On finite free Fisher information for eigenvectors of a modular operator

Abstract: Suppose $M$ is a von Neumann algebra equipped with a faithful normal state $\varphi$ and generated by a finite set $G=G^*$, $|G|\geq 2$. We show that if $G$ consists of eigenvectors of the modular operator $\Delta_\varphi$ with finite free Fisher information, then the centralizer $M^\varphi$ is a $\mathrm{II}1$ factor and $M$ is either a type $\mathrm{II}_1$ factor or a type $\mathrm{III}\lambda$ factor, $0<\lambda\leq 1$, depending on the eigenvalues of $G$. Furthermore, $(M^\varphi)'\cap M=\mathbb{C}$, $M^\varphi$ does not have property $\Gamma$, and $M$ is full provided it is type $\mathrm{III}_\lambda$, $0<\lambda<1$.