Ultraproducts of von Neumann algebras

Ando, Hiroshi; Haagerup, Uffe

doi:10.1016/j.jfa.2014.03.013

Cited by 99 publications

(163 citation statements)

References 30 publications

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“…Throughout this section, we fix a free ultrafilter ω on N. Here we recall a few definitions. For complete treatments, we refer the reader to [2]. Let M be a von Neumann algebra.…”

Section: Non-commutative Dynamical Systemsmentioning

confidence: 99%

Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

Suzuki

2019

Commun. Math. Phys.

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Practically and intrinsically, inclusions of operator algebras are of fundamental interest. The subject of this paper is intermediate operator algebras of inclusions. There are two previously known theorems which naturally and completely describe all intermediate operator algebras: the Galois Correspondence Theorem and the Tensor Splitting Theorem. Here we establish the third, new complete description theorem which gives a canonical bijective correspondence between intermediate operator algebras and intermediate extensions of dynamical systems. One can also regard this theorem as a crossed product splitting theorem, analogous to the Tensor Splitting Theorem. We then give concrete applications, particularly to maximal amenability problem and a new realization result of intermediate operator algebra lattice.2000 Mathematics Subject Classification. Primary 46L05, 46L10, Secondary 46L55, 54H20.

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“…Throughout this section, we fix a free ultrafilter ω on N. Here we recall a few definitions. For complete treatments, we refer the reader to [2]. Let M be a von Neumann algebra.…”

Section: Non-commutative Dynamical Systemsmentioning

confidence: 99%

Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

Suzuki

2019

Commun. Math. Phys.

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“…The proof of Theorem A given in Section 4 (see Theorems 4.1 and 4.2) uses a combination of Ozawa's C * -algebraic techniques [Oz03,Oz04,Is12], ultraproduct von Neumann algebraic techniques [Oc85,AH12] and the recent generalization of Popa's intertwining-by-bimodules to the framework of type III von Neumann algebras developed by the authors in [HI15]. The interesting feature of the proof of Theorem A is that it does not rely on Connes-Tomita-Takesaki modular theory.…”

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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“…The von Neumann algebra M ω GR is very large (not separable and not even σ-finite in general). The main interest in this ultraproduct comes from the fact that, as explained in [AH12], there is a natural identification L 2 (M ω GR ) = L 2 (M ) ω . Choose a faithful state ϕ ∈ M * .…”

Section: Preliminariesmentioning

confidence: 99%

Structure of bicentralizer algebras and inclusions of type $$\mathrm{III}$$ factors

et al. 2019

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We investigate the structure of the relative bicentralizer algebra B(N ⊂ M, ϕ) for inclusions of von Neumann algebras with normal expectation where N is a type III1 subfactor and ϕ ∈ N * is a faithful state. We first construct a canonical flow β ϕ : R * + B(N ⊂ M, ϕ) on the relative bicentralizer algebra and we show that the W * -dynamical system (B(N ⊂ M, ϕ), β ϕ ) is independent of the choice of ϕ up to a canonical isomorphism. In the case when N = M , we deduce new results on the structure of the automorphism group of B(M, ϕ) and we relate the period of the flow β ϕ to the tensorial absorption of Powers factors. For general irreducible inclusions N ⊂ M , we relate the ergodicity of the flow β ϕ to the existence of irreducible hyperfinite subfactors in M that sit with normal expectation in N . When the inclusion N ⊂ M is discrete, we prove a relative bicentralizer theorem and we use it to solve Kadison's problem when N is amenable.2010 Mathematics Subject Classification. 46L10, 46L30, 46L36, 46L37, 46L55.

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Ultraproducts of von Neumann algebras

Cited by 99 publications

References 30 publications

Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

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

Structure of bicentralizer algebras and inclusions of type $$\mathrm{III}$$ factors

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