Structure of extensions of free Araki–Woods factors
Annales de l'Institut Fourier, Volume 71 (2021) no. 2, pp. 473-514.

We investigate the structure of crossed product von Neumann algebras arising from Bogoljubov actions of countable groups on Shlyakhtenko’s free Araki–Woods factors. Among other results, we settle the questions of factoriality and Connes’ type classification. We moreover provide general criteria regarding fullness and strong solidity. As an application of our main results, we obtain examples of type III 0 factors that are prime, have no Cartan subalgebra and possess a maximal amenable abelian subalgebra. We also obtain a new class of strongly solid type III factors with prescribed Connes’ invariants that are not isomorphic to any free Araki–Woods factors.

Nous étudions la structure de produits croisés d’algèbres de von Neumann provenant des actions Bogoljubov de groupes dénombrables sur les facteurs d’Araki–Woods libres. Nous résolvons notamment les questions concernant la factorialité et la classification du type de Connes. Nous donnons également des critères généraux concernant le caractère plein et la solidité forte. Comme application de nos résultats, nous obtenons des exemples de facteurs de type III 0 qui sont premiers, sans sous-algèbre de Cartan et qui possèdent une sous-algèbre maximale moyennable abélienne. Nous obtenons aussi une nouvelle classe de facteurs fortement solides de type III avec des invariants de Connes prescrits et qui ne sont pas isomorphes à des facteurs d’Araki–Woods libres.

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DOI: 10.5802/aif.3391
Classification: 46L10,  46L40,  46L54,  46L55
Keywords: Free Araki–Woods factors; Fullness; Maximal amenable subalgebras; Popa’s deformation/rigidity theory; Type III factors; Ultraproduct von Neumann algebras
Houdayer, Cyril 1, 2; Trom, Benjamin 2

1 Institut Universitaire de France
2 Université Paris-Saclay, CNRS Laboratoire de mathématiques d’Orsay 91405 Orsay (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Houdayer, Cyril; Trom, Benjamin. Structure of extensions of free Araki–Woods factors. Annales de l'Institut Fourier, Volume 71 (2021) no. 2, pp. 473-514. doi : 10.5802/aif.3391. https://aif.centre-mersenne.org/articles/10.5802/aif.3391/

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