Automatic continuity of new generalized derivations

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be two algebras and let $n$ be a positive integer. A linear mapping $D:\mathcal{A}\to \mathcal{B}$ is called a \emph{strongly generalized derivation of order $n$} if there exist families of linear mappings $\{E_k:\mathcal{A}\to \mathcal{B}{\}}_{k=1}^n$, $\{F_k:\mathcal{A}\to \mathcal{B}{\}}_{k=1}^n$, $\{G_k:\mathcal{A}\to \mathcal{B}{\}}_{k=1}^n$ and $\{H_k:\mathcal{A}\to \mathcal{B}{\}}_{k=1}^n$ which satisfy $D(ab)={\sum }_{k=1}^n\left[E_k(a)F_k(b)+G_k(a)H_k(b)\right]$ for all $a,b\in \mathcal{A}$. The purpose of this article is to study the automatic continuity of such derivations on Banach algebras and $C^{\ast }$-algebras.

15 pages