Inner Amenability and Fullness

Choda, Marie

doi:10.2307/2043605

Cited by 20 publications

(29 citation statements)

References 2 publications

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“…Murray and von Neumann's proof that the group von Neumann algebra of the free group on two generators has no nontrivial asymptotically central sequences [32]. 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].…”

supporting

confidence: 59%

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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supporting

confidence: 59%

CAT(0) cube complexes and inner amenability

Duchesne

Tucker-Drob

Wesolek³

2021

Groups Geom. Dyn.

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“…Since lim n→ω W (ξ)x n − x n W (ξ) 2 = 0, a combination of (5) and (6) yields Proposition 2.6 yields W (ξ)P X 1 (H⊖Cξ)⊗ℓ 2 (G) (y n ) 2 = ξ P X 1 (H⊖Cξ)⊗ℓ 2 (G) (y n ) 2 . By (5) and (7), we get lim n→ω y n 2 = 0, whence lim n→ω x n − E L(G) (x n ) 2 = 0. This shows that M ′ ∩ M ω ⊂ L(G) ω and finishes the proof of Proposition 6.1.…”

Section: Proof Ofmentioning

confidence: 92%

Structure of II 1 factors arising from free Bogoljubov actions of arbitrary groups

Houdayer

2014

Advances in Mathematics

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In this paper, we investigate several structural properties for crossed product II1 factors M arising from free Bogoljubov actions associated with orthogonal representations π : G → O(HR) of arbitrary countable discrete groups. Under fairly general assumptions on the orthogonal representation π : G → O(HR), we show that M does not have property Gamma of Murray and von Neumann. Then we show that any regular amenable subalgebra A ⊂ M can be embedded into L(G) inside M . Finally, when G is assumed to be amenable, we locate precisely any possible amenable or Gamma extension of L(G) inside M .2010 Mathematics Subject Classification. 46L10; 46L54; 46L55; 22D25.

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“…Unless the action is strongly ergodic, the von Neumann algebra Γ L ∞ (X, ) cannot be full. We note that in the case where the strongly ergodic action Γ (X, µ) is probability measure preserving, Choda [Cho82] obtained in 1982 the rather satisfactory result that the factor Γ L ∞ (X) is full whenever Γ is not inner amenable. In this note, we combine Choda's proof with Zimmer's notion of amenable action [Zim77] and prove Houdayer and Isono's property for a wider class of groups, notably including SL(3, Z), which is not biexact [Sak09].…”

Section: Introductionmentioning

confidence: 90%