On finite free Fisher information for eigenvectors of a modular operator

Brent Nelson

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arXiv · 1604.03900

On finite free Fisher information for eigenvectors of a modular operator

Brent Nelson

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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 $\mathrm{III}_\lambda$, $0<\lambda<1$.

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Brent Nelson. 2016-04-13. On finite free Fisher information for eigenvectors of a modular operator. https://arxiv.org/abs/1604.03900

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