Bilocal Lie derivations on nest algebras. (English) Zbl 1504.47115
Summary: Let \(\mathcal{N}\) be a nest on a complex separable Hilbert space \(H\) and \(\mathcal{N}\) be the associated nest algebra. In this paper, we prove that every bilocal Lie derivation from \(\operatorname{Alg} \mathcal{N}\) into itself is of the form \(A \to [A, T] + \lambda A+f(A)\), where \(T \in \operatorname{Alg} \mathcal{N}\), \(\lambda \in \mathbb{C}\) and \(f: \operatorname{Alg} \mathcal{N} \to \mathbb{C}I\) is a linear map vanishing on each commutator. Moreover, we show that every bilocal Lie derivation from \(\operatorname{Alg} \mathcal{N}\) into itself is a Lie derivation if \(\mathcal{N}\) is a non-atomic nest or there exists an atom \(E\) of \(\mathcal{N}\) with \(\dim E >1\).
MSC:
| 47L35 | Nest algebras, CSL algebras |
| 47B47 | Commutators, derivations, elementary operators, etc. |
| 17B40 | Automorphisms, derivations, other operators for Lie algebras and super algebras |
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