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Operator algebras in dynamical systems. The theory of unbounded derivations in $$C^*$$-algebras. (English) Zbl 0743.46078
Encyclopedia of Mathematics and Its Applications. 41. Cambridge etc.: Cambridge University Press. xi, 219 p. (1991).
The contents of this monograph are based in large parts on the author’s lecture notes on the theory of unbounded derivations in $$C^*$$-algebras given at the Universities of Copenhagen and Newcastle upon Tyne in 1977. The material is concentrated on topics involving local quantum field theory, quantum statistical mechanics and differentiations on commutative manifolds [with respect to non-commutative manifolds consult the book of O. Bratteli, Derivations, dissipations and group actions on $$C^*$$-algebras, Lect. Notes Math. 1229, Berlin (1986; Zbl 0607.46035)].
Some elementary facts of $$C^*$$-algebras and $$W^*$$-algebras are stated in the first chapter, while the second chapter is devoted to the application of bounded derivations on $$C^*$$-algebras, especially to uniformly continuous dynamical systems and $$C^*$$-dynamical systems. Ground states and some special unbounded derivations arising in local quantum field theory are also discussed in chapter 2.
The third chapter is concentrated on the general theory of unbounded derivations, their closability, the domain problem of closed derivations and the theory of generators. A detailed discussion of the properties of unbounded closable $$^*$$-derivations in commutative $$C^*$$-algebras, especially derivations in $$C([0,1])$$ and transformation groups in commutative $$C^*$$-algebras as $$C^*$$-dynamical systems are closing the third chapter.
Chapter 4 is devoted to $$C^*$$-dynamical systems arising in quantum statistical mechanics, a unified axiomatic treatment of quantum lattice systems and quasi-free dynamics in Fermionic field theory as $$C^*$$- dynamical systems is discussed. In these systems the time evolution, ground states and thermal equilibrium states (KMS states), stability under bounded derivations and the problem of phase transitions are treated in detail, especially approximate inner $$C^*$$-dynamical systems and the Power-Sakai-conjecture are investigated. Finally the author generalizes the notion of $$C^*$$-dynamical systems to a wide class of interacting models in continuous quantum statistical systems and he discusses the existence of time evolutions and KMS states.
Reviewer: U.Grimmer (Berlin)

##### MSC:
 46L60 Applications of selfadjoint operator algebras to physics 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 82-02 Research exposition (monographs, survey articles) pertaining to statistical mechanics 46L57 Derivations, dissipations and positive semigroups in $$C^*$$-algebras 46L55 Noncommutative dynamical systems 82B26 Phase transitions (general) in equilibrium statistical mechanics 82B10 Quantum equilibrium statistical mechanics (general) 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics