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2010 $(\delta‎, ‎\varepsilon)$-double derivations on Banach algebras
Shirin Hejazian, ‎Hussein Mahdavian Rad, Madjid Mirzavaziri
Ann. Funct. Anal. 1(2): 103-111 (2010). DOI: 10.15352/afa/1399900592

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

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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

Information

Published: 2010
First available in Project Euclid: 12 May 2014

zbMATH: 1239.47027
MathSciNet: MR2772043
Digital Object Identifier: 10.15352/afa/1399900592

Subjects:
Primary: 47B47
Secondary: 46H40

Keywords: ‎$(\delta‎, ‎\varepsilon)$-double derivation , ‎automatic continuity , derivation‎

Rights: Copyright © 2010 Tusi Mathematical Research Group

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