A characterization of derivations on uniformly mean value Banach algebras

Authors: AMIN HOSSEINI

Abstract: In this paper, a uniformly mean value Banach algebra (briefly UMV-Banach algebra) is defined as a new class of Banach algebras, and we characterize derivations on this class of Banach algebras. Indeed, it is proved that if $\mathcal{A}$ is a unital UMV-Banach algebra such that either $a = 0$ or $b = 0$ whenever $ab = 0$ in $\mathcal{A}$, and if $\delta:\mathcal{A} \rightarrow \mathcal{A}$ is a derivation such that $a \delta(a) = \delta(a)a$ for all $a \in \mathcal{A}$, then the following assertions are equivalent:\\ (i) $\delta$ is continuous; \\(ii) $\delta(e^a) = e^a\delta(a)$ for all $a \in \mathcal{A}$; \\(iii) $\delta$ is identically zero.

Keywords: Derivation, mean value property, uniformly mean value property, classical mean value theorem, Gelfand transform

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