Integration in Abelian C^*-Dynamical Systems

Abstract

Let \mathcal{A}=C₀(X) be an abelian C^*-algebra and t∈R{\mapsto}σ_t a strongly continuous group of *-automorphisms with generator δ₀. We consider derivations δ=λδ₀, where λ is a multiplication operator on C₀(X), and establish conditions on λ which ensure that δ has a unique generator extension. As a corollary we deduce that each derivation δ from ∩_n{≥}1D(δ₀ⁿ) into D(δ₀) 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 \mathcal{A}. Some results on local dissipations are also given.