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Free quasi-free states and factors of type III. (États quasi-libres libres et facteurs de type III.) (French) Zbl 1091.46037
Bourbaki seminar. Volume 2003/2004. Exposes 924–937. Paris: Société Mathématique de France (ISBN 2-85629-173-2/pbk). Astérisque 299, 329-350 (2005).
The quasi-free states on the CAR algebra give representations generated by the Araki-Woods factors. In the framework of free probability of Voiculescu, Shlyakhtenko found a free analog of these Araki-Woods factors. The construction of Shlyakhtenko is connected with one-parameter groups of orthogonal transformations $$(U_t )$$ of a real Hilbert space. The associated factors give many new examples of factors of type $$III_1$$ in the Connes classification.
The main results of classification and non-isomorphism of free Araki-Woods factors, due to Shlyakhtenko, presented in the paper are: 1. Complete classification of free Araki-Woods factors associated with an almost-periodic representation $$(U_t )$$. 2. Construction of an uncountable family of free and mutually non-isomorphic Araki-Woods factors. These factors are distinguished by the invariant $$\tau$$ of Connes. 3. Construction of two free non-isomorphic Araki-Woods factors with the same invariant $$\tau$$. 4. Demonstration that the class of automorphisms of free Araki-Woods factors can depend on the multiplicity of the representation $$(U_t )$$.
For the entire collection see [Zbl 1066.00008].

##### MSC:
 46L35 Classifications of $$C^*$$-algebras 46L54 Free probability and free operator algebras 46L30 States of selfadjoint operator algebras
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