Characterizations of (Jordan) derivation on Banach algebra with local actions

Abstract

Let $\mathcal{A}$ be a unital Banach $\ast$-algebra and $\mathcal{M}$ be a unital $\ast$-$\mathcal{A}$-bimodule. If $W$ is a left separating point of $\mathcal{M}$, we show that every $\ast$-derivable mapping at $W$ is a Jordan derivation, and every $\ast$-left derivable mapping at $W$ is a Jordan left derivation under the condition $W\mathcal{A}=\mathcal{A}W$. Moreover we give a complete description of linear mappings $\delta$ and $\tau$ from $\mathcal{A}$ into $\mathcal{M}$ satisfying $\delta (A)B^{\ast }+A\tau (B{)}^{\ast }=0$ for any $A,B\in \mathcal{A}$ with $A{B}^{\ast }=0$ or $\delta (A)\circ B^{\ast }+A\circ \tau (B{)}^{\ast }=0$ for any $A,B\in \mathcal{A}$ with $A\circ B^{\ast }=0$, where $A\circ B=AB+BA$ is the Jordan product.