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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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