Unique Cartan decomposition for II₁ factors arising from arbitrary actions of free groups

Original · EN

We prove that for any free ergodic probability measure preserving action ₙ (X,μ) of a free group on n generators ₙ, 2 ≤ n ≤ ∞, the associated group measure space II₁ factor L∞(X) ₙ has L∞(X) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II₁ factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II₁ factors arising from arbitrary actions of a rather large family of groups, including all free products of amenable groups and their direct products.