A Note about Proving Non-γ under a Finite Non-microstates Free Fisher Information Assumption

We prove that if X1, ..., Xn(n ≥ 2) are selfadjoints in a W ∗-probability space with finite nonmicrostates 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.