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 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 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 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 avec des invariants de Connes prescrits et qui ne sont pas isomorphes à des facteurs d’Araki–Woods libres.
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Keywords: Free Araki–Woods factors; Fullness; Maximal amenable subalgebras; Popa’s deformation/rigidity theory; Type $\mathrm{III}$ factors; Ultraproduct von Neumann algebras
Mot clés : Facteurs d’Araki–Woods libres ; Facteurs pleins ; Sous-algèbres maximales moyennables ; Théorie déformation/rigidité de Popa ; Facteurs de type $\mathrm{III}$ ; Ultraproduits d’algèbres de von Neumann
Houdayer, Cyril 1, 2; Trom, Benjamin 2
@article{AIF_2021__71_2_473_0, author = {Houdayer, Cyril and Trom, Benjamin}, title = {Structure of extensions of free {Araki{\textendash}Woods} factors}, journal = {Annales de l'Institut Fourier}, pages = {473--514}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {71}, number = {2}, year = {2021}, doi = {10.5802/aif.3391}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3391/} }
TY - JOUR AU - Houdayer, Cyril AU - Trom, Benjamin TI - Structure of extensions of free Araki–Woods factors JO - Annales de l'Institut Fourier PY - 2021 SP - 473 EP - 514 VL - 71 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3391/ DO - 10.5802/aif.3391 LA - en ID - AIF_2021__71_2_473_0 ER -
%0 Journal Article %A Houdayer, Cyril %A Trom, Benjamin %T Structure of extensions of free Araki–Woods factors %J Annales de l'Institut Fourier %D 2021 %P 473-514 %V 71 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3391/ %R 10.5802/aif.3391 %G en %F AIF_2021__71_2_473_0
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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