William Arveson

Noncommutative Dynamics and E-Semigroups (eBook, PDF)

• Format: PDF

Produktbeschreibung

The term Noncommutative Dynamics can be interpreted in several ways. It is used in this book to refer to a set of phenomena associated with the dynamics of quantum systems of the simplest kind that involve rigorous mathematical structures associated with infinitely many degrees of freedom. The dynamics of such a system is represented by a one-parameter group of automorphisms of a noncommutative algebra of observables, and the author focuses primarily on the most concrete case in which that algebra consists of all bounded operators on a Hilbert space.

This subject overlaps with several mathematical areas of current interest, including quantum field theory, the dynamics of open quantum systems, noncommutative geometry, and both classical and noncommutative probability theory. This is the first book to give a systematic presentation of progress during the past fifteen years on the classification of E-semigroups up to cocycle conjugacy. There are many new results that cannot be found in the existing literature, as well as significant reformulations and simplifications of the theory as it exists today.

William Arveson is Professor of Mathematics at the University of California, Berkeley. He has published two previous books with Springer-Verlag, An Invitation to C*-algebras (1976) and A Short Course on Spectral Theory (2001).

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Produktdetails

• Verlag: Springer New York
• Seitenzahl: 434
• Erscheinungstermin: 6. November 2012
• Englisch
• ISBN-13: 9780387215242
• Artikelnr.: 43998938

Inhaltsangabe

Preface Dynamical Origins Part 1: Index and Perturbation Theory E-semigroups Continuous Tensor Products Spectral C -algebras Part 2: Classification: Type I Cases Path Spaces DecomposableProduct Systems Part 3: Noncommutative Laplacians CP-semigroups C -Generators and Dilation Theory Index Theory for CP-Semigroups Bounded Generators Part 4: Causality and Dynamics PurePerturbations of CAR/CCR Flows Interaction Theory Part 5: Type IIIExamples Powers' Examples Tsirelson-Vershik Product Systems Bibliography Index

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1. Dynamical Origins.- 1.1. The Flow of Time in Quantum Theory.- 1.2. Causality and Interactions.- 1.3. Semigroups of Endomorphisms.- 1.4. Existence of Dynamics.- 1. Index and Perturbation Theory.- 2. E-Semigroups.- 3. Continuous Tensor Products.- 4. Spectral C*-Algebras.- 2. Classification: Type I Cases.- 5. Path Spaces.- 6. Decomposable Product Systems.- 3. Noncommutative Laplacians.- 7. CP-Semigroups.- 8. C*-Generators and Dilation Theory.- 9. Index Theory for CP-Semigroups.- 10. Bounded Generators.- 4. Causality and Dynamics.- 11. Pure Perturbations of CAR/CCR Flows.- 12. Interaction Theory.- 5. Type III Examples.- 13. Powers' Examples.- 14. Tsirelson-Vershik Product Systems.

Rezensionen

Aus den Rezensionen: "... Die vorliegende ausgezeichnete Monographie ist die erste, die sich diesem Thema widmet und die Entwicklung des Gebietes einschließlich der neuesten Ergebnisse zusammenfasst. Der Autor ist ein führender Experte ... und schließt mit dieser Publikation eine echte Lücke. ... Das Buch verfügt über ein ausführliches Inhalts- und Literaturverzeichnes [sic] und einen guten Index. Am Ende eines jeden der ... Kapitel finden sich wertvolle Hintergrundinformationen und Hinweise auf weiterführende Entwicklungen. ... die Konzepte werden sehr gut motiviert. Viele Beweise sind gegenüber den Orginalquellen [sic] erheblich vereinfacht ..." (J. Zacharias, in: Jahresbericht der Deutschen Mathematiker-Vereinigung, 2006, Vol. 108, Issue 4, S. 39 f.)

From the reviews:

"This book serves as a good introduction to the currently very active field of E0-semigroups. ... This reviewer thinks that this is a timely publication in an area that has advanced rapidly in recent years ... . the reviewer is happy to recommend this book." (Masamichi Takesaki, SIAM Review, Vol. 46 (4), 2004)

"The book is suitable for researchers and students who have had a first course in functional analysis ... . For traditional researchers in operator algebras the book should be of interest because it looks at problems in a new way ... . As someone who has been studying E0-semigroups for just shy of twenty years I am extremely pleased with the appearance of this valuable book, which I virtually carry with me." (Robert T. Powers, Mathematical Reviews, 2004 g)

"The author of this book is one of the great names in functional analysis ... . The book has clearly been written with the goal of teaching the subject; the exposition is crisp, clear, and will be accessible to graduate students. ... It is superbly written ... . In addition to being the essential reference for those working on E0-semigroups, this magnificent book will be useful and inspirational to a wide range of mathematicians and mathematical physicists." (D. P. Blecher, Proceedings of the Edinburgh Mathematical Society, Vol. 49, 2006)