Relative and quasi-entropies in semifinite von Neumann algebras

Abstract

We prove that Araki’s relative entropy equals Umegaki’s information between normal states on semifinite von Neumann algebras, as well as derive formulae for quasi-entropies and Rényi’s relative entropy in semifinite von Neumann algebras, generalizing thus the results known in finite dimension. Our investigations rest on a thorough analysis of the relative modular operator in such an algebra.