Lie derivations on Banach algebras
A Lie derivation \(\Delta\) on a unital complex Banach algebra \(A\) is considered. The main result of the article is the following Theorem. Let \(D\) be a Lie derivation on a unital complex Banach algebra \(A\). Then for every primitive ideal \(P\) of \(A\), except for a finite set of them which have finite codimension greater than one, there exist a derivation \(d\) from \(A/P\) to itself and a linear functional \(\tau\) on \(A\) such that \[ Q_p\Delta(a)= d(a+ P)+ \tau(a) \] for all \(a\in A\) (where \(Q_p\) denotes the quotient map from \(A\) onto \(A(p)\). Moreover, the preceding decomposition holds for all primitive ideals in the case where \(\Delta\) is continuous.
scientific article; zbMATH DE number 1441815