the group measure space decomposition of a II_1 factor

Abstract

We provide a unified and self-contained treatment of several of the recent uniqueness theorems for the group measure space decomposition of a II_1 factor. We single out a large class of groups Γ, characterized by a one-cohomology property, and prove that for every free ergodic probability measure preserving action of Γthe associated II_1 factor has a unique group measure space Cartan subalgebra up to unitary conjugacy. Our methods follow closely a recent article of Chifan-Peterson, but we replace the usage of Peterson's unbounded derivations by Thomas Sinclair's dilation into a one-parameter group of automorphisms.

v2: minor changes, final version, to appear in Mathematische Annalen

References in corpus (4)

• Strong solidity of group factors from lattices in SO(n,1) and SU(n,1)
• Some unique group-measure space decomposition results
• Examples of group actions which are virtually W*-superrigid
• A class of groups for which every action is W*-superrigid

Cited by in corpus (7)

• On the structural theory of II_1 factors of negatively curved groups, II: Actions by product groups
• W*-superrigidity for group von Neumann algebras of left-right wreath products
• Unique Cartan decomposition for II_1 factors arising from arbitrary actions of hyperbolic groups
• A class of superrigid group von Neumann algebras
• W*-superrigidity of mixing Gaussian actions of rigid groups
• A class of II_1 factors with many non conjugate Cartan subalgebras
• Uniqueness of group-measure space Cartan subalgebras