Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 5

Special issue on “Subfactors and Related Topics” in memory of Vaughan Jones

Guest Editors: Dietmar Bisch, Arthur Jaffe, Yasuyuki Kawahigashi, and Zhengwei Liu

A note on continuous entropy

Pages: 2501 – 2523

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n5.a5

Authors

Roberto Longo (Dipartimento di Matematica, Università di Roma “Tor Vergata”, Rome, Italy)

Edward Witten (Institute for Advanced Study, School of Natural Sciences, Princeton, New Jersey, U.S.A.)

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.

2010 Mathematics Subject Classification

Primary 46L10, 94A17. Secondary 81T05.

Received 5 February 2022

Accepted 27 April 2022

Published 30 January 2024