**Contemporary Mathematics**

Volume: 335;
2003;
328 pp;
Softcover

MSC: Primary 46; 47; 60;

Print ISBN: 978-0-8218-3215-8

Product Code: CONM/335

List Price: $92.00

Individual Member Price: $73.60

**Electronic ISBN: 978-0-8218-7925-2
Product Code: CONM/335.E**

List Price: $92.00

Individual Member Price: $73.60

# Advances in Quantum Dynamics

Share this page *Edited by *
*Geoffrey L. Price; B. Mitchell Baker; Palle E.T. Jorgensen; Paul S. Muhly*

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

# Table of Contents

## Advances in Quantum Dynamics

- Contents v6 free
- Preface vii8 free
- Four lectures on noncommutative dynamics 110 free
- Construction of E0-semigroups of B(h) from CP-flows 5766
- Atomic dilations 99108
- Strong solutions to the Dirichlet problem for differential forms: A quantum dynamical semigroup approach 109118
- Modular invariants and their fusion rules 119128
- A decomposition of E0-semigroups 131140
- A duality between extension and dilation 139148
- On the structure of spectral algebras and their generalizations 149158
- Outer actions of a countable discrete amenable group on an AFD factor 163172
- A construction of C* -algebras from C* -correspondences 173182
- Classification of operator algebraic conformal field theories 183192
- Rohlin property for flows 195204
- Survey on a quantum stochastic extension of Stone's theorem 209218
- Quantized convolution semigroups 223232
- A model for quantum Markov semigroups 235244
- A predual characterization of semi-finite von Neumann algebras 243252
- Pure states on C* -algebras 247256
- Commutants of von Neumann modules, representations of Ba(E) and other topics related to product systems of Hilbert modules 253262
- Non-commutative Brownian motions 263272
- Non-isomorphic product systems 273282
- Introduction 274283
- 1. Basic notions 274283
- 2. Some invariants 281290
- 3. Continuous products of measure classes 287296
- 4. Continuous products of probability spaces 290299
- 5. Random sets, and type I I0 291300
- 6. Constructing random sets 294303
- 7. Time reversal 301310
- 8. FHS space: Logarithm of a Hilbert space 303312
- 9. Continuous sums and off-white noises 308317
- 10. Type III 312321
- 11. The invariant via the logarithm 313322
- 12. Ensuring asymptotic orthogonality 318327
- 13. Calculating the invariant 320329
- References 326335
- Index 327336 free

#### Readership

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