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.

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. |