Characterizations of Additive Jordan Left $*$-Derivations on $C^*$-Algebras |
Received:July 25, 2020 Revised:April 27, 2021 |
Key Words:
additive mapping Jordan left $*$-derivation left $*$-derivable mapping $C^{*}$-algebra
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Fund Project: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). |
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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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