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

Abstract

For a self-symmetric tracial von Neumann algebra $A$, we study rescalings of $A^{\ast n}\ast L{\mathbb{F}}_r$ 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 $1-s<r\le \infty$. 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 $(A^{\ast n}{)}^t$ for $0<t<1$, and the Murray-von Neumann fundamental group of $A^{\ast \infty }$. When $A$ is also non-separable and abelian, this answers two questions in Section 4.3 of recent work of Boutonnet-Drimbe-Ioana-Popa.

21 pages