Relative modular operator in semifinite von Neumann algebras and its use

TL;DR

This paper explores the properties of the relative modular operator in semifinite von Neumann algebras, establishing formulas for trace, entropy equivalences, and quasi-entropies, extending finite-dimensional results to infinite-dimensional settings.

Contribution

It introduces new results on the relative modular operator in semifinite von Neumann algebras, linking various entropy measures and trace formulas.

Findings

Established basic trace formulae for semifinite von Neumann algebras.

Proved the equivalence between Araki's relative entropy and Umegaki's information.

Derived formulas for quasi-entropies and Rényi's relative entropy in this setting.

Abstract

We present some results concerning the relative modular operator in semifinite von Neumann algebras. These results allow one to prove some basic formula for trace, to obtain equivalence between Araki's relative entropy and Umegaki's information as well as to derive some formulae for quasi-entropies, and R\'enyi's relative entropy known in finite dimension.