A Class of Freely Complemented von Neumann Subalgebras of $L{\mathbb{F}}_n$

Abstract

We prove that if $A_1,A_2,\dots ,A_n$ are tracial abelian von Neumann algebras for $2\le n\le \infty$ and $M=A_1\ast \cdots \ast A_n$ is their free product, then any subalgebra $A\subset M$ of the form $A={\sum }_{i=1}^n{u}_i{A}_i{p}_i{u}_i^{\ast }$, for some projections $p_i\in A_i$ and unitaries $u_i\in U(M)$, for $1\le i\le n$, such that ${\sum }_i{u}_i{p}_i{u}_i^{\ast }=1$, is freely complemented (FC) in $M$. Moreover, if $A_1,A_2,\dots ,A_n$ are purely non-separable abelian, and $M=A_1\ast \cdots \ast A_n$, 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, i.e., there exist Haar unitaries $u\in L{\mathbb{F}}_n$ that are free independent to $A$.