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

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 $n\ge 2$, 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 Jones and Schmidt (Am J Math 109(1):91–114, 1987). 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 Jones and Schmidt (1987). 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.