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.

Classification : 46L54, 46L30, 46L35
Mot clés : probabilités libres, facteurs d'Araki-Woods libres, classification de facteurs non-injectifs
Keywords: free probability theory, free Araki-Woods factors, classification of non-injective factors

[1] H. Araki - “On quasifree states of CAR and Bogoliubov automorphisms”, Publ. RIMS, Kyoto Univ. 6 (1970), p. 385-442. | MR | Zbl

[2] H. Araki & E. J. Woods - “A classification of factors”, Publ. RIMS, Kyoto Univ. 4 (1968), p. 51-130. | MR | Zbl

[3] L. Barnett - “Free product von Neumann algebras of type III”, Proc. Amer. Math. Soc. 123 (1995), no. 2, p. 543-553. | MR | Zbl

[4] F. Combes - “Les facteurs de von Neumann de type III (d'après A. Connes)”, in Sém. Bourbaki, Lect. Notes in Math., vol. 514, Springer, Berlin, 1976, Exp. no 461, p. 124-137. | Numdam | MR | Zbl

[5] A. Connes - “Une classification des facteurs de type III” 6 (1973), p. 133-252. | Numdam | MR | Zbl

[6] -, “Almost periodic states and factors of type III${}_1$”, J. Funct. Anal. 16 (1974), p. 415-445. | MR | Zbl

[7] -, “Classification of injective factors”, Ann. of Math. 104 (1976), p. 73-115. | MR | Zbl

[8] J. Dixmier - Les C${}^{*}$-algèbres et leurs représentations, Gauthier-Villars, Paris, 1964. | MR | Zbl

[9] É. Ghys & P. De La Harpe - Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Math., vol. 83, Birkhäuser, Boston, 1990. | MR | Zbl

[10] U. Haagerup - “Connes’ bicentralizer problem and uniqueness of the injective factor of type III${}_1$”, Acta Math. 158 (1987), p. 95-148. | MR | Zbl

[11] F. J. Murray & J. Von Neumann - “On rings of operators. IV”, Ann. of Math. (2) 44 (1943), p. 716-808. | MR | Zbl

[12] N. Ozawa - “Solid von Neumann algebras”, Acta Math. 192 (2004), p. 111-117. | MR | Zbl

[13] G. Pisier & D. Shlyakhtenko - “Grothendieck's theorem for operator spaces”, Invent. Math. 150 (2002), p. 185-217. | MR | Zbl

[14] R. T. Powers & E. Størmer - “Free states of the canonical anticommutation relations”, Comm. Math. Phys. 16 (1970), p. 1-33. | MR | Zbl

[15] F. Rădulescu - “A one-parameter group of automorphisms of $L\left({𝔽}_{\infty }\right)\otimes B\left(H\right)$ scaling the trace”, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), p. 1027-1032. | MR | Zbl

[16] -, “A type III${}_{\lambda }$ factor with core isomorphic to the von Neumann algebra of a free group, tensor $B\left(H\right)$”, in Recent advances in operator algebras (Orléans, 1992), Astérisque, vol. 232, Société Mathématique de France, 1995, p. 203-209. | Numdam | MR | Zbl

[17] M. A. Rieffel & A. Van Daele - “A bounded operator approach to Tomita-Takesaki theory”, Pacific J. Math. 69 (1977), p. 187-221. | MR | Zbl

[18] D. Shale & W. F. Stinespring - “States of the Clifford algebra”, Ann. of Math. 80 (1964), p. 365-381. | MR | Zbl

[19] D. Shlyakhtenko - “On multiplicity and free absorption for free Araki-Woods factors”, Prépublication.

[20] -, “Free quasi-free states”, Pacific J. Math. 177 (1997), p. 329-368. | MR

[21] -, “Some applications of freeness with amalgamation”, J. reine angew. Math. 500 (1998), p. 191-212. | MR | Zbl

[22] -, “$A$-valued semicircular systems”, J. Funct. Anal. 166 (1999), p. 1-47. | MR | Zbl

[23] -, “Prime type III factors”, Proc. Nat. Acad. Sci. U.S.A. 97 (2000), p. 12439-12441. | MR

[24] -, “On the Classification of Full Factors of Type III”, Trans. Amer. Math. Soc. 356 (2004), p. 4143-4159. | MR | Zbl

[25] G. Skandalis - “Algèbres de von Neumann de groupes libres et probabilités non commutatives (d'après Voiculescu, etc.)”, in Sém. Bourbaki, 1992/93, Astérisque, vol. 216, Société Mathématique de France, 1993, p. 87-102. | Numdam | MR | Zbl

[26] Ş. Strătilă & L. Zsidó - Lectures on von Neumann algebras, Abacus Press, Tunbridge Wells, 1979. | Zbl

[27] S. Vaes - “Strictly outer actions of groups and quantum groups”, J. reine angew. Math. 578 (2005), p. 147-184. | MR | Zbl

[28] D. V. Voiculescu - “The analogues of entropy and of Fisher's information measure in free probability theory. II”, Invent. Math. 118 (1994), p. 411-440. | MR | Zbl

[29] -, “The analogues of entropy and of Fisher's information measure in free probability theory. III”, Geom. Funct. Anal. 6 (1996), p. 172-199. | MR | Zbl

[30] D. V. Voiculescu, K. J. Dykema & A. Nica - Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. | MR | Zbl