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

Abstract

We provide a fairly large class of II1 factors N such that M = N ⊗R has a unique McDuff decomposition, up to isomorphism, where R denotes the hyperfinite II1 factor. This class includes all II1 factors N = L ∞ (X) ⋊ Γ associated to free ergodic probability measure preserving (p.m.p.) actions Γ (X, µ) such that either (a) Γ is a free group, Fn, for some n ≥ 2, or (b) Γ is a non-inner amenable group and the orbit equivalence relation of the action Γ (X, µ) satisfies a property introduced in [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 [JS85]. We also prove that if R is a countable strongly ergodic p.m.p. equivalence relation and T is a hyperfinite ergodic p.m.p. equivalence relation, then R × T has a unique stable decomposition, up to isomorphism. Finally, we provide new characterisations of property Gamma for II1 factors and of strong ergodicity for countable p.m.p. equivalence relations.

II$_1$ factors with exotic central sequence algebras

“…Jones and Schmidt asked whether there is always such a quotient with the additional property that Theorem A allows us to strengthen the negative solution to [JS85,Problem 4.3] given in [IS18]. More precisely, in the context of Theorem A, assume that Γ is not inner amenable and let R be the equivalence relation associated to the action Γ…”

CAT(0) cube complexes and inner amenability

“…Similar connections between inner amenability and central sequences were later found by Choda [12] and Jones and Schmidt [21] in the context of ergodic theory. These connections to operator algebras and ergodic theory have continued to provide a rich context and motivation for the study of inner amenability; see, e.g., [38,23,25,24,11,26,27,37,19,33,15,20,3,22,28]. Perhaps because of this, inner amenability has been studied primarily by virtue of its relevance to these two fields (with a few exceptions, e.g., [1,2,36,18]).…”