Integration In Abelian $C^*$-Dynamical Systems

Abstract

Let JL = C 0 (X) be an abelian C*-algebra and t^R^a t a strongly continuous group of ""-automorphisms with generator do. We consider derivations d = AO Q , where X is a multiplication operator on C 0 (X), and establish conditions on 2 w^hich ensure that a has a unique generator extension. As a corollary we deduce that each derivation d from r\ n^i D(8") into D(do) is closable and its closure is a generator. An analogous result is established for derivations defined on the smooth elements associated with the action of a compact Lie group on <JL Some results on local dissipations are also given. § 1. Introduction Our aim is to analyze derivations d defined on the smooth elements of an abelian C*-dynamical system as generators of Co-groups of ^-automorphisms. The basic question is whether d has a unique generator extension, and then a subsidiary problem is to relate this extension to d. Since each abelian C*-dynamical system determines a topological dynamical system, based on the spectrum of the abelian algebra, this question can be viewed as a problem of integration in topological dynamics. This approach appears particularly useful. In order to be more precise we introduce the following definition and notation. Let (JL, R, a] denote an abelian C*-dynamical system. Thus J. is an abelian O-algebra and t^R^a t^A utJi is a strongly continuous one-parameter group of ^-automorphisms of JL Next let d 0 denote the generator of a and define JL n =D(dft, and Communicated by H. Araki, April 1, 1985. t Permanent Address:

Smooth derivations on abelian C*-dynamical systems

“…Second the differentiability assumption, X e D(8 0 ), which was necessary in Theorem 2.12 of [3] in order to identify the unique generator extension of X8 0 with its closure, is not necessary in the present context. The advantage of the present result is that it suffices for the discussion of derivations from A^ -> A 1 / 2 , it is considerably easier to prove than the analogous results of [3], and it has potential extensions to non-abelian systems.…”

“…It is only used to ensure that 8 is densely defined. Although the following result could be partly deduced from Theorems 2.6 and 2.10 of [3], we give an almost independent proof based on resolvent convergence arguments. But to apply this technique it is convenient to use a density result from [3] which essentially allows one to avoid high frequencies.…”

“…Finally, in [3] it was proved that if a maps A^ into A 1( then 8 is automatically the generator of a strongly continuous one-parameter group of *-automorphisms. The principal aim of this paper is to generalize and optimize this last statement.…”

“…But first note that since \\(o t f -f)/t\\ < 2||/||/|f|, one can equally well define A 1/2 by e A; sup \\(a,f-f)/t\\< +00 of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700028238 [3] Abelian C*-dynamical systems 249…”

“…Our aim is to continue the investigation [1], [2], [3] of the structure of *-derivations from A x into A. To describe the elements of this investigation, it is necessary to introduce a number of additional concepts.…”