Continuity and Structure of Generalized $\varvec{(\phi ,\psi )}$-Derivations

Abstract

Let ${\mathcal {X}}$ be a Banach algebra, let $\phi ,\psi$ be mappings on ${\mathcal {X}}$, let $\delta$ be a $(\phi ,\psi )$-derivation on ${\mathcal {X}}$ and let d be a generalized $(\phi ,\psi )$-derivation related to $\delta$. If ${\mathcal {X}}$ is simple, we determine some sufficient conditions under which every generalized $(\phi ,\psi )$-derivation on ${\mathcal {X}}$ is continuous (without continuity of $\delta$). In addition, we show that if d is inner on $\mathcal {F}_1(X)$ (the set of all rank one operators on ${\mathcal {X}})$ and $\phi ,\psi : \mathcal {B}({\mathcal {X}})\longrightarrow \mathcal {B}({\mathcal {X}})$ are homomorphisms and surjective on $\mathcal {F}_1(X)$ then d is inner on $\mathcal {B}({\mathcal {X}})$. Finally, we characterize the linear mappings on $\mathcal {B}({\mathcal {X}})$ which behave like generalized $(\phi ,\psi )$-derivations when acting on zero products.