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

Ioana, Adrian; Spaas, Pieter

doi:10.1007/s00208-019-01862-z

Cited by 8 publications

(10 citation statements)

References 29 publications

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“…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]).…”

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confidence: 99%

CAT(0) cube complexes and inner amenability

Duchesne

Tucker-Drob

Wesolek³

2021

Groups Geom. Dyn.

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We here consider inner amenability from a geometric and group theoretical perspective. We prove that for every non-elementary action of a group G on a finite dimensional irreducible CAT(0) cube complex, there is a nonempty G-invariant closed convex subset such that every conjugation invariant mean on G gives full measure to the stabilizer of each point of this subset. Specializing our result to trees leads to a complete characterization of inner amenability for HNN-extensions and amalgamated free products. One novelty of the proof is that it makes use of the existence of certain idempotent conjugation-invariant means on G.We additionally obtain a complete characterization of inner amenability for permutational wreath product groups. One of the main ingredients used for this is a general lemma which we call the location lemma, which allows us to "locate" conjugation invariant means on a group G relative to a given normal subgroup N of G. We give several further applications of the location lemma beyond the aforementioned characterization of inner amenable wreath products.

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confidence: 99%

CAT(0) cube complexes and inner amenability

Duchesne

Tucker-Drob

Wesolek³

2021

Groups Geom. Dyn.

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“…To put this into context, recall from [JS85] that a countable ergodic p.m.p. equivalence relation R is called stable if R admits a decomposition R ∼ = S × R hyp for some equivalence relation S. In recent years, there has been a lot of interest in the study of stable equivalence relations, see for instance [Ki12,TD14,Ki16,Ma17,IS18]. In [JS85], Jones and Schmidt also establish a characterization of stability in terms of central sequences, providing a criterion to check when a decomposition as above exists.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Stable decompositions and rigidity for products of countable equivalence relations

Spaas¹

2021

Preprint

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We show that the "stabilization" of any countable ergodic p.m.p. equivalence relation which is not Schmidt, i.e. admits no central sequences in its full group, always gives rise to a stable equivalence relation with a unique stable decomposition, providing the first non-strongly ergodic such examples. In the proof, we moreover establish a new local characterization of the Schmidt property. We also prove some new structural results for product equivalence relations and orbit equivalence relations of diagonal product actions.

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“…If such a quotient exists, then following [21, Definition 1.3], we say that R has the Jones-Schmidt property. If R has the Jones-Schmidt property and we let M = L(R), A = L ∞ (X ), then there exists a decreasing sequence of von Neumann subalgebras [21,Proposition 5.3 and the proof of Lemma 6.1]).…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…In [21,Theorems E and F], the authors settled in the negative [23,Problem 4.3] by providing examples of equivalence relations R without the Jones-Schmidt property. This was achieved by showing that for certain R, in the above notation, M ∩ A ω is not equal to ∩ n B ω n , for any decreasing sequence of von Neumann subalgebras (B n ) n∈N of A with B n+1 ⊂ B n of finite index for every n ∈ N.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%