# Left derivable or Jordan left derivable mappings on Banach algebras

Document Type : Research Paper

Authors

Department of Mathematics‎, ‎East China University of Science and Technology‎, ‎Shanghai‎, ‎China.

Abstract

‎Let $\mathcal{A}$ be a unital Banach algebra‎, ‎$\mathcal{M}$ be a left $\mathcal{A}$-module‎, ‎and $W$ in $\mathcal{Z}(\mathcal{A})$ be a left separating point of $\mathcal{M}$‎. ‎We show that if $\mathcal{M}$ is a unital left $\mathcal{A}$-module and $\delta$ is a linear mapping from $\mathcal{A}$ into $\mathcal{M}$‎, ‎then the following four conditions are equivalent‎: ‎(i) $\delta$ is a Jordan left derivation; (ii)$\delta$ is left derivable at $W$; (iii) $\delta$ is Jordan left derivable at $W$; (iv)$A\delta(B)+B\delta(A)=\delta(W)$ for each $A,B$ in $\mathcal{A}$ with $AB=BA=W$‎.
‎Let $\mathcal{A}$ have property ($\mathbb{B}$) (see Definition \ref{Prop_B})‎, ‎$\mathcal{M}$ be a Banach left $\mathcal{A}$-module‎, ‎and $\delta$ be a continuous linear operator from $\mathcal{A}$ into $\mathcal{M}$‎. ‎Then $\delta$ is a generalized Jordan left derivation if and only if $\delta$ is Jordan left derivable at zero‎. ‎In addition‎, ‎if there exists an element $C\in\mathcal{Z}(\mathcal{A})$ which is a left separating point of $\mathcal{M}$‎, ‎and $Rann_{\mathcal{M}}(\mathcal{A})=\{0\}$‎, ‎then $\delta$ is a generalized left derivation if and only if $\delta$ is left derivable at zero.

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Main Subjects

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### History

• Receive Date: 09 July 2015
• Revise Date: 27 November 2015
• Accept Date: 29 November 2015