Integration as a generalization of the integral operator

Document Type : Research Paper


1 Department of Mathematics, University of Birjand, Birjand, Iran

2 Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad


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.