États quasi-libres libres et facteurs de type III  [ Free quasi-free states and factors of type III ]
Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 937, p. 329-350

Quasi-free states on the algebra of canonical anticommutation relations give rise to representations generating the injective Araki-Woods factors. In the framework of Voiculescu’s free probability theory, Shlyakhtenko has developed a free analogue of these Araki-Woods factors. Shlyakhtenko’s construction starts with a one-parameter group of orthogonal transformations of a real Hilbert space. The associated factors provide a rich class of new examples of non-injective factors of type III 1 in the classification of Connes.

Les états quasi-libres sur l’algèbre des relations d’anticommutation canoniques donnent lieu à des représentations qui engendrent les facteurs moyennables d’Araki et Woods. Dans le cadre des probabilités libres de Voiculescu, Shlyakhtenko a trouvé un analogue libre de ces facteurs Araki-Woods. La construction de Shlyakhtenko part d’un groupe à un paramètre de transformations orthogonales d’un espace de Hilbert réel. Les facteurs associés fournissent une richesse de nouveaux exemples de facteurs de type III 1 dans la classification de Connes.

Classification:  46L54,  46L30,  46L35
Keywords: free probability theory, free Araki-Woods factors, classification of non-injective factors
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     author = {Vaes, Stefaan},
     title = {\'Etats quasi-libres libres et facteurs de type III},
     booktitle = {S\'eminaire Bourbaki : volume 2003/2004, expos\'es 924-937},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France},
     address = {Paris},
     number = {299},
     year = {2005},
     note = {talk:937},
     pages = {329-350},
     zbl = {1091.46037},
     mrnumber = {2167212},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2003-2004__46__329_0}
}
Vaes, Stefaan. États quasi-libres libres et facteurs de type III, in Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 937, pp. 329-350. http://www.numdam.org/item/SB_2003-2004__46__329_0/

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