Noncommutative Dynamics and E-Semigroups

These days, the term Noncommutative Dynamics has several interpretations. It is used in this book to refer to a set of phenomena associated with the dynamical evo lution 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 non commutative algebra of observables, and we focus primarily on the most concrete case in which that algebra consists of all bounded operators on a Hilbert space. If one introduces a natural causal structure into such a dynamical system, then a pair of one-parameter semigroups of endomorphisms emerges, and it is useful to think of this pair as representing the past and future with respect to the given causality. These are both Eo-semigroups, and to a great extent the problem of understanding such causal dynamical systems reduces to the problem of under standing Eo-semigroups. The nature of these connections is discussed at length in Chapter 1. The rest of the book elaborates on what the author sees as the impor tant aspects of what has been learned about Eo-semigroups during the past fifteen years. Parts of the subject have evolved into a satisfactory theory with effective toolsj other parts remain quite mysterious. Like von Neumann algebras, Eo-semigroups divide naturally into three types: 1,11,111.

List of contents

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.

Summary

These days, the term Noncommutative Dynamics has several interpretations. It is used in this book to refer to a set of phenomena associated with the dynamical evo lution 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 non commutative algebra of observables, and we focus primarily on the most concrete case in which that algebra consists of all bounded operators on a Hilbert space. If one introduces a natural causal structure into such a dynamical system, then a pair of one-parameter semigroups of endomorphisms emerges, and it is useful to think of this pair as representing the past and future with respect to the given causality. These are both Eo-semigroups, and to a great extent the problem of understanding such causal dynamical systems reduces to the problem of under standing Eo-semigroups. The nature of these connections is discussed at length in Chapter 1. The rest of the book elaborates on what the author sees as the impor tant aspects of what has been learned about Eo-semigroups during the past fifteen years. Parts of the subject have evolved into a satisfactory theory with effective toolsj other parts remain quite mysterious. Like von Neumann algebras, Eo-semigroups divide naturally into three types: 1,11,111.

Additional text

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)

Report

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)

Product details

• Authors: William Arveson
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 01.03.2012
• EAN: 9781441918031
• ISBN: 978-1-4419-1803-1
• No. of pages: 434
• Dimensions: 156 mm x 23 mm x 234 mm
• Weight: 675 g
• Illustrations: X, 434 p.
• Series: Springer Monographs in Mathematics; Springer Monographs in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; Algebra, B, Field, Mathematics and Statistics, Commutative algebra, Operator Theory, perturbation theory, spectral theory, perturbation, index theory, semigroup, automorphism