Characterizations of Lie triple derivations on the algebra of operators in Hilbert \(C^\ast\)-modules. (English) Zbl 07892715
Summary: Let \(\mathcal A\) be a commutative unital \(C^\ast\)-algebra with the unit element \(e\) and \(\mathcal{M}\) be a full Hilbert \(\mathcal{A}\)-module. Denote by \(\mathrm{End}_{\mathcal{A}}(\mathcal{M})\) the algebra of all bounded \(\mathcal{A}\)-linear mappings on \(\mathcal{M}\) and by \(\mathcal{M}^\prime\) 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 \(\mathcal{M}^\prime\) such that \(f_0(x_0) = e\), then every \(\mathcal{A}\)-linear Lie triple derivation on \(\mathrm{End}_{\mathcal{A}}(\mathcal{M})\) is standard.
MSC:
| 47B48 | Linear operators on Banach algebras |
| 47L35 | Nest algebras, CSL algebras |
| 47C15 | Linear operators in \(C^*\)- or von Neumann algebras |
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