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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 \({\mathcal S}(D)\) the separating space of the map \(D\) and by \({\mathcal Z}(A)\) the centre of \(A\). The authors prove the following
Theorem. Let \(D\) be a derivation on a semisimple Banach algebra \(A\). Then \({\mathcal S}(D)\subset {\mathcal Z}(A)\).
They also give an example of a discontinuous derivation on a semisimple Banach algebra whose centre is \(\mathbb C\). For an extension of these results, see also B. Aupetit and M. Mathieu, Stud. Math. 138, No. 2, 193-199 (2000; Zbl 0962.46038).

MSC:
46H40 Automatic continuity
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
46H70 Nonassociative topological algebras
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