Neshveyev / Størmer Dynamical Entropy in Operator Algebras

1. Auflage 2006
ISBN: 978-3-540-34673-9
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 296 Seiten

The book addresses mathematicians and physicists, including graduate students, who are interested in quantum dynamical systems and applications of operator algebras and ergodic theory. It is the only monograph on this topic. Although the authors assume a basic knowledge of operator algebras, they give precise definitions of the notions and in most cases complete proofs of the results which are used.

Autoren/Hrsg.

Neshveyev, Sergey

Størmer, Erling

Inhaltsverzeichnis

1;Preface;5
2;Contents;7
3;General Theory;10
3.1;1 Classical Dynamical Systems;11
3.1.1;1.1 Measure Entropy;11
3.1.2;1.2 Topological Entropy;17
3.1.3;1.3 Entropy via Partitions of Unity;19
3.1.4;1.4 Notes;22
3.2;2 Relative Entropy;23
3.2.1;2.1 Relative Entropy for Matrix Algebras;23
3.2.2;2.2 Von Neumann Entropy;29
3.2.3;2.3 Relative Entropy for General C*-algebras;34
3.2.4;2.4 Notes;38
3.3;3 Dynamical Entropy;40
3.3.1;3.1 Mutual Entropy;40
3.3.2;3.2 Entropy of Dynamical Systems;55
3.3.3;3.3 Type I Algebras;61
3.3.4;3.4 Notes;64
3.4;4 Maximality of Entropy and Commutativity;68
3.4.1;4.1 Maximal Entropy of Subalgebras;68
3.4.2;4.2 Independent Algebras;73
3.4.3;4.3 Entropic K-systems;76
3.4.4;4.4 Notes;81
3.5;5 Dynamical Abelian Models;83
3.5.1;5.1 Entropy via Stationary Couplings;83
3.5.2;5.2 Zero Entropy Systems;92
3.5.3;5.3 Notes;97
3.6;6 Topological Entropy;99
3.6.1;6.1 Rank of a Completely Positive Approximation;99
3.6.2;6.2 Topological Entropy;102
3.6.3;6.3 Notes;109
3.7;7 Dynamics on the State Space;113
3.7.1;7.1 Measure Entropy;113
3.7.2;7.2 Topological Entropy;116
3.7.3;7.3 Notes;125
3.8;8 Crossed Products;127
3.8.1;8.1 Crossed Products by Discrete Amenable Groups;127
3.8.2;8.2 Generalizations;131
3.8.3;8.3 Notes;136
3.9;9 Variational Principle;139
3.9.1;9.1 Pressure;139
3.9.2;9.2 The variational principle;144
3.9.3;9.3 KMS-states;150
3.9.4;9.4 Notes;157
4;Special Topics;160
4.1;10 Relative Entropy and Subfactors;161
4.1.1;10.1 Relative Entropy;161
4.1.2;10.2 Index of Subfactors;168
4.1.3;10.3 Generators and Relative Entropy;171
4.1.4;10.4 The Canonical Shift;175
4.1.5;10.5 Shifts on Temperley-Lieb Algebras;184
4.1.6;10.6 Notes;187
4.2;11 Systems of Algebraic Origin;189
4.2.1;11.1 Twisted Group C*-algebras;189
4.2.2;11.2 Estimates of Topological Entropy;191
4.2.3;11.3 K-systems;198
4.2.4;11.4 Zero Entropy Systems;205
4.2.5;11.5 Automorphisms of Noncommutative Tori;210
4.2.6;11.6 Notes;213
4.3;12 Binary Shifts;215
4.3.1;12.1 The C*-algebra of a Bitstream;215
4.3.2;12.2 Entropy of Binary Shifts;221
4.3.3;12.3 Notes;228
4.4;13 Bogoliubov Automorphisms;230
4.4.1;13.1 Canonical Anticommutation Relations;230
4.4.2;13.2 Topological Entropy;232
4.4.3;13.3 Classical Bernoullian Subsystems;236
4.4.4;13.4 Dynamical Entropy;241
4.4.5;13.5 Notes;251
4.5;14 Free Products;253
4.5.1;14.1 Free Products of Algebras and Maps;253
4.5.2;14.2 Free Shifts;256
4.5.3;14.3 Free Product Automorphisms;259
4.5.4;14.4 Notes;265
5;A Completely Positive Maps;267
6;B Operator Inequalities;272
7;C Direct Integrals;276
8;References;279
9;List of Symbols;290
10;Index;293