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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
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