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

TL;DR

This paper investigates the structure of certain II$_1$ factors, showing that they have at most one Cartan subalgebra, with implications for their amenability and uniqueness properties.

Contribution

It proves that the normalizer of any diffuse amenable subalgebra in free group factors generates an amenable algebra and establishes uniqueness of Cartan subalgebras in specific group measure space factors.

Findings

Normalizer of diffuse amenable subalgebras generates an amenable algebra

Certain II$_1$ factors have no Cartan subalgebras

Profinite actions lead to unique Cartan subalgebras

Abstract

We prove that the normalizer of any diffuse amenable subalgebra of a free group factor $L(\Bbb F_r)$ generates an amenable von Neumann subalgebra. Moreover, any II$_1$ factor of the form $Q \vt L(\Bbb 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 $\Bbb F_r$, $2\leq r \leq \infty$, on a probability space $(X,\mu)$ is profinite then the group measure space factor $L^\infty(X)\rtimes \Bbb F_r$ has unique Cartan subalgebra, up to unitary conjugacy.