Free products and rescalings involving non-separable abelian von Neumann algebras

TL;DR

This paper investigates rescalings and free products involving non-separable abelian von Neumann algebras, providing formulas and computations that address open questions in the structure of such algebras.

Contribution

It introduces an interpolation framework for rescalings of free products with abelian von Neumann algebras, solving specific open problems in the field.

Findings

Formulas for free products involving these algebras

Computed compressions of certain free products

Determined the fundamental group of specific free products

Abstract

For a self-symmetric tracial von Neumann algebra $A$, we study rescalings of $A^{*n} * L\mathbb{F}_r$ for $n \in \mathbb{N}$ and $r \in (1, \infty]$ and use them to obtain an interpolation $\mathcal{F}_{s,r}(A)$ for all real numbers $s>0$ and $1-s < r \leq \infty$. We get formulas for their free products, and free products with finite-dimensional or hyperfinite von Neumann algebras. In particular, for any such $A$, we can compute compressions $(A^{*n})^t$ for $0<t<1$, and the Murray-von Neumann fundamental group of $A^{*\infty}$. When $A$ is also non-separable and abelian, this answers two questions in Section 4.3 of recent work of Boutonnet-Drimbe-Ioana-Popa.