Generalisations of the Haagerup approximation property to arbitrary von Neumann algebras

Martijn Caspers; Rui Okayasu; Adam Skalski; Reiji Tomatsu

doi:10.1016/j.crma.2014.04.003

Functional analysis

Generalisations of the Haagerup approximation property to arbitrary von Neumann algebras
[Généralisations de la propriété d'approximation de Haagerup pour les algèbres de von Neumann arbitraires]

Présenté par : Gilles Pisier

Martijn Caspers ¹ ; Rui Okayasu ² ; Adam Skalski ^3,⁴ ; Reiji Tomatsu ⁵

¹ Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany
² Department of Mathematics Education, Osaka Kyoiku University, Osaka 582-8582, Japan
³ Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland
⁴ Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
⁵ Department of Mathematics, Hokkaido University, Hokkaido 060-0810, Japan

Comptes Rendus. Mathématique, Volume 352 (2014) no. 6, pp. 507-510.

Résumé
Abstract

La notion de propriété d'approximation de Haagerup, introduite à l'origine pour les algèbres de von Neumann ayant une trace finie, normale, et fidèle, est généralisée pour les algèbres de von Neumann arbitraires. Nous discutons deux caractérisations équivalentes : une du point de vue de la représentation standard et une autre du point de vue des applications linéaires approximantes liées à un poids fidèle, normal, semifini. Quelques propriétés de permanence, en particulier celles concernant les produits croisés, sont établies et certains exemples sont introduits.

The notion of the Haagerup approximation property, originally introduced for von Neumann algebras equipped with a faithful normal tracial state, is generalised to arbitrary von Neumann algebras. We discuss two equivalent characterisations, one in term of the standard form and the other in term of the approximating maps with respect to a fixed faithful normal semifinite weight. Several stability properties, in particular regarding the crossed product construction are established and certain examples are introduced.

Reçu le : 2014-02-24
Accepté le : 2014-04-08
Publié le : 2014-04-24

DOI : 10.1016/j.crma.2014.04.003

Affiliations des auteurs :

Martijn Caspers ¹ ; Rui Okayasu ² ; Adam Skalski ^{3, 4} ; Reiji Tomatsu ⁵

¹ Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany
² Department of Mathematics Education, Osaka Kyoiku University, Osaka 582-8582, Japan
³ Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland
⁴ Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
⁵ Department of Mathematics, Hokkaido University, Hokkaido 060-0810, Japan

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     author = {Martijn Caspers and Rui Okayasu and Adam Skalski and Reiji Tomatsu},
     title = {Generalisations of the {Haagerup} approximation property to arbitrary von {Neumann} algebras},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {507--510},
     publisher = {Elsevier},
     volume = {352},
     number = {6},
     year = {2014},
     doi = {10.1016/j.crma.2014.04.003},
     language = {en},
}

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Martijn Caspers; Rui Okayasu; Adam Skalski; Reiji Tomatsu. Generalisations of the Haagerup approximation property to arbitrary von Neumann algebras. Comptes Rendus. Mathématique, Volume 352 (2014) no. 6, pp. 507-510. doi : 10.1016/j.crma.2014.04.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.04.003/

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