Characterization of linear mappings on Banach algebras

Abstract

Let $\mathcal{A}$ be a unital Banach algebra, $\mathcal{M}$ be a unital $\mathcal{A}$-bimodule, and $W$ be a separating point of $\mathcal{M}$. We show that if linear mappings $\delta$ and $\tau$ from $\mathcal{A}$ into $\mathcal{M}$ satisfying $\delta(AB)=\delta(A)B+A\tau(B)$ for each $A,B$ in $\mathcal{A}$ with $AB=W$, then $\tau$ is a Jordan derivation and $\delta$ is a generalized Jordan derivation.
Based on this result, if linear mappings $\delta$ and $\tau$ from a unital semisimple Banach algebra $\mathcal{A}$ into itself satisfying $\delta(W)=\delta(A)B+A\tau(B)$ for each $A,B\in \mathcal{A}$ with $AB=W$, then $\tau$ is a Jordan derivation and $\delta(A)=\tau(A)+\delta(I)A$ for every $A$ in $\mathcal{A}$. As an application, we present a characterization of linear mappings $\delta$ and $\tau$ on a unital semisimple Banach $*$-algebra $\mathcal{A}$ satisfying $\delta(W)=\delta(A)B^*+A\tau(B)^*$ for each $A,B\in \mathcal{A}$ with $AB^*=W$.