Characterizations of Additive Jordan Left $*$-Derivations on $C^*$-Algebras

Abstract:

      An additive mapping $\delta$ from a $*$-algebra $\mathcal{A}$ into a left $\mathcal{A}$-module $\mathcal{M}$ is called an additive Jordan left $*$-derivation if $\delta(A^2)=A\delta(A)+A^*\delta(A)$ for every $A$ in $\mathcal A$. In this paper, we prove that every additive Jordan left $*$-derivation from a complex unital $C^*$-algebra into its unital Banach left module is equal to zero. An additive mapping $\delta$ from a $*$-algebra $\mathcal{A}$ into a left $\mathcal{A}$-module $\mathcal{M}$ is called left $*$-derivable at $G$ in $\mathcal{A}$ if $\delta(AB)=A\delta(B)+B^*\delta(A)$ for each $A,B$ in $\mathcal{A}$ with $AB=G$. We prove that every continuous additive left $*$-derivable mapping at the unit element $I$ from a complex unital $C^*$-algebra into its unital Banach left module is equal to zero.