Noncommutative Dynamics and E-Semigroups

1. Auflage. Softcover version of original hardcover Auflage 2003
ISBN: 978-1-4419-1803-1
Verlag: Springer

Buch, Englisch, 434 Seiten, Previously published in hardcover, Format (B × H): 155 mm x 235 mm, Gewicht: 674 g

Reihe: Springer Monographs in Mathematics

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.

Zielgruppe

Research

Autoren/Hrsg.

Arveson, William

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Moderne Anwendungen der Analysis
• Mathematik | Informatik Mathematik Geometrie Dynamische Systeme
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

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.

Über Autor(innen)

Arveson, William
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).