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Lie derivations on Banach algebras. (English) Zbl 0957.46032

A Lie derivation Δ 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 τ on A such that

Q p Δ(a)=d(a+P)+τ(a)

for all aA (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 Δ is continuous.

46H05General theory of topological algebras