Free products and rescalings involving non-separable abelian von Neumann algebras
Ken Dykema, Junchen Zhao
TL;DR
This work constructs a flexible interpolation family $\mathcal{F}_{s,r}(A)$ of II$_1$ factors from a self-symmetric tracial von Neumann algebra $A$, extending interpolated free group factors to non-separable settings. The authors prove precise rescaling and free-product addition formulas: $(\mathcal{F}_{s,r})^t \cong \mathcal{F}_{s/t,\,(s+r-1)/t^2 - s/t + 1}$ and $\mathcal{F}_{s,r} * \mathcal{F}_{s',r'} \cong \mathcal{F}_{s+s',r+r'}$, and establish well-definedness via compression techniques, enabling explicit compressions $(A^{*n})^t$ and compositions with finite-dimensional or hyperfinite algebras. They derive a rich array of applications, including the determination of the fundamental group of $A^{*\infty}$ as $\mathbb{R}_+^*$ for all self-symmetric $A$, and concrete results for non-separable abelian algebras that address questions in recent literature. The paper also extends to countable free products and finite-sum decompositions, providing a unified framework for assembling complex free-product structures from simpler pieces. These contributions deepen the understanding of free product phenomena, rescalings, and non-separable von Neumann algebra invariants.
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.