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On a class of II$$_1$$ factors with at most one Cartan subalgebra. (English) Zbl 1201.46054
Summary: 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 II$$_1$$ factor of the form $$Q \overline {\otimes } L(\mathbb F_r)$$, 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 a unique Cartan subalgebra, up to unitary conjugacy.

##### MSC:
 46L10 General theory of von Neumann algebras 37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
##### Keywords:
free group factors; profinite actions; Cartan subalgebras
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##### References:
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