Noncommutative Dynamics and E-Semigroups

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

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.

• C*-algebra
• Hilbert space
• Mathematica
• algebra
• automorphism
• commutative algebra
• field
• index theory
• perturbation
• perturbation theory
• semigroup
• spectral theory

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)

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

  William Arveson

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