Characterizations of Additive Jordan Left $*$Derivations on $C^*$Algebras 
Received:July 25, 2020 Revised:April 27, 2021 
Key Word:
additive mapping Jordan left $*$derivation left $*$derivable mapping $C^{*}$algebra

Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11801342), the Natural Science Foundation of Shaanxi Province (Grant No.2020JQ693) and the Scientific Research Plan Projects of Shannxi Education Department (Grant No.19JK0130). 

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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. 
Citation: 
DOI:10.3770/j.issn:20952651.2021.05.008 
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