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Noncommutative Dynamics and E-Semigroups

  • Book
  • © 2003

Overview

Authors:
  1. William Arveson

    1. Department of Mathematics, University of California at Berkeley, Berkeley, USA

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Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-x
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  2. Dynamical Origins

    1. Dynamical Origins

      • William Arveson
      Pages 1-15

  3. Index and Perturbation Theory

    1. Front Matter

      Pages 17-17
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    2. E-Semigroups

      • William Arveson
      Pages 18-65

    3. Continuous Tensor Products

      • William Arveson
      Pages 66-100

    4. Spectral C*-Algebras

      • William Arveson
      Pages 101-159

  4. Classification: Type I Cases

    1. Front Matter

      Pages 161-161
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    2. Path Spaces

      • William Arveson
      Pages 162-198

    3. Decomposable Product Systems

      • William Arveson
      Pages 199-234

  5. Noncommutative Laplacians

    1. Front Matter

      Pages 235-235
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    2. CP-Semigroups

      • William Arveson
      Pages 236-253

    3. C*-Generators and Dilation Theory

      • William Arveson
      Pages 254-303

    4. Index Theory for CP-Semigroups

      • William Arveson
      Pages 304-323

    5. Bounded Generators

      • William Arveson
      Pages 324-354

  6. Causality and Dynamics

    1. Front Matter

      Pages 355-355
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    2. Pure Perturbations of CAR/CCR Flows

      • William Arveson
      Pages 356-373

    3. Interaction Theory

      • William Arveson
      Pages 374-387

  7. Type III Examples

    1. Front Matter

      Pages 389-389
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    2. Powers’ Examples

      • William Arveson
      Pages 390-411

    3. Tsirelson—Vershik Product Systems

      • William Arveson
      Pages 412-426
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Keywords

About this book

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.

Reviews

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)

Authors and Affiliations

  • Department of Mathematics, University of California at Berkeley, Berkeley, USA

    William Arveson

Bibliographic Information

  • Book Title: Noncommutative Dynamics and E-Semigroups

  • Authors: William Arveson

  • Series Title: Springer Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-0-387-21524-2

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 2003

  • Hardcover ISBN: 978-0-387-00151-7Published: 12 May 2003

  • Softcover ISBN: 978-1-4419-1803-1Published: 14 December 2011

  • eBook ISBN: 978-0-387-21524-2Published: 06 November 2012

  • Series ISSN: 1439-7382

  • Series E-ISSN: 2196-9922

  • Edition Number: 1

  • Number of Pages: X, 434

  • Topics: Operator Theory

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