Characterizations of Lie Triple Derivations on the Algebra of Operators in Hilbert $C^*$-Modules

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.