A note about proving non-Γ under a finite non-microstates free Fisher information assumption

We prove that if X1,…,Xn (n>1) are self-adjoints in a W∗-probability space with finite non-microstates free Fisher information, then the von Neumann algebra W∗(X1,…,Xn) they generate doesn't have property Γ (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.