A Note about proving non-$\Gamma$ under a finite non-microstates free Fisher information Assumption

TL;DR

This paper demonstrates that in a W*-probability space, self-adjoint variables with finite non-microstates free Fisher information generate a von Neumann algebra lacking property Γ and is non-amenable, extending Voiculescu's results.

Contribution

It establishes that finite non-microstates free Fisher information implies the generated algebra is non-Γ and non-amenable, generalizing previous microstates entropy results.

Findings

Von Neumann algebra lacks property Γ

Algebra is non-amenable

Factoriality under finite non-microstates entropy

Abstract

We prove that if $X_{1},...,X_{n} (n >1)$ are selfadjoints in a $W^{*}$-probability space with finite non-microstates free Fisher information, then the von Neumann algebra $W^{*}(X_{1},...,X_{n})$ they generate doesn't have property $\Gamma$ (especially is not amenable). This is an analog of a well-known result of Voiculescu for microstates free entropy. We also prove factoriality under finite non-microstates entropy.