On finite free Fisher information for eigenvectors of a modular operator

Suppose M is a von Neumann algebra equipped with a faithful normal state φ and generated by a finite set G=G⁎, |G|≥2. We show that if G consists of eigenvectors of the modular operator Δφ with finite free Fisher information, then the centralizer Mφ is a II1 factor and M is either a type II1 factor or a type IIIλ factor, 0<λ≤1, depending on the eigenvalues of G. Furthermore, (Mφ)′∩M=C, Mφ does not have property Γ, and M is full provided it is type IIIλ, 0<λ<1.