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Edited by: Geoffrey L. Price, Editor-in-Chief and B. Mitchell Baker, U.S. Naval Academy, Annapolis, MD, and Palle E.T. Jorgensen and Paul S. Muhly, University of Iowa, Iowa City, IA
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Contemporary Mathematics
2003; 328 pp; softcover
Volume: 335
ISBN-10: 0-8218-3215-8
ISBN-13: 978-0-8218-3215-8
List Price: US$92 Member Price: US$73.60
Order Code: CONM/335

This volume contains the proceedings of the conference on Advances in Quantum Dynamics. The purpose of the conference was to assess the current state of knowledge and to outline future research directions of quantum dynamical semigroups on von Neumann algebras.

Since the appearance of the landmark papers by F. Murray and J. von Neumann, On the Rings of Operators, von Neumann algebras have been used as a mathematical model in the study of time evolution of quantum mechanical systems. Following the work of M. H. Stone, von Neumann, and others on the structure of one-parameter groups of unitary transformations, many researchers have made fundamental contributions to the understanding of time-reversible dynamical systems. This book deals with the mathematics of time-irreversible systems, also called dissipative systems. The time parameter is the half-line, and the transformations are now endomorphisms as opposed to automorphisms.

For over a decade, W. B. Arveson and R. T. Powers have pioneered the effort to understand the structure of irreversible quantum dynamical systems on von Neumann algebras. Their papers in this volume serve as an excellent introduction to the theory. Also included are contributions in other areas which have had an impact on the theory, such as Brownian motion, dilation theory, quantum probability, and free probability.

The volume is suitable for graduate students and research mathematicians interested in the dynamics of quantum systems and corresponding topics in the theory of operator algebras.

Graduate students and research mathematicians interested in the dynamics of quantum systems and corresponding topics in the theory of operator algebras.

• W. Arveson -- Four lectures on noncommutative dynamics
• R. T. Powers -- Construction of $$E_0$$-semigroups of $$\mathfrak B(\mathfrak h)$$ from $$CP$$-flows
• B. V. R. Bhat -- Atomic dilations
• F. Cipriani and J.-L. Sauvageot -- Strong solutions to the Dirichlet problem for differential forms: A quantum dynamical semigroup approach
• D. E. Evans and P. R. Pinto -- Modular invariants and their fusion rules
• R. Floricel -- A decomposition of $$E_0$$-semigroups
• R. Gohm -- A duality between extension and dilation
• I. Hirshberg and J. Zacharias -- On the structure of spectral algebras and their generalizations
• Y. Katayama and M. Takesaki -- Outer actions of a countable discrete amenable group on an AFD factor
• T. Katsura -- A construction of $$C^*$$-algebras from $$C^*$$-correspondences
• Y. Kawahigashi -- Classification of operator algebraic conformal field theories
• A. Kishimoto -- Rohlin property for flows
• C. Köstler -- Survey on a quantum stochastic extension of Stone's theorem
• D. Markiewicz -- Quantized convolution semigroups
• P. S. Muhly and B. Solel -- A model for quantum Markov semigroups
• T. Oikhberg, H. P. Rosenthal, and E. Størmer -- A predual characterization of semi-finite von Neumann algebras
• S. Sakai -- Pure states on $$C^*$$-algebras
• M. Skeide -- Commutants of von Neumann modules, representations of $$\mathcal{B}^a(E)$$ and other topics related to product systems of Hilbert modules
• R. Speicher -- Non-commutative Brownian motions
• B. Tsirelson -- Non-isomorphic product systems