Non-isomorphism of $A^{\ast n},2\le n\le \infty$, for a non-separable abelian von Neumann algebra $A$

Abstract

We prove that if $A$ is a non-separable abelian tracial von Neuman algebra then its free powers $A^{\ast n},2\le n\le \infty$, are mutually non-isomorphic and with trivial fundamental group, $\mathcal{F}(A^{\ast n})=1$, whenever $2\le n<\infty$. This settles the non-separable version of the free group factor problem.