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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, (cyril.houdayer@umpa.ens-lyon.fr)
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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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