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Derivations, dissipations and group actions on $$C^ *$$-algebras. (English) Zbl 0607.46035
Lecture Notes in Mathematics, 1229. Berlin etc.: Springer-Verlag. VI, 277 p. DM 42.50 (1986).
Let G be a Lie group acting on a $$C^*$$-algebra A. Let $$A_ n$$, $$A_ F$$ be the subalgebras of n times G-differentiable and of G-finite elements respectively. The book under review is mainly concerned with the following two questions:
1. When does every $${}^*$$-derivation $$\delta$$ from $$A_ n$$ or $$A_ F$$ into A have a decomposition $$\delta =a_ 1\delta_ 1+a_ 2\delta_ 2+...+a_ k\delta_ k+{\tilde \delta}$$ where $$\delta_ 1,...,\delta_ k$$ is a basis for the Lie algebra of G, $$a_ 1,...,a_ k$$ are real functions on Prim A and $${\tilde \delta}$$ is approximately inner or bounded ?
2. Are all $${}^*$$-derivations from $$A_ F$$ to $$A_ F$$, or from $$A_{\infty}$$ to $$A_ 1$$ pregenerators ? ($$\delta$$ is a pregenerator if its closure generates a one-parameter automorphism group).
The author assembles a large number of positive results in special cases (e.g. A commutative or G compact $$+ additional$$ conditions) due to various authors. He also discusses the question of when a derivation that commutes with the action of G is a generator and studies analogs of the 2 questions above for dissipations in the place of derivations.
The book is very complete and gives a state of the art account of this subject which has been investigated by a fair number of mathematicians and is still under active research.
Reviewer: J.Cuntz

##### MSC:
 46L55 Noncommutative dynamical systems 47B47 Commutators, derivations, elementary operators, etc. 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 47B44 Linear accretive operators, dissipative operators, etc.