Noncommutative Dynamics and E-Semigroups

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Summary: Dynamics, quantum theory, and operator algebras.
Review: The landmark papers by F. Murray and J. von Neumann introduced Rings of Operators (now called von Neumann algebras) as a model for quantum observables, and as a tool in mathematics. Thanks to the conservation laws of physics, one-parameter groups of automorphisms have been used sucessfully as a mathematical model for time evolution in quantum mechanical systems. This begins with work by Eugene Wigner. While the mathematical structure of one-parameter groups is now well understood, thanks to the theorems of Stone and von Neumann, it is only in recent years that systematic results have emerged for the different, but related dissipative or time-irreversible dynamical systems. The dissipative framework includes the current models for quantum information theory, e.g., decoherence due to build-up of errors in quantum channels. This book deals with the mathematics of these dissipative systems. The time parameter is the half-line, and the transformations are now endomorphisms as opposed to automorphisms. The issues include type-classification of the semigroups, and the tools are operator algebraic. A new class of C*-algebras, called spectral algebras, is introduced, and they are of independent interest.

For over a decade, W. B. Arveson and others have pioneered research on the structure of irreversible quantum dynamical systems on von Neumann algebras, and the related operator theory. This book serves as an excellent introduction to the theory. Also included are other areas which have had an impact on the theory, such as Brownian motion, dilation theory, quantum probability, and free probability.

The book is suitable for graduate students and research mathematicians interested in the dynamics of quantum systems and corresponding topics in the theory of operator algebras.
Review by Palle Jorgensen, September 2003.