Ultraproducts of von Neumann algebras

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Ultraproducts of von Neumann algebras. / Ando, Hiroshi; Haagerup, Uffe.

I: Journal of Functional Analysis, Bind 266, Nr. 12, 2014, s. 6842-6913.

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt

Harvard

Ando, H & Haagerup, U 2014, 'Ultraproducts of von Neumann algebras', Journal of Functional Analysis, bind 266, nr. 12, s. 6842-6913. https://doi.org/10.1016/j.jfa.2014.03.013

APA

Ando, H., & Haagerup, U. (2014). Ultraproducts of von Neumann algebras. Journal of Functional Analysis, 266(12), 6842-6913. https://doi.org/10.1016/j.jfa.2014.03.013

Vancouver

Ando H, Haagerup U. Ultraproducts of von Neumann algebras. Journal of Functional Analysis. 2014;266(12):6842-6913. https://doi.org/10.1016/j.jfa.2014.03.013

Author

Ando, Hiroshi ; Haagerup, Uffe. / Ultraproducts of von Neumann algebras. I: Journal of Functional Analysis. 2014 ; Bind 266, Nr. 12. s. 6842-6913.

Bibtex

@article{0b3e73e039264f4ea7cb6b3e4fad2a93,
title = "Ultraproducts of von Neumann algebras",
abstract = "We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M  , the ultraproduct MωMω introduced by Ocneanu is a corner of the ultraproduct ∏ωM∏ωM introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M   on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state (σtφω=(σtφ)ω). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower MωMω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M   in MωMω and Connes' asymptotic centralizer algebra MωMω.",
author = "Hiroshi Ando and Uffe Haagerup",
year = "2014",
doi = "10.1016/j.jfa.2014.03.013",
language = "English",
volume = "266",
pages = "6842--6913",
journal = "Journal of Functional Analysis",
issn = "0022-1236",
publisher = "Academic Press",
number = "12",

}

RIS

TY - JOUR

T1 - Ultraproducts of von Neumann algebras

AU - Ando, Hiroshi

AU - Haagerup, Uffe

PY - 2014

Y1 - 2014

N2 - We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M  , the ultraproduct MωMω introduced by Ocneanu is a corner of the ultraproduct ∏ωM∏ωM introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M   on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state (σtφω=(σtφ)ω). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower MωMω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M   in MωMω and Connes' asymptotic centralizer algebra MωMω.

AB - We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M  , the ultraproduct MωMω introduced by Ocneanu is a corner of the ultraproduct ∏ωM∏ωM introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M   on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state (σtφω=(σtφ)ω). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower MωMω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M   in MωMω and Connes' asymptotic centralizer algebra MωMω.

U2 - 10.1016/j.jfa.2014.03.013

DO - 10.1016/j.jfa.2014.03.013

M3 - Journal article

VL - 266

SP - 6842

EP - 6913

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 12

ER -

ID: 137751676