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Noncommutative measures and Lp and Orlicz Spaces, with Applications to Quantum Physics
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Series: Oxford Graduate Texts in Mathematics Author: Goldstein, Stanislaw (Professor, Head of Department of Applied Computer Science, Professor, Head of Department of Applied Computer Science, University of Lodz, Faculty of Mathematics and Computer Science) / Labuschagne, Louis (Director: Focus Area for Pur Publisher: Oxford University Press ISBN: 9780198950202 Cover: HARDCOVER Date: 2025年06月 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'etre 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 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:???? 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: ???? and Orlicz spaces for general von Neumann algebras Part 4: Advanced Theory and Applications 11:Complex interpolation of noncommutative ???? spaces 12:Extensions of maps to ????(M) spaces and applications 13:Haagerup's reduction theorem 14:Applications to quantum physics
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