A linear map L on an algebra is said to be Lie derivable at zero if L([A,B]) = [L(A),B] + [A,L(B)] whenever [A,B] = 0. It is shown that, for a 𝒥-subspace lattice ℒ on a Banach space X satisfying dim K ≠ 2 whenever K ∈ 𝒥(ℒ), every linear map on ℱ(ℒ) (the subalgebra of all finite rank operators in the JSL algebra Alg ℒ) Lie derivable at zero is of the standard form A ↦ δ (A) + ϕ(A), where δ is a generalized derivation and ϕ is a center-valued linear map. A characterization of linear maps Lie derivable at zero on Alg ℒ is also obtained, which are not of the above standard form in general.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm197-2-3,
author = {Xiaofei Qi and Jinchuan Hou},
title = {Linear maps Lie derivable at zero on J-subspace lattice algebras},
journal = {Studia Mathematica},
volume = {196},
year = {2010},
pages = {157-169},
zbl = {1190.47088},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm197-2-3}
}
Xiaofei Qi; Jinchuan Hou. Linear maps Lie derivable at zero on 𝒥-subspace lattice algebras. Studia Mathematica, Tome 196 (2010) pp. 157-169. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm197-2-3/