A Class of Freely Complemented von Neumann Subalgebras of L𝔽n

Abstract We prove that if $A_{1}, A_{2}, \dots , A_{n}$ are diffuse abelian tracial von Neumann algebras for $2\leq n \leq \infty $ and $M = A_{1} * \cdots * A_{n}$ is their free product, then any subalgebra $A \subset M$ of the form $A = \sum _{i=1}^{n} u_{i} A_{i} p_{i} u_{i}^{*}$, for some projections $p_{i} \in A_{i}$ and unitaries $u_{i} \in \mathcal{U}(M)$, for $1 \leq i \leq n$, such that $\sum _{i} u_{i} p_{i} u_{i}^{*} = 1$, is freely complemented (FC) in $M$. We also generalize this result in situations where the $A_{i}$ are not assumed abelian but the projections $p_{i}$ are instead assumed central, and subsequently without assuming the $p_{i}$ are central, but with the weaker result of the existence of free Haar unitaries. Moreover, we prove that if $A_{1}, A_{2}, \dots , A_{n}$ are purely non-separable abelian, and $M = A_{1} * \cdots * A_{n}$, then any purely non-separable singular MASA in $M$ is FC. We also show that any of the known maximal amenable MASAs $A\subset L\mathbb{F}_{n}$ (notably the radial MASA) satisfies Popa’s weak FC conjecture, that is, there exist Haar unitaries $u\in L\mathbb{F}_{n}$ that are freely independent from $A$.