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).
Author NameAffiliation
Guangyu AN Department of Mathematics, Shaanxi University of Science and Technology, Shaanxi 710021, P. R. China 
Jun SHENG Department of Mathematics, Shaanxi University of Science and Technology, Shaanxi 710021, P. R. China 
Jun HE Department of Mathematics, Anhui Polytechnic University, Anhui 241000, P. R. China 
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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:2095-2651.2024.03.009
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