Abstract We prove that if A₁, A₂, , A₍ are diffuse abelian tracial von Neumann algebras for 2 n and M = A₁ * * A₍ is their free product, then any subalgebra A M of the form A = ₈=₁^n u₈ A₈ p₈ u₈^*, for some projections p₈ A₈ and unitaries u₈ U (M), for 1 i n, such that ₈ u₈ p₈ u₈^* = 1, is freely complemented (FC) in M. We also generalize this result in situations where the A₈ are not assumed abelian but the projections p₈ are instead assumed central, and subsequently without assuming the p₈ are central, but with the weaker result of the existence of free Haar unitaries. Moreover, we prove that if A₁, A₂, , A₍ are purely non-separable abelian, and M = A₁ * * A₍, then any purely non-separable singular MASA in M is FC. We also show that any of the known maximal amenable MASAs A LF₍ (notably the radial MASA) satisfies Popa’s weak FC conjecture, that is, there exist Haar unitaries u LF₍ that are freely independent from A.
Boschert et al. (Wed,) studied this question.