Noncommutative measures and Lp and Orlicz Spaces, with Applications to Quantum Physics
Series: Oxford Graduate Texts in Mathematics; 33;
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Product details:
- Publisher OUP Oxford
- Date of Publication 19 June 2025
- ISBN 9780198950202
- Binding Hardback
- No. of pages666 pages
- Size 37x156x234 mm
- Weight 1104 g
- Language English 700
Categories
Short description:
The theory of noncommutative Haagerup -------- and Orlicz spaces is an important tool in both Quantum Harmonic Analysis and Mathematical Physics. 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.
MoreLong description:
The theory of noncommutative Haagerup -------- and Orlicz spaces is an important tool in both Quantum Harmonic Analysis and Mathematical Physics. Indeed, noncommutativity is arguably the raison-d'?tre 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.
Table of Contents:
Preface
Introduction
Preliminaries
Part 1: Foundational Examples
Abelian von Neumann algebras
The Schatten-von Neumann classes
Part 2: Tracial case
Noncommutative measure theory U+02014 tracial case
Weights and densities
Basic theory of decreasing rearrangements
-------- and Orlicz spaces in the tracial case
Real interpolation and monotone spaces
Part 3: General case
Basic elements of modular theory
Crossed products
L^p: -------- and Orlicz spaces for general von Neumann algebras
Part 4: Advanced Theory and Applications
Complex interpolation of noncommutative -------- spaces
Extensions of maps to --------(M) spaces and applications
Haagerup's reduction theorem
Applications to quantum physics
Bibliography
Notation Index
Subject Index
