Characterizations of (Jordan) derivation on Banach algebra with local actions (2205.01352v1)

Abstract: Let $\mathcal{A}$ be a unital Banach $$-algebra and $\mathcal{M}$ be a unital $$-$\mathcal{A}$-bimodule. If $W$ is a left separating point of $\mathcal{M}$, we show that every $$-derivable mapping at $W$ is a Jordan derivation, and every $$-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 $\delta(A)B^+A\tau(B)^=0$ for any $A, B\in \mathcal{A}$ with $AB^*=0$ or $\delta(A)\circ B^+A\circ\tau(B)^=0$ for any $A, B\in \mathcal{A}$ with $A\circ B^*=0$, where $A\circ B=AB+BA$ is the Jordan product.