On finite free Fisher information for eigenvectors of a modular operator

TL;DR

This paper investigates von Neumann algebras generated by eigenvectors of a modular operator with finite free Fisher information, revealing their type classification, properties of the centralizer, and conditions for fullness.

Contribution

It establishes that such algebras have a centralizer that is a $ ext{II}_1$ factor and classifies the algebra type based on eigenvalues, also analyzing properties like property $ ext{Gamma}$ and fullness.

Findings

Centralizer $M^$ is a $ ext{II}_1$ factor.

$M$ is either type $ ext{II}_1$ or type $ ext{III}_l$ depending on eigenvalues.

$M^$ has trivial relative commutant and lacks property $ ext{Gamma}$.

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