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Continuity of Lie derivations on Banach algebras. (English) Zbl 0957.46035
A linear map $D$ from a Banach algebra $A$ to itself which satisfies $D([a,b])=[D(a),b]+[a,D(b)]$, where $[a,b]:=ab-ba$ is the usual Lie bracket, is called a Lie derivation. Denote by ${\cal S}(D)$ the separating space of the map $D$ and by ${\cal Z}(A)$ the centre of $A$. The authors prove the following Theorem. Let $D$ be a derivation on a semisimple Banach algebra $A$. Then ${\cal S}(D)\subset {\cal Z}(A)$. They also give an example of a discontinuous derivation on a semisimple Banach algebra whose centre is $\Bbb C$. For an extension of these results, see also {\it B. Aupetit} and {\it M. Mathieu}, Stud. Math. 138, No. 2, 193-199 (2000; Zbl 0962.46038).

##### MSC:
 46H40 Automatic continuity 16W10 Associative rings with involution, etc. 46H70 Nonassociative topological algebras
##### Keywords:
Lie derivation; Lie product; automatic continuity
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