A note on continuous entropy

TL;DR

The paper extends the notion of entropy to semifinite von Neumann algebras via Segal's construction and relates it to Araki's relative entropy $S(\varphi||\tau)$. It develops the bound $S(\varphi||\varphi\circ\varepsilon) \le \log[\mathcal{A}:\mathcal{B}]_\varepsilon$ for trace-preserving inclusions and shows that the bound equals the Jones index in the infinite-dimensional case. Using Kosaki's variational formula, Connes' spatial derivative, and the Pimsner–Popa inequality, the authors treat finite, semifinite, and Type II settings in a unified framework. The results connect entropy renormalization in quantum field theory to subfactor index theory, offering a rigorous quantitative tool for entropy production in noncommutative probability spaces.

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.

Theorems & Definitions (21)

• Lemma 2.1
• Proposition 2.2
• Proposition 2.3
• Corollary 2.4
• Proposition 2.5
• Proposition 2.6
• Proposition 2.7
• Lemma 3.1
• Proposition 3.2
• Proposition 3.3