Type III factors with unique Cartan decomposition

TL;DR

This paper demonstrates that for a broad class of nonamenable group actions, the associated group measure space factors have a unique Cartan subalgebra, extending previous measure-preserving results and establishing primeness and indecomposability.

Contribution

It proves the uniqueness of Cartan subalgebras in group measure space factors for nonamenable actions, generalizing prior measure-preserving results and analyzing their structural properties.

Findings

Uniqueness of Cartan subalgebra in the factors for a large class of groups

Primeness and indecomposability of the crossed products

Extension of measure-preserving case to nonsingular actions

Abstract

We prove that for any free ergodic nonsingular nonamenable action \Gamma\ \actson (X,\mu) of all \Gamma\ in a large class of groups including all hyperbolic groups, the associated group measure space factor $L^\infty(X) \rtimes \Gamma$ has L^\infty(X) as its unique Cartan subalgebra, up to unitary conjugacy. This generalizes the probability measure preserving case that was established in [PV12]. We also prove primeness and indecomposability results for such crossed products, for the corresponding orbit equivalence relations and for arbitrary amalgamated free products $M_1 *_B M_2$ over a subalgebra B of type I.