Classification and rigidity for von Neumann algebras

Ioana, Adrian

doi:10.4171/120-1/37

Cited by 36 publications

(38 citation statements)

References 102 publications

(226 reference statements)

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“…actions of countable groups (cf. the surveys in [11,24,34]). In comparison, our understanding of group von Neumann algebras LG is much more limited.…”

Section: Introduction and Statements Of The Main Resultsmentioning

confidence: 99%

W*-superrigidity for group von Neumann algebras of left-right wreath products

Berbec

Vaes

2013

Proceedings of the London Mathematical Society

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We prove that for many nonamenable groups Γ, including all hyperbolic groups and all nontrivial free products, the left–right wreath product group 𝒢 ≔ (ℤ/2ℤ)(Γ) ⋊ (Γ×Γ) is W*‐superrigid. This means that the group von Neumann algebra L 𝒢 entirely remembers 𝒢. More precisely, if L 𝒢 is isomorphic with L Λ for an arbitrary countable group Λ, then Λ must be isomorphic with 𝒢.

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“…actions of countable groups (cf. the surveys in [11,24,34]). In comparison, our understanding of group von Neumann algebras LG is much more limited.…”

Section: Introduction and Statements Of The Main Resultsmentioning

confidence: 99%

W*-superrigidity for group von Neumann algebras of left-right wreath products

Berbec

Vaes

2013

Proceedings of the London Mathematical Society

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“…This question has attracted a lot of attention during the last 15 years and several important developments regarding the structure and the rigidity of group measure space factors have been made possible thanks to Popa's deformation/rigidity theory [Po06a]. We refer the reader to [Ga10,Va10,Io12b] for recent surveys on this topic.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Bi-Exact Groups, Strongly Ergodic Actions and Group Measure Space Type III Factors with No Central Sequence

Houdayer

Isono

2016

Commun. Math. Phys.

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Abstract. We investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebras M = B ⋊Γ arising from arbitrary actions Γ B of bi-exact discrete groups (e.g. free groups) on amenable von Neumann algebras. We prove a spectral gap rigidity result for the central sequence algebra N ′ ∩ M ω of any nonamenable von Neumann subalgebra with normal expectation N ⊂ M . We use this result to show that for any strongly ergodic essentially free nonsingular action Γ (X, µ) of any bi-exact countable discrete group on a standard probability space, the corresponding group measure space factor L ∞ (X) ⋊ Γ has no nontrivial central sequence. Using recent results of Boutonnet-Ioana-Salehi Golsefidy [BISG15], we construct, for every 0 < λ ≤ 1, a type III λ strongly ergodic essentially free nonsingular action F∞ (X λ , µ λ ) of the free group F∞ on a standard probability space so that the corresponding group measure space type III λ factor L ∞ (X λ , µ λ ) ⋊ F∞ has no nontrivial central sequence by our main result. In particular, we obtain the first examples of group measure space type III factors with no nontrivial central sequence.

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“…Γ and Λ admit stably orbit equivalent actions, see Definition 2.8), then m ≥ n, and if m = n then, after a permutation of indices, Γ i is measure equivalent to Λ i , for any 1 ≤ i ≤ n (see [MS02,Theorem 1.16] and [Sa09,Theorem 3]). See the surveys [Sh04,Po07,Fu09,Ga10,Va10,Io12,Io17] for an overview on orbit equivalence rigidity results and related topics.…”

Section: Introductionmentioning

confidence: 99%

Orbit Equivalence Rigidity for Product Actions

Drimbe

2019

Commun. Math. Phys.

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Let Γ1, . . . , Γn be hyperbolic, property (T) groups, for some n ≥ 1. We prove that if a product Γ1 × · · · × Γn X1 × · · · × Xn of measure preserving actions is stably orbit equivalent to a measure preserving action Λ Y , then there exists a direct product decomposition Λ = Λ1×· · ·×Λn into n infinite groups. Moreover, there exists a measure preserving action ΛiYi which is stably orbit equivalent to Γi Xi, for any 1 ≤ i ≤ n, and the product action Λ1 × · · · × Λn Y1 × · · · × Yn is isomorphic to Λ Y .

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Classification and rigidity for von Neumann algebras

Cited by 36 publications

References 102 publications

W*-superrigidity for group von Neumann algebras of left-right wreath products

W*-superrigidity for group von Neumann algebras of left-right wreath products

Bi-Exact Groups, Strongly Ergodic Actions and Group Measure Space Type III Factors with No Central Sequence

Orbit Equivalence Rigidity for Product Actions

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