A Lie derivation on a unital complex Banach algebra is considered. The main result of the article is the following
Theorem. Let be a Lie derivation on a unital complex Banach algebra . Then for every primitive ideal of , except for a finite set of them which have finite codimension greater than one, there exist a derivation from to itself and a linear functional on such that
for all (where denotes the quotient map from onto .
Moreover, the preceding decomposition holds for all primitive ideals in the case where is continuous.