Strongly generalized derivations on C*-algebras

Abstract: Let $\mathcal{A}$ and $\mathcal{B}$ be two algebras, let $\mathcal{M}$ be a $\mathcal{B}$-bimodule and let $n$ be a positive integer. A linear mapping $D_n:\mathcal{A} \rightarrow \mathcal{M}$ is called a strongly generalized derivation of order $n$, if there exist the families $${E_k:\mathcal{A} \rightarrow \mathcal{M}}<em>{k = 1}<sup>{n}$$, $${H_k:\mathcal{A} \rightarrow \mathcal{M}}</em>{k = 1}<sup>{n}$$, $${F_k:\mathcal{A} \rightarrow \mathcal{B}}<em>{k = 1}<sup>{n}$$ and $${G_k:\mathcal{A} \rightarrow \mathcal{B}}</em>{k = 1}<sup>{n}$$ of mappings which satisfy $$D_n(ab) = \sum_{k = 1}^{{n}\left[E_k(a)} F_k(b) + G_k(a)H_k(b)\right]$$ for all $a, b \in \mathcal{A}$. In this paper, we prove that every strongly generalized derivation of order one from a $C<sup>{\ast}$-algebra into a Banach bimodule is automatically continuous under certain conditions. The main theorem of this paper extends some celebrated results in this regard.