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Lie derivations on Banach algebras. (English) Zbl 0957.46032
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.

MSC:
46H05 General theory of topological algebras
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