On a class of II₁ factors with at most one Cartan subalgebra

الأصل · EN

We prove that the normalizer of any diffuse amenable subalgebra of a free group factor L(Fᵣ) generates an amenable von Neumann subalgebra. Moreover, any II₁ factor of the form Q L(Fᵣ), 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 Fᵣ, 2≤ r ≤ ∞, on a probability space (X,μ) is profinite then the group measure space factor L∞(X) Fᵣ has unique Cartan subalgebra, up to unitary conjugacy.