%0 Journal Article %T Left derivable or Jordan left derivable mappings on Banach algebras %J Bulletin of the Iranian Mathematical Society %I Iranian Mathematical Society (IMS) %Z 1017-060X %A Ding, Y. %A Li, J. %D 2017 %\ 04/01/2017 %V 43 %N 2 %P 427-437 %! Left derivable or Jordan left derivable mappings on Banach algebras %K (Jordan) left derivation‎ %K ‎generalized (Jordan) left derivation‎ %K ‎(Jordan) left derivable mapping‎ %R %X ‎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. %U http://bims.iranjournals.ir/article_940_2fb4eda367c76aee7ee5aa3cf6360896.pdf