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.2020JQ-693) and the Scientific Research Plan Projects of Shannxi Education Department (Grant No.19JK0130).
Author NameAffiliation
Ying YAO Department of Mathematics, Shaanxi University of Science and Technology, Shaanxi 710021, P. R. China 
Guangyu AN Department of Mathematics, Shaanxi University of Science and Technology, Shaanxi 710021, P. R. China 
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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:2095-2651.2021.05.008
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