Linear maps Lie derivable at zero on J-subspace lattice algebras

摘要

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 ￡ on a Banach space X satisfying dim K ≠ 2 whenever K ∈ J(￡), every linear map on F(￡) (the subalgebra of all finite rank operators in the JSL algebra Alg ￡) Lie derivable at zero is of the standard form A δ(A)+?(A), where δ is a generalized derivation and ? is a center-valued linear map. A characterization of linear maps Lie derivable at zero on Alg ￡ is also obtained, which are not of the above standard form in general.