An introduction to KMS weights

TL;DR

This paper provides a comprehensive introduction to KMS weights, covering fundamental theorems, their proofs, and recent developments, with minimal prerequisites and connections to modular theory of von Neumann algebras.

Contribution

It offers a detailed exposition of KMS weights, including new results and recent collaborations, bridging foundational theorems with advanced modular theory applications.

Findings

Proofs of fundamental theorems requiring minimal prerequisites

Presentation of recent results by the author and collaborators

Discussion of factor types for KMS weights and states

Abstract

' The theory of KMS weights is based on a theorem of Combes and a theorem of Kustermans. In applications to KMS states for flows on a unital $C^*$-algebra the relation to KMS weights of the stabilized algebra has proved useful and this relation hinges on a theorem of Laca and Neshveyev. The first three chapters present proofs of these fundamental results that require a minimum of prerequisites; in particular, they do not depend on the modular theory of von Neumann algebras. In contrast, starting with chapter four the presented material draws heavily on the modular theory of von Neumann algebras. Most results are known from the work of N. V. Pedersen, J. Quaegebeur, J. Verding, J. Kustermans, S. Vaes, A. Kishimoto, A. Kumjian and J. Christensen, but new ones begin to surface. In chapter nine and the Appendices D and E the reader can find a presentation of results obtained recently by the author, partly in collaboration with G. A. Elliott and Y. Sato. This material is a natural culmination of methods developed around 1980 by Bratteli, Elliott, Herman and Kishimoto. Finally, in chapter ten there is a short presentation of the notion of factor types for KMS weights and states.