On a class of $\mathrm{II}_1$ factors with at most one Cartan subalgebra

Abstract

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.