Advances in Quantum Dynamics: (Contemporary Mathematics)

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.

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