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 $\{E_k:\mathcal{A} \rightarrow \mathcal{M}\}_{k = 1}^{n}$, $\{H_k:\mathcal{A} \rightarrow \mathcal{M}\}_{k = 1}^{n}$, $\{F_k:\mathcal{A} \rightarrow \mathcal{B}\}_{k = 1}^{n}$ and $\{G_k:\mathcal{A} \rightarrow \mathcal{B}\}_{k = 1}^{n}$ 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 $C^{\ast}$-algebra into a Banach bimodule is automatically continuous under certain conditions. The main theorem of this paper extends some celebrated results in this regard.