Characterizations of Lie Triple Derivations on the Algebra of Operators in Hilbert $C^*$Modules 
Received:June 24, 2023 Revised:September 23, 2023 
Key Words:
Lie triple derivation standard derivation Hilbert $C^*$module

Fund Project:Supported by the Shaanxi College Students Innovation and Entrepreneurship Training Program (Grant No.S202110708069). 

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Abstract: 
Let $\mathcal A$ be a commutative unital $C^*$algebra with the unit element $e$ and $\mathcal M$ be a full Hilbert $\mathcal A$module. Denote by End$_{\mathcal A}(\mathcal M)$ the algebra of all bounded $\mathcal A$linear mappings on $\mathcal M$ and by $\mathcal M'$ the set of all bounded $\mathcal A$linear mappings from $\mathcal M$ into $\mathcal A$. In this paper, we prove that if there exists $x_0$ in $\mathcal M$ and $f_0$ in $\mathcal M'$ such that $f_0(x_0)=e$, then every $\mathcal A$linear Lie triple derivation on End$_{\mathcal A}(\mathcal M)$ is standard. 
Citation: 
DOI:10.3770/j.issn:20952651.2024.03.009 
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