Some unique group-measure space decomposition results

TL;DR

This paper introduces a new method using closable derivations on von Neumann algebras to identify classes of groups with unique measure space decompositions, leading to new examples of superrigid actions.

Contribution

It establishes a class of groups with unique group measure space Cartan subalgebras for their actions, advancing the understanding of W*-superrigidity.

Findings

Identification of a class of groups with unique Cartan subalgebras

Construction of new W*-superrigid actions

Application of derivation techniques to von Neumann algebras

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.