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

TL;DR

This work identifies a broad class of freely complemented MASAs in free products $M=A_1*\cdots*A_n$ by reassembling corners into $\mathcal A=\sum_i u_i A_i p_i u_i^*$, proving these reassembled algebras are FC in $M$ (and hence maximal amenable in the diffuse separable case). It further proves that when the $A_i$ are purely non-separable, any purely non-separable singular MASA in $M$ is FC, establishing an equivalence between singularity, maximal amenability, and FC in this non-separable regime. The paper also verifies Popa's weak FC conjecture for several well-known maximal amenable MASAs in $L\mathbb F_n$, including the radial MASA, using freeness arguments and structural decompositions. These results illuminate how the free product structure enforces FC for a wide array of MASAs and demonstrate robust methods to produce Haar unitaries free to given amenable subalgebras, with implications for automorphism groups via non-separable FC phenomena.

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 $A$.

Theorems & Definitions (32)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Remark 2.4