We provide a class of separable II1 factors
$M$
whose central sequence algebra is not the ‘tail’ algebra associated with any decreasing sequence of von Neumann subalgebras of
$M$
. This settles a question of McDuff [On residual sequences in a II1 factor, J. Lond. Math. Soc. (2) (1971), 273–280].
We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We construct a large family of II 1 factors whose Cartan subalgebras up to unitary conjugacy are not classifiable by countable structures, providing the first such examples. Additionally, we construct examples of II 1 factors whose Cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures. Finally, we show directly that the Cartan subalgebras of the hyperfinite II 1 factor up to unitary conjugacy are not classifiable by countable structures, and deduce that the same holds for any McDuff II 1 factor with at least one Cartan subalgebra.
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.
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