Abstract
Let $\mathcal A$ be an algebra and let $\delta, \varepsilon :\mathcal A \to \mathcal A$ be two linear mappings. A ($\delta, \varepsilon$)-double derivation is a linear mapping $d: \mathcal A\rightarrow \mathcal A$ satisfying $d(ab)=d(a)b+ad(b)+\delta(a)\varepsilon(b)+\varepsilon(a)\delta(b)\;(a,b \in \mathcal{A})$. We study some algebraic properties of these mappings and give a formula for calculating $d^n(ab)$. We show that if $\mathcal A$ is a Banach algebra such that either is semi-simple or every derivation from $\mathcal A$ into any Banach $\mathcal A$-bimodule is continuous then every ($\delta, \varepsilon$)-double derivation on $\mathcal A$ is continuous whenever so are $\delta$ and $\varepsilon$. We also discuss the continuity of $\varepsilon$ when $d$ and $\delta$ are assumed to be continuous.
Citation
Shirin Hejazian. Hussein Mahdavian Rad. Madjid Mirzavaziri. "$(\delta, \varepsilon)$-double derivations on Banach algebras." Ann. Funct. Anal. 1 (2) 103 - 111, 2010. https://doi.org/10.15352/afa/1399900592
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