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

Xiaofei Qi¹ ; Jinchuan Hou²

¹Department of Mathematics Shanxi University Taiyuan 030006, P.R. China
²Department of Mathematics Taiyuan University of Technology Taiyuan 030024, P.R. China

Studia Mathematica, Tome 197 (2010) no. 2, pp. 157-169

Résumé

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.

Détail

DOI : 10.4064/sm197-2-3

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