A note on continuous entropy

TL;DR

This paper explores the extension of von Neumann entropy to semifinite von Neumann algebras, relating it to relative entropy and establishing an optimal bound on entropy increase tied to the Jones index.

Contribution

It extends von Neumann entropy to semifinite von Neumann algebras and connects it with relative entropy, providing an optimal bound for entropy increase in factor inclusions.

Findings

Entropy increase bounded by the logarithm of the Jones index.

Extension of von Neumann entropy to semifinite von Neumann algebras.

Optimal bound achieved in infinite-dimensional cases.

Abstract

Von Neumann entropy has a natural extension to the case of an arbitrary semifinite von Neumann algebra, as was considered by I. E. Segal. We relate this entropy to the relative entropy and show that the entropy increase for an inclusion of von Neumann factors is bounded by the logarithm of the Jones index. The bound is optimal if the factors are infinite dimensional.