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.
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.
Accepted:
Published online:
Martijn Caspers 1; Rui Okayasu 2; Adam Skalski 3, 4; Reiji Tomatsu 5
@article{CRMATH_2014__352_6_507_0,
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},
}
TY - JOUR AU - Martijn Caspers AU - Rui Okayasu AU - Adam Skalski AU - Reiji Tomatsu TI - Generalisations of the Haagerup approximation property to arbitrary von Neumann algebras JO - Comptes Rendus. Mathématique PY - 2014 SP - 507 EP - 510 VL - 352 IS - 6 PB - Elsevier DO - 10.1016/j.crma.2014.04.003 LA - en ID - CRMATH_2014__352_6_507_0 ER -
%0 Journal Article %A Martijn Caspers %A Rui Okayasu %A Adam Skalski %A Reiji Tomatsu %T Generalisations of the Haagerup approximation property to arbitrary von Neumann algebras %J Comptes Rendus. Mathématique %D 2014 %P 507-510 %V 352 %N 6 %I Elsevier %R 10.1016/j.crma.2014.04.003 %G en %F CRMATH_2014__352_6_507_0
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/
[1] Approximation properties for free orthogonal and free unitary quantum groups, J. Reine Angew. Math., Volume 672 (2012), pp. 223-251
[2] The Haagerup property for arbitrary von Neumann algebras (preprint) | arXiv
[3] Groups with the Haagerup Property. Gromov's a-T-menability, Progress in Mathematics, vol. 197, Basel, Switzerland, 2001
[4] Group factors of the Haagerup type, Proc. Jpn. Acad., Ser. A, Math. Sci., Volume 59 (1983) no. 5, pp. 174-177
[5] Property T for von Neumann algebras, Bull. Lond. Math. Soc., Volume 17 (1985), pp. 57-62
[6] The Haagerup property for locally compact quantum groups, J. Reine Angew. Math. (2014) (in press) | arXiv
[7] CCAP for the discrete quantum groups , Commun. Math. Phys. (2014) (in press) | arXiv
[8] The standard form of von Neumann algebras, Math. Scand., Volume 37 (1975) no. 2, pp. 271-283
[9] An example of non-nuclear -algebra which has the metric approximation property, Invent. Math., Volume 50 (1979) no. 3, pp. 279-293
[10] Haagerup approximation property for finite von Neumann algebras, J. Oper. Theory, Volume 48 (2002) no. 3, pp. 549-571 (suppl.)
[11] On a characterization of the tensor product of self-dual cones associated with the standard von Neumann algebras, Sci. Rep. Niigata Univ., Ser. A, Volume 20 (1984), pp. 1-11
[12] Haagerup approximation property for arbitrary von Neumann algebras (preprint) | arXiv
[13] Haagerup approximation property and positive cones associated with a von Neumann algebra | arXiv
[14] Characterization of matrix-ordered standard forms of -algebras, Math. Scand., Volume 51 (1982) no. 2, pp. 241-260
[15] Theory of Operator Algebras II, Springer, 2000
[16] A.M. Torpe, A characterisation of semidiscrete von Neumann algebras in terms of matrix ordered Hilbert spaces, preprint, Matematisk Institut, Odense Universitet, 1981.
Cited by Sources:
Comments - Policy