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

Abstract

We prove that if $X_1,...,X_n(n>1)$ are selfadjoints in a $W^{\ast }$-probability space with finite non-microstates free Fisher information, then the von Neumann algebra $W^{\ast }(X_1,...,X_n)$ 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.

12 pages; New results with similar techniques; cf. abstracts for details

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