Integration as a generalization of the integral operator

Abstract

Let A be an algebra. A derivation on A is a linear mapping δ : A → A such that δ(ab) = δ(a)b+aδ(b) for every a, b ∈ A. As a dual to this notion, we consider a linear mapping ∆ : A → A with the property ∆(a)∆(b) = ∆(∆(a)b + a∆(b)) for every a, b ∈ A and we call it an integration. In this paper, we give some examples, counterexamples and facts concerning integrations on algebras. Furthermore, we state and prove a characterization for integrations on finite dimensional matrix algebras.