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 $D_n:\mathcal{A}\to \mathcal{M}$ is called a strongly generalized derivation of order $n$, if there exist the families $\{E_k:\mathcal{A}\to \mathcal{M}{\}}_{k=1}^n$, $\{H_k:\mathcal{A}\to \mathcal{M}{\}}_{k=1}^n$, $\{F_k:\mathcal{A}\to \mathcal{B}{\}}_{k=1}^n$ and $\{G_k:\mathcal{A}\to \mathcal{B}{\}}_{k=1}^n$ 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.

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