A Class of Freely Complemented von Neumann Subalgebras of L𝔽n

We prove that if A₁, A₂, , Aₙ are diffuse abelian tracial von Neumann algebras for 2≤ n ≤ ∞ and M = A₁ * ⋯ * Aₙ is their free product, then any subalgebra A ⊂ M of the form A = ∑ ᵢ₌₁ⁿ uᵢ Aᵢ pᵢ uᵢ*, for some projections pᵢ ∈ Aᵢ and unitaries uᵢ ∈ U(M), for 1 ≤ i ≤ n, such that ∑ ᵢ uᵢ pᵢ uᵢ* = 1, is freely complemented (FC) in M. We also generalize this result in situations where the Aᵢ are not assumed abelian but the projections pᵢ are instead assumed central, and subsequently without assuming the pᵢ are central, but with the weaker result of the existence of free Haar unitaries. Moreover, we prove that if A₁, A₂, , Aₙ are purely non-separable abelian, and M = A₁ * ⋯ * Aₙ, then any purely non-separable singular MASA in M is FC. We also show that any of the known maximal amenable MASAs A⊂ LFₙ (notably the radial MASA) satisfies Popa’s weak FC conjecture, that is, there exist Haar unitaries u∈ LFₙ that are freely independent from A.