Dynamical Entropy in Operator Algebras (eBook)

Sergey Neshveyev, Erling Størmer (Autoren)

eBook Download: PDF

2006 | 2006
X, 296 Seiten
Springer Berlin (Verlag)
978-3-540-34673-9 (ISBN)

Infos

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.

Inhalt

Preface 5
Contents 7
General Theory 10
1 Classical Dynamical Systems 11
1.1 Measure Entropy 11
1.2 Topological Entropy 17
1.3 Entropy via Partitions of Unity 19
1.4 Notes 22
2 Relative Entropy 23
2.1 Relative Entropy for Matrix Algebras 23
2.2 Von Neumann Entropy 29
2.3 Relative Entropy for General C*-algebras 34
2.4 Notes 38
3 Dynamical Entropy 40
3.1 Mutual Entropy 40
3.2 Entropy of Dynamical Systems 55
3.3 Type I Algebras 61
3.4 Notes 64
4 Maximality of Entropy and Commutativity 68
4.1 Maximal Entropy of Subalgebras 68
4.2 Independent Algebras 73
4.3 Entropic K-systems 76
4.4 Notes 81
5 Dynamical Abelian Models 83
5.1 Entropy via Stationary Couplings 83
5.2 Zero Entropy Systems 92
5.3 Notes 97
6 Topological Entropy 99
6.1 Rank of a Completely Positive Approximation 99
6.2 Topological Entropy 102
6.3 Notes 109
7 Dynamics on the State Space 113
7.1 Measure Entropy 113
7.2 Topological Entropy 116
7.3 Notes 125
8 Crossed Products 127
8.1 Crossed Products by Discrete Amenable Groups 127
8.2 Generalizations 131
8.3 Notes 136
9 Variational Principle 139
9.1 Pressure 139
9.2 The variational principle 144
9.3 KMS-states 150
9.4 Notes 157
Special Topics 160
10 Relative Entropy and Subfactors 161
10.1 Relative Entropy 161
10.2 Index of Subfactors 168
10.3 Generators and Relative Entropy 171
10.4 The Canonical Shift 175
10.5 Shifts on Temperley-Lieb Algebras 184
10.6 Notes 187
11 Systems of Algebraic Origin 189
11.1 Twisted Group C*-algebras 189
11.2 Estimates of Topological Entropy 191
11.3 K-systems 198
11.4 Zero Entropy Systems 205
11.5 Automorphisms of Noncommutative Tori 210
11.6 Notes 213
12 Binary Shifts 215
12.1 The C*-algebra of a Bitstream 215
12.2 Entropy of Binary Shifts 221
12.3 Notes 228
13 Bogoliubov Automorphisms 230
13.1 Canonical Anticommutation Relations 230
13.2 Topological Entropy 232
13.3 Classical Bernoullian Subsystems 236
13.4 Dynamical Entropy 241
13.5 Notes 251
14 Free Products 253
14.1 Free Products of Algebras and Maps 253
14.2 Free Shifts 256
14.3 Free Product Automorphisms 259
14.4 Notes 265
A Completely Positive Maps 267
B Operator Inequalities 272
C Direct Integrals 276
References 279
List of Symbols 290
Index 293

Produkt-Details

• Erscheint lt. Verlag: 22.9.2006
• Reihe/Serie: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
• Reihe/Serie: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
• Zusatzinfo: X, 296 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: Algebra • C*-algebra • C*-algebras • differential equation • Dynamical Systems • Entropy • ergodic theory • Maximum • measure • von neumann algebras
• ISBN-10: 3-540-34673-2 / 3540346732
• ISBN-13: 978-3-540-34673-9 / 9783540346739