Linear maps Lie derivable at zero on $\mathcal J$-subspace lattice algebras
Tom 197 / 2010
Studia Mathematica 197 (2010), 157-169
MSC: 47L35, 17B40.
DOI: 10.4064/sm197-2-3
Streszczenie
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 $\mathcal{J}$-subspace lattice $\mathcal L$ on a Banach space $X$ satisfying $\dim K\not=2$ whenever $K\in{\mathcal J}({\mathcal L})$, every linear map on ${\mathcal F}({\mathcal L})$ (the subalgebra of all finite rank operators in the JSL algebra ${\rm Alg} {\mathcal L}$) Lie derivable at zero is of the standard form $A\mapsto \delta (A)+\phi(A)$, where $\delta $ is a generalized derivation and $\phi$ is a center-valued linear map. A characterization of linear maps Lie derivable at zero on ${\rm Alg} {\mathcal L}$ is also obtained, which are not of the above standard form in general.
