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Structural results for free Araki–Woods factors and their continuous cores

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  18 February 2010

Cyril Houdayer
Cyril Houdayer
Affiliation:
Centre National de la Recherche Scientifique-École Normale Supérieure Lyon, 46, allée d'Italie, UMPA UMR 5669, 69364 Lyon Cedex 7, France, ([email protected])
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Abstract

We show that for any type III1 free Araki–Woods factor = (HR, Ut)″ associated with an orthogonal representation (Ut) of R on a separable real Hilbert space HR, the continuous core M = ⋊σR is a semisolid II∞ factor, i.e. for any non-zero finite projection q ∈ M, the II1 factor qM q is semisolid. If the representation (Ut) is moreover assumed to be mixing, then we prove that the core M is solid. As an application, we construct an example of a non-amenable solid II1 factor N with full fundamental group, i.e. (N) = R*+, which is not isomorphic to any interpolated free group factor L(Ft), for 1 < t ≤ = +∞.

Keywords

free Araki–Woods factors(semi-)solid factorsdeformation/rigidity techniquesintertwining techniquesspectral gap property

MSC classification

Secondary: 46L10: General theory of von Neumann algebras 46L54: Free probability and free operator algebras
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 9 , Issue 4 , October 2010 , pp. 741 - 767
Copyright
Copyright © Cambridge University Press 2010

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References

1.Anantharaman-Delaroche, C., Amenable correspondences and approximation properties for von Neumann algebras, Pac. J. Math. 171 (1995), 309–341.CrossRefGoogle Scholar
2.Antoniou, I. and Shkarin, S. A., Decay measures on locally compact abelian topological groups, Proc. R. Soc. Edinb. A 131 (2001), 1257–1273.CrossRefGoogle Scholar
3.Barnett, L., Free product von Neumann algebras of type III, Proc. Am. Math. Soc. 123 (1995), 543–553.Google Scholar
4.Chifan, I. and Houdayer, C., Bass Serre rigidity results in von Neumann algebras, Duke Math. J., in press (arXiv:0805.1566).Google Scholar
5.Chifan, I. and Ioana, A., Ergodic subequivalence relations induced by a Bernoulli action, preprint (arXiv:0802.2353).Google Scholar
6.Connes, A., Une classification des facteurs de type III, Annales Scient. Éc. Norm. Sup. 6 (1973), 133–252.CrossRefGoogle Scholar
7.Connes, A., Almost periodic states and factors of type III1, J. Funct. Analysis 16 (1974), 415–445.CrossRefGoogle Scholar
8.Dykema, K., Interpolated free group factors, Pac. J. Math. 163 (1994), 123–135.CrossRefGoogle Scholar
9.Erdös, P., On a family of symmetric Bernoulli convolutions, Am. J. Math. 61 (1939), 974–976.CrossRefGoogle Scholar
10.Haagerup, U., Connes' bicentralizer problem and uniqueness of the injective factor of type III1, Acta Math. 69 (1986), 95–148.Google Scholar
11.Houdayer, C., On some free products of von Neumann algebras which are free Araki–Woods factors, Int. Math. Res. Not. 2007 (2007), rnm098.Google Scholar
12.Houdayer, C., Free Araki–Woods factors and Connes' bicentralizer problem, Proc. Am. Math. Soc. 137 (2009), 3749–3755.CrossRefGoogle Scholar
13.Houdayer, C., Construction of type II1 factors with prescribed countable fundamental group, J. Reine Angew. Math. 634 (2009), 169–207.Google Scholar
14.Ioana, A., Peterson, J. and Popa, S., Amalgamated free products of w-rigid factors and calculation of their symmetry groups, Acta Math. 200 (2008), 85–153.CrossRefGoogle Scholar
15.Ozawa, N., Solid von Neumann algebras, Acta Math. 192 (2004), 111–117.CrossRefGoogle Scholar
16.Ozawa, N., A Kurosh-type theorem for type II1 factors, Int. Math. Res. Not. 2006 (2006), 97560.Google Scholar
17.Ozawa, N., An example of a solid von Neumann algebra, Hokkaido Math. J. 38 (2009), 557–561.CrossRefGoogle Scholar
18.Ozawa, N. and Popa, S., On a class of II1 factors with at most one Cartan subalgebra, Annals Math., in press (arXiv:0706.3623).Google Scholar
19.Popa, S., Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I, Invent. Math. 165 (2006), 369–408.CrossRefGoogle Scholar
20.Popa, S., On a class of type II1 factors with Betti numbers invariants, Annals Math. 163 (2006), 809–899.CrossRefGoogle Scholar
21.Popa, S., Some rigidity results for non-commutative Bernoulli Shifts, J. Funct. Analysis 230 (2006), 273–328.CrossRefGoogle Scholar
22.Popa, S., On Ozawa's property for free group factors, Int. Math. Res. Not. 2007 (2007), rnm036.Google Scholar
23.Popa, S., On the superrigidity of malleable actions with spectral gap, J. Am. Math. Soc. 21 (2008), 981–1000.CrossRefGoogle Scholar
24.Popa, S. and Vaes, S., Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups, Adv. Math. 217 (2008), 833–872.CrossRefGoogle Scholar
25.Rădulescu, F., A one-parameter group of automorphisms of L(F∞)⊗B(H) scaling the trace, C. R. Acad. Sci. Paris Sér. I 314 (1992), 1027–1032.Google Scholar
26.Rădulescu, F., Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index, Invent. Math. 115 (1994), 347–389.CrossRefGoogle Scholar
27.Shlyakhtenko, D., Free quasi-free states, Pac. J. Math. 177 (1997), 329–368.CrossRefGoogle Scholar
28.Shlyakhtenko, D., Some applications of freeness with amalgamation, J. Reine Angew. Math. 500 (1998), 191–212.CrossRefGoogle Scholar
29.Shlyakhtenko, D., A-valued semicircular systems, J. Funct. Analysis 166 (1999), 1–47.CrossRefGoogle Scholar
30.Shlyakhtenko, D., Some estimates for non-microstates free entropy dimension, with applications to q-semicircular families, Int. Math. Res. Not. 51 (2004), 2757–2772.CrossRefGoogle Scholar
31.Shlyakhtenko, D., On the classification of full factors of type III, Trans. Am. Math. Soc. 356 (2004), 4143–4159.CrossRefGoogle Scholar
32.Shlyakhtenko, D., On multiplicity and free absorption for free Araki–Woods factors, preprint (math.OA/0302217).Google Scholar
33.Takesaki, M., Duality for crossed products and structure of von Neumann algebras of type III, Acta Math. 131 (1973), 249–310.CrossRefGoogle Scholar
34.Ueda, Y., Amalgamated free products over Cartan subalgebra, Pac. J. Math. 191 (1999), 359–392.CrossRefGoogle Scholar
35.Vaes, S., États quasi-libres libres et facteurs de type III (d'après D. Shlyakhtenko), Séminaire Bourbaki, Exposé 937, Astérisque 299 (2005), 329–350.Google Scholar
36.Vaes, S., Rigidity results for Bernoulli actions and their von Neumann algebras (after S. Popa), Séminaire Bourbaki, Exposé 961, Astérisque 311 (2007), 237–294.Google Scholar
37.Vaes, S. and Vergnioux, R., The boundary of universal discrete quantum groups, exactness and factoriality, Duke Math. J. 140 (2007), 35–84.CrossRefGoogle Scholar
38.Voiculescu, D.-V., Dykema, K. J. and Nica, A., Free random variables, CRM Monograph Series, Volume 1 (American Mathematical Society, Providence, RI, 1992).Google Scholar