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

TL;DR

This paper demonstrates that certain subalgebras within free product von Neumann algebras are freely complemented and confirms Popa's weak FC conjecture for known maximal amenable MASAs in free group factors.

Contribution

It establishes conditions under which subalgebras are freely complemented in free product von Neumann algebras and verifies Popa's weak FC conjecture for specific MASAs.

Findings

Subalgebras of the form $A = igoplus_{i=1}^n u_i A_i p_i u_i^*$ are freely complemented in free product algebras.

Purely non-separable singular MASAs are freely complemented in free product algebras.

Known maximal amenable MASAs, including the radial MASA, satisfy Popa's weak FC conjecture.

Abstract

We prove that if $A_1, A_2, \dots, A_n$ are tracial abelian 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 $u_i \in U(M)$, for $1 \leq i \leq n$, such that $\sum_i u_i p_i u_i^* = 1$, is freely complemented (FC) in $M$. Moreover, 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, i.e., there exist Haar unitaries $u\in L\mathbb{F}_n$ that are free independent to $A$.