Geodesic
On a class of $\mathrm{II}_1$ factors with at most one Cartan subalgebra
Narutaka Ozawa1 ; Sorin Popa2
1 University of California Los Angeles, Department of Mathematics, Los Angeles, CA 90095-1555, United States and Department of Mathematical Sciences, University of Tokyo, Komaba, 153-8914, Japan
2 University of California Los Angeles, Department of Mathematics, Los Angeles, CA 90095-1555, United States
Annals of mathematics, Tome 172 (2010) no. 1, pp. 713-749.

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We prove that the normalizer of any diffuse amenable subalgebra of a free group factor $L(\mathbb{F}_r)$ generates an amenable von Neumann subalgebra. Moreover, any ${\rm II}_1$ factor of the form $Q \bar{\otimes} L(\mathbb{F}_r) $, with $Q$ an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure-preserving action of a free group $\mathbb{F}_r$, $2\leq r \leq \infty$, on a probability space $(X,\mu)$ is profinite then the group measure space factor $L^\infty(X) \rtimes \mathbb{F}_r$ has unique Cartan subalgebra, up to unitary conjugacy.
DOI : 10.4007/annals.2010.172.713

Narutaka Ozawa 1 ; Sorin Popa 2

1 University of California Los Angeles, Department of Mathematics, Los Angeles, CA 90095-1555, United States and Department of Mathematical Sciences, University of Tokyo, Komaba, 153-8914, Japan
2 University of California Los Angeles, Department of Mathematics, Los Angeles, CA 90095-1555, United States 
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Narutaka Ozawa; Sorin Popa. On a class of $\mathrm{II}_1$ factors with at most  one Cartan subalgebra. Annals of mathematics, Tome 172 (2010) no. 1, pp. 713-749. doi : 10.4007/annals.2010.172.713. http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.713/

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