Free Araki-Woods factors and Connes’ bicentralizer problem
Author:
Cyril Houdayer
Journal:
Proc. Amer. Math. Soc. 137 (2009), 3749-3755
MSC (2000):
Primary 46L10, 46L54
DOI:
https://doi.org/10.1090/S0002-9939-09-09923-7
Published electronically:
May 21, 2009
MathSciNet review:
2529883
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Abstract | References | Similar Articles | Additional Information
Abstract: We show that for any type $\mathrm {III}_1$ free Araki-Woods factor $\mathcal {M} = \Gamma (H_{\mathbf {R}}, U_t)''$, the bicentralizer of the free quasi-free state $\varphi _U$ is trivial. Using Haagerup’s Theorem, it follows that there always exists a faithful normal state $\psi$ on $\mathcal {M}$ such that $(\mathcal {M}^\psi )’ \cap \mathcal {M} = \mathbf {C}$.
- Lance Barnett, Free product von Neumann algebras of type ${\rm III}$, Proc. Amer. Math. Soc. 123 (1995), no. 2, 543–553. MR 1224611, DOI https://doi.org/10.1090/S0002-9939-1995-1224611-7
- Alain Connes, Une classification des facteurs de type ${\rm III}$, Ann. Sci. École Norm. Sup. (4) 6 (1973), 133–252 (French). MR 341115
- A. Connes, Almost periodic states and factors of type ${\rm III}_{1}$, J. Functional Analysis 16 (1974), 415–445. MR 0358374, DOI https://doi.org/10.1016/0022-1236%2874%2990059-7
- A. Connes, Factors of type ${\rm III}_1$, property $L_\lambda ’$ and closure of inner automorphisms, J. Operator Theory 14 (1985), no. 1, 189–211. MR 789385
- Alain Connes and Erling Størmer, Homogeneity of the state space of factors of type ${\rm III}_{1}$, J. Functional Analysis 28 (1978), no. 2, 187–196. MR 0470689, DOI https://doi.org/10.1016/0022-1236%2878%2990085-x
- Kenneth J. Dykema, Free products of finite-dimensional and other von Neumann algebras with respect to non-tracial states, Free probability theory (Waterloo, ON, 1995) Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 41–88. MR 1426835
- Uffe Haagerup, Connes’ bicentralizer problem and uniqueness of the injective factor of type ${\rm III}_1$, Acta Math. 158 (1987), no. 1-2, 95–148. MR 880070, DOI https://doi.org/10.1007/BF02392257
- Narutaka Ozawa, Solid von Neumann algebras, Acta Math. 192 (2004), no. 1, 111–117. MR 2079600, DOI https://doi.org/10.1007/BF02441087
- Florin Rădulescu, A one-parameter group of automorphisms of ${\scr L}(F_\infty )\otimes B(H)$ scaling the trace, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), no. 13, 1027–1032 (English, with French summary). MR 1168529
- D. Shlyakhtenko, On multiplicity and free absorption for free Araki-Woods factors. arXiv:math.OA/0302217
- Dimitri Shlyakhtenko, On the classification of full factors of type III, Trans. Amer. Math. Soc. 356 (2004), no. 10, 4143–4159. MR 2058841, DOI https://doi.org/10.1090/S0002-9947-04-03457-9
- Dimitri Shlyakhtenko, $A$-valued semicircular systems, J. Funct. Anal. 166 (1999), no. 1, 1–47. MR 1704661, DOI https://doi.org/10.1006/jfan.1999.3424
- Dimitri Shlyakhtenko, Some applications of freeness with amalgamation, J. Reine Angew. Math. 500 (1998), 191–212. MR 1637501, DOI https://doi.org/10.1515/crll.1998.066
- Dimitri Shlyakhtenko, Free quasi-free states, Pacific J. Math. 177 (1997), no. 2, 329–368. MR 1444786, DOI https://doi.org/10.2140/pjm.1997.177.329
- Stefaan Vaes, États quasi-libres libres et facteurs de type III (d’après D. Shlyakhtenko), Astérisque 299 (2005), Exp. No. 937, ix, 329–350 (French, with French summary). Séminaire Bourbaki. Vol. 2003/2004. MR 2167212
- D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR 1217253
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Additional Information
Cyril Houdayer
Affiliation:
Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, California 90095
Address at time of publication:
CNRS-ENS Lyon, UMPA UMR 5669, 69364 Lyon cedex 7, France
Email:
cyril@math.ucla.edu, cyril.houdayer@umpa.ens-lyon.fr
Keywords:
Free Araki-Woods factors,
Connes’ bicentralizer problem
Received by editor(s):
October 7, 2008
Received by editor(s) in revised form:
February 16, 2009
Published electronically:
May 21, 2009
Communicated by:
Marius Junge
Article copyright:
© Copyright 2009
American Mathematical Society