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 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. © 2010 Elsevier Inc. All rights reserved