États quasi-libres libres et facteurs de type III

Vaes, Stefaan

Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Exposé no. 937, pp. 329-350.

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.

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.

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