# Maximal amenable MASAs of radial type in the free group factors

## TL;DR

This paper constructs a broad family of maximal amenable MASAs in free group factors by analyzing von Neumann algebras generated by weighted radial elements formed from semicircular elements in free products.

## Contribution

It introduces a new class of maximal amenable MASAs in free group factors using weighted radial elements with at least two non-zero weights, expanding known examples.

## Key findings

- Maximal amenability of the constructed MASAs is proven.
- Unitary conjugacy of these MASAs corresponds to proportional weights.
- Provides a large family of maximal amenable MASAs in free group factors.

## Abstract

We prove that if $\{(M_j, \tau_j)\}_{j\in J}$ are tracial von Neumann algebras, $s_j \in M_j$ are selfadjoint semicircular elements and $t=(t_j)_j$ is a square summable $J$-tuple of real numbers with at least two non-zero entries, then the von Neumann algebra $A(t)$ generated by the ``weighted radial element'' $\sum_j t_j s_j\in M:=*_{j\in J} M_j$ is maximal amenable in $M$, with $A(t)$, $A(t')$ unitary conjugate in $M$ iff $t, t'$ are proportional. Letting $M_j$ be diffuse amenable, $\forall j$, this provides a large family of maximal amenable MASAs in the free group factor $L\mathbb F_n$.