The theory of noncommutative Haagerup 𝐿𝑝 and Orlicz spaces is an important tool in both Quantum Harmonic Analysis and Mathematical Physics. Indeed, noncommutativity is arguably the raison-d'être of the Heisenberg approach to quantum mechanics. Just as classical analysis formed the foundation for classical mechanics, a mature response to the challenges posed by quantum mechanics (from the Heisenberg perspective) similarly needs to be built on a well-developed foundation of noncommutative analysis.
In the passage from the classical to the quantum setting, functions get replaced with (possibly noncommuting) operators. Von Neumann himself realised early on that some sort of noncommutative integral calculus tailored to this setting is therefore needed to meet this challenge. This book seeks to help address this need. The noncommutative Orlicz spaces presented here help in dealing with observable quantities and entropy.
Goldstein and Labuschagne provide a detailed account of the current theories in a way that is useful and accessible to a wide range of readers, from graduate students to advanced users. Beginning with some foundational examples intended to build intuition for the theory to follow, including the theory of noncommutative decreasing arrangements, as developed by Fack and Kosaki, and of Orlicz spaces for general von Neumann algebras. The authors then present the theory of the more accessible tracial case, followed by that of the more demanding general (type III) case. The final part of the book is devoted to advanced theory and applications.
Preface
Introduction
Preliminaries
Part 1: Foundational Examples
1: Abelian von Neumann algebras
2: The Schatten-von Neumann classes
Part 2: Tracial case
3: Noncommutative measure theory U+02014 tracial case
4: Weights and densities
5: Basic theory of decreasing rearrangements
6: 𝐿𝑝 and Orlicz spaces in the tracial case
7: Real interpolation and monotone spaces
Part 3: General case
8: Basic elements of modular theory
9: Crossed products
10: Lp: 𝐿𝑝 and Orlicz spaces for general von Neumann
algebras
Part 4: Advanced Theory and Applications
11: Complex interpolation of noncommutative 𝐿𝑝 spaces
12: Extensions of maps to 𝐿𝑝(M) spaces and applications
13: Haagerup's reduction theorem
14: Applications to quantum physics
Bibliography
Notation Index
Subject Index
Stanisaw Goldstein began working at the University of Lodz in 1977 and continues there till this day, holding a chair at the Faculty of Mathematics and Computer Science. He earned his PhD in 1978 and became a full professor in 2001. In 1989-1990 he spent a year in Bielefeld-Bochum-Stochastik, King's College London and Nottingham University as a Humboldt Fellow. His research interests are primarily operator algebras and noncommutative measure theory.
Louis Labuschagne obtained his PhD in 1988 in the field of Single Linear Operator theory. He started his professional academic career in the same year at Stellenbosch University, moving to the University of Pretoria in 1992. This move also coincided with a shift in his research focus to Operator Algebras and their application to Quantum Theory. After spending 19 years in Pretoria, first at the University of Pretoria and then UNISA from 2001, he took up an appointment at North-West University in January 2011, where he currently serves as director of the Focus Area for Pure and Applied Analytics.
