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Dynamical Entropy in Operator Algebras

:	Sergey Neshveyev, Erling Størmer
:	Dynamical Entropy in Operator Algebras
:	Springer-Verlag
:	9783540346739
:	1
:	CHF 104.70
:

:	Analysis
:	English

:	296
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

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.

	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