A linear map from a Banach algebra to itself which satisfies , where is the usual Lie bracket, is called a Lie derivation. Denote by the separating space of the map and by the centre of . The authors prove the following
Theorem. Let be a derivation on a semisimple Banach algebra . Then .
They also give an example of a discontinuous derivation on a semisimple Banach algebra whose centre is . For an extension of these results, see also B. Aupetit and M. Mathieu, Stud. Math. 138, No. 2, 193-199 (2000; Zbl 0962.46038).