Free Araki-Woods factors and Connes’ bicentralizer problem

Houdayer, Cyril

doi:10.1090/S0002-9939-09-09923-7

Free Araki-Woods factors and Connes’ bicentralizer problem
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Proc. Amer. Math. Soc. 137 (2009), 3749-3755 Request permission

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 $(\mathcal {M}^\psi )’ \cap \mathcal {M} = \mathbf {C}$.

References

Lance Barnett, Free product von Neumann algebras of type $\textrm {III}$, Proc. Amer. Math. Soc. 123 (1995), no. 2, 543–553. MR 1224611, DOI 10.1090/S0002-9939-1995-1224611-7
Alain Connes, Une classification des facteurs de type $\textrm {III}$, Ann. Sci. École Norm. Sup. (4) 6 (1973), 133–252 (French). MR 341115
A. Connes, Almost periodic states and factors of type $\textrm {III}_{1}$, J. Functional Analysis 16 (1974), 415–445. MR 0358374, DOI 10.1016/0022-1236(74)90059-7
A. Connes, Factors of type $\textrm {III}_1$, 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 $\textrm {III}_{1}$, J. Functional Analysis 28 (1978), no. 2, 187–196. MR 0470689, DOI 10.1016/0022-1236(78)90085-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 $\textrm {III}_1$, Acta Math. 158 (1987), no. 1-2, 95–148. MR 880070, DOI 10.1007/BF02392257
Narutaka Ozawa, Solid von Neumann algebras, Acta Math. 192 (2004), no. 1, 111–117. MR 2079600, DOI 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 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 10.1006/jfan.1999.3424
Dimitri Shlyakhtenko, Some applications of freeness with amalgamation, J. Reine Angew. Math. 500 (1998), 191–212. MR 1637501, DOI 10.1515/crll.1998.066
Dimitri Shlyakhtenko, Free quasi-free states, Pacific J. Math. 177 (1997), no. 2, 329–368. MR 1444786, DOI 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, DOI 10.1090/crmm/001