Left derivable or Jordan left derivable mappings on Banach algebras

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)$Adelta(B)+Bdelta(A)=delta(W)$ for each $A,B$ in $mathcal{A}$ with $AB=BA=W$‎.<br /> ‎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 $Cinmathcal{Z}(mathcal{A})$ 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.