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Lie derivations of reflexive algebras. (English) Zbl 1201.47078

Let \(X\) be a Banach space and \(B(X)\) be the algebra of all bounded linear operators on \(X\). Let \(\mathcal{L}\) be a subspace lattice on \(X\). The authors show that, if \(\mathcal{L}\) is a nest or has a nontrivial smallest element or has a nontrivial greatest element, then a Lie derivation from \(\text{Alg}\mathcal{L}\) into \(B(X)\) is the sum of a derivation and a linear map with image in the center which vanishes on commutators.

MSC:

47L35 Nest algebras, CSL algebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
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