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Annales scientifiques de l'ENS - Parutions - série 4, 48 (2015)

Parutions < série 4, 48

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 48, fascicule 1 (2015)

Adrian Ioana
Cartan subalgebras of amalgamated free product II_1 factors
Annales scientifiques de l'ENS 48, fascicule 1 (2015), 71-130

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Résumé :
Sous-algèbres de Cartan de produit amalgamé de facteurs de type II_1
Nous étudions les sous-algèbres de Cartan dans le contexte du produit amalgamé de facteurs de type II_1 et nous obtenons plusieurs résultats d'unicité et de non-existence. Nous démontrons que, si appartient à une grande classe de produits amalgamés de groupes (qui contient le produit libre de deux groupes infinis), alors tout facteur de type II_1 associé à une action libre ergodique de a une sous-algèbre de Cartan unique, à conjugaison unitaire. Nous démontrons aussi que, si R=R_1*R_2 est le produit libre de toute relation d'équivalence ergodique non-hyperfinie dénombrable, alors le facteur de type II_1 L(R) a une sous-algèbre de Cartan unique, à conjugaison unitaire. Enfin, nous démontrons que le produit libre M = M_1 * M_2 de tout facteur de type II_1 n'a pas de sous-algèbre de Cartan. Plus généralement, nous démontrons que, si AM est une sous-algèbre de von Neumann amenable et non-atomique et si PM désigne l'algèbre engendrée par son normalisateur, alors soit P est amenable, soit un coin de P peut être unitairement conjugué dans M_1 ou M_2.

Mots-clefs : Facteur de type II_1, sous-algèbre de Cartan, produit amalgamé.

Abstract:
We study Cartan subalgebras in the context of amalgamated free product II_1 factors and obtain several uniqueness and non-existence results. We prove that if belongs to a large class of amalgamated free product groups (which contains the free product of any two infinite groups) then any II_1 factor L^(X) arising from a free ergodic probability measure preserving action of  has a unique Cartan subalgebra, up to unitary conjugacy. We also prove that if R=R_1*R_2 is the free product of any two non-hyperfinite countable ergodic probability measure preserving equivalence relations, then the II_1 factor L(R) has a unique Cartan subalgebra, up to unitary conjugacy. Finally, we show that the free product M=M_1*M_2 of any two II_1 factors does not have a Cartan subalgebra. More generally, we prove that if AM is a diffuse amenable von Neumann subalgebra and PM denotes the algebra generated by its normalizer, then either P is amenable, or a corner of P can be unitarily conjugate into M_1 or M_2.

Keywords: II_1 factor, Cartan subalgebra, amalgamated free product.

Class. math. : 46L10, 46L36, 37A20.


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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