Cartan subalgebras  of amalgamated free product II$_1$ factors

Ioana, Adrian

doi:10.24033/asens.2239

Cartan subalgebras of amalgamated free product II

_{1}

factors
[Sous-algèbres de Cartan de produit amalgamé de facteurs de type II

_{1}

]

Ioana, Adrian

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 1, pp. 71-130.

Résumé
Abstract

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 $Γ$ 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 $Γ$ 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 $A \subset M$ est une sous-algèbre de von Neumann amenable et non-atomique et si $P \subset M$ 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 $M_{2}$ .

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 $Γ$ 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 $L^{\infty} (X) ⋊ Γ$ arising from a free ergodic probability measure preserving action of $Γ$ 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 $A \subset M$ is a diffuse amenable von Neumann subalgebra and $P \subset M$ 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 $M_{2}$ .

Publié le : 2015-09-24

DOI : 10.24033/asens.2239

Classification : 46L10, 46L36, 37A20.
Keywords: II$_1$ factor, Cartan subalgebra, amalgamated free product.
Mot clés : Facteur de type II$_1$, sous-algèbre de Cartan, produit amalgamé.

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     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
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     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Ioana, Adrian. Cartan subalgebras  of amalgamated free product II$_1$ factors. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 1, pp. 71-130. doi : 10.24033/asens.2239. http://www.numdam.org/articles/10.24033/asens.2239/

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