Some unique group-measure space decomposition results

Abstract

Using an approach emerging from the theory of closable derivations on von Neumann algebras, we exhibit a class of groups CR satisfying the following property: given any groups G_1, G_2 in CR, then any free, ergodic, measure preserving action on a probability space G_1 x G_2 on X gives rise to a von Neumann algebra with unique group measure space Cartan subalgebra. Pairing this result with Popa's Orbit Equivalence Superrigidity Theorem we obtain new examples of W*-superrigid actions.

Revised proofs in Section 4

References in corpus (6)

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• On a class of ${\mathrm{II}}_1$ factors with at most one Cartan subalgebra
• Examples of group actions which are virtually W*-superrigid
• Cocycle superrigidity for profinite actions of irreducible lattices
• A non-commutative Path Space approach to stationary free Stochastic Differential Equations

Cited by in corpus (17)

• On the structural theory of II_1 factors of negatively curved groups, II: Actions by product groups
• W*-superrigidity for arbitrary actions of central quotients of braid groups
• OE and W* superrigidity results for actions by surface braid groups
• Unique Cartan decomposition for II_1 factors arising from arbitrary actions of hyperbolic groups
• Primeness results for von Neumann algebras associated with surface braid groups
• One-cohomology and the uniqueness of the group measure space decomposition of a II_1 factor
• Some $OE$ and $W^{\ast }$-rigidity results for actions by wreath product groups
• 1-bounded entropy and regularity problems in von Neumann algebras
• Cocycle superrigidity for coinduced actions
• Uniqueness of the group measure space decomposition for Popa's $\Cal H\Cal T$ factors
• A random matrix approach to absorption in free products
• A class of II_1 factors with many non conjugate Cartan subalgebras
• Classification of Tensor Decompositions of II${}_1$ Factors Associated With Poly-Hyperbolic Groups
• W${}^{\ast }$-superrigidity for wreath products with groups having positive first ${\ell }^2$-Betti number
• Classification of tensor decompositions for II${}_1$ factors
• Some classification results for generalized q-gaussian algebras
• Uniqueness of group-measure space Cartan subalgebras