Integration as a generalization of the integral operator

Let \mathfrak{A} be an algebra. A \textit{derivation} on \mathfrak{A} is a linear mapping \delta:\mathfrak{A}\to\mathfrak{A} such that \delta(ab)=\delta(a)b+a\delta(b) for every a,b\in\mathfrak{A}. As a dual to this notion, we consider a linear mapping \Delta:\mathfrak{A}\to\mathfrak{A} with the property \Delta(a)\Delta(b)=\Delta(\Delta(a)b+a\Delta(b)) for every a,b\in\mathfrak{A} and we call it an \textit{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.

Keywords:

Derivation , integration , C*-algebra , matrix algebra