On finite free Fisher information for eigenvectors of a modular operator

Suppose $M$ is a von Neumann algebra equipped with a faithful normal state $\phi$ and generated by a finite set $G=G^{\ast }$, $\mathrm{\mid }G\mathrm{\mid }\ge 2$. We show that if $G$ consists of eigenvectors of the modular operator ${\mathrm{\Delta }}_{\phi }$ with finite free Fisher information, then the centralizer $M^{\phi }$ is a ${\mathrm{I}\mathrm{I}}_1$ factor and $M$ is either a type ${\mathrm{I}\mathrm{I}}_1$ factor or a type ${\mathrm{I}\mathrm{I}\mathrm{I}}_{\lambda }$ factor, $0<\lambda \le 1$, depending on the eigenvalues of $G$. Furthermore, $(M^{\phi }{)}^{\mathrm{\prime }}\cap M=\mathbb{C}$, $M^{\phi }$ does not have property $\mathrm{\Gamma }$, and $M$ is full provided it is type ${\mathrm{I}\mathrm{I}\mathrm{I}}_{\lambda }$, $0<\lambda <1$.