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

TL;DR

This paper proves that for non-separable abelian von Neumann algebras, their free powers are mutually non-isomorphic and have trivial fundamental group, resolving a non-separable case of the free group factor problem.

Contribution

It establishes the non-isomorphism and trivial fundamental group of free powers of non-separable abelian von Neumann algebras, addressing a key open problem.

Findings

Mutual non-isomorphism of free powers for non-separable algebras

Trivial fundamental group of these free powers

Resolution of the non-separable free group factor problem

Abstract

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