This book is concerned with the theory of unbounded derivations in C*-algebras, a subject whose study was motivated by questions in quantum physics and statistical mechanics, and to which the author has made considerable contributions. This is an active area of research, and one of the most ambitious aims of the theory is to develop quantum statistical mechanics within the framework of C*-theory. The presentation concentrates on topics involving quantum statistical mechanics and differentiations on manifolds. One of the goals is to formulate the absence theorem of phase transitions in its most general form within the C* setting. For the first time, the author globally constructs, within that setting, derivations for a fairly wide class of interacting models, and presents a new axiomatic treatment of the construction of time evolutions and KMS states.
Author(s): Shtichirt Sakai
Series: Encyclopedia of Mathematics and its Applications
Publisher: CUP
Year: 1991
Language: English
Pages: 232
Cover......Page 1
About......Page 2
Encyclopedia of Mathematics and its Applications 41......Page 4
Operator Algebras in Dynamical Systems: The theory of unbounded derivations in C*-algebras......Page 6
Goto 6 /FitH 555521400961......Page 7
Contents......Page 8
Preface......Page 10
1.1 Banach algebras, C*-algebras and W*-algebras......Page 14
1.5 Positive linear functionals and states......Page 15
1.7 Concrete C*- and W*-algebras......Page 16
1.11 Gelfand-Naimark-Segal representations......Page 18
1.12 Factorial and pure states......Page 19
1.15 Perturbation expansion theorem (t in mathbb{R})......Page 20
1.16 Corollary (the complex version)......Page 22
1.17 Theorem (convergence on geometric vectors)......Page 24
1.18 Proposition (the restricted C*-system)......Page 27
2 Bounded derivations......Page 29
2.2 The commutation relation ab - ba= 1......Page 30
2.3 Continuity of everywhere-defined derivations......Page 35
2.4 Quantum field-theoretic observations (some unbounded derivations)......Page 37
2.5 Application to bounded derivations......Page 47
2.6 Uniformly continuous dynamical systems......Page 54
2.7 C*-dynamical systems and ground states......Page 61
3 Unbounded derivations......Page 68
3.1 Definition of derivations......Page 69
3.2 Closability of derivations......Page 71
3.3 The domain of closed *-derivations......Page 78
3.4 Generators......Page 85
3.5 Unbounded derivations in commutative C*-algebras......Page 92
3.6 Transformation groups and unbounded derivations......Page 107
4.1 Approximately inner C*-dynamics......Page 114
4.2 Ground states......Page 120
4.3 KMS states......Page 127
4.4 Bounded perturbations......Page 143
4.5 UHF algebras and normal *-derivations......Page 168
4.6 Commutative normal *-derivations in UHF algebras......Page 181
4.7 Phase transitions......Page 188
4.8 Continuous quantum systems......Page 196
References......Page 220
Index......Page 230
