A class of II$_1$ factors with a unique McDuff decomposition

TL;DR

This paper identifies a broad class of II$_1$ factors with a unique McDuff decomposition, introduces new examples of ergodic equivalence relations, and offers characterizations of property Gamma and strong ergodicity.

Contribution

It establishes conditions for the uniqueness of McDuff decompositions in II$_1$ factors and provides the first examples of ergodic relations lacking a specific property.

Findings

Class of II$_1$ factors with unique McDuff decomposition identified.

First examples of ergodic p.m.p. equivalence relations without the JS85 property.

New characterizations of property Gamma and strong ergodicity.

Abstract

We provide a fairly large class of II$_1$ factors $N$ such that $M=N\bar{\otimes}R$ has a unique McDuff decomposition, up to isomorphism, where $R$ denotes the hyperfinite II$_1$ factor. This class includes all II$_1$ factors $N=L^{\infty}(X)\rtimes\Gamma$ associated to free ergodic probability measure preserving (p.m.p.) actions $\Gamma\curvearrowright (X,\mu)$ such that either (a) $\Gamma$ is a free group, $\mathbb F_n$, for some $n\geq 2$, or (b) $\Gamma$ is a non-inner amenable group and the orbit equivalence relation of the action $\Gamma\curvearrowright (X,\mu)$ satisfies a property introduced in \cite{JS85}. On the other hand, settling a problem posed by Jones and Schmidt in 1985, we give the first examples of countable ergodic p.m.p. equivalence relations which do not satisfy the property of \cite{JS85}. We also prove that if $\mathcal R$ is a countable strongly ergodic p.m.p. equivalence relation and $\mathcal T$ is a hyperfinite ergodic p.m.p. equivalence relation, then $\mathcal R\times\mathcal T$ has a unique stable decomposition, up to isomorphism. Finally, we provide new characterisations of property Gamma for II$_1$ factors and of strong ergodicity for countable p.m.p. equivalence relations.