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

We prove that if $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n(n>1)$ are selfadjoints in a $W^{\ast }$-probability space with finite non-microstates free Fisher information, then the von Neumann algebra $W^{\ast }(X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n)$ they generate doesn’t have property $\mathrm{\Gamma }$ (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.