Communications of the
Korean Mathematical Society CKMS

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.

Keywords: Banach algebra, derivation, Jordan derivation, separating point

MSC numbers: Primary 47B47, 15A86

Supported by: This work was financially supported by the National
Natural Science Foundation of China (Grant No.11871021).