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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 𝒮(D) the separating space of the map D and by 𝒵(A) the centre of A. The authors prove the following

Theorem. Let D be a derivation on a semisimple Banach algebra A. Then 𝒮(D)𝒵(A).

They also give an example of a discontinuous derivation on a semisimple Banach algebra whose centre is . 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:
46H40Automatic continuity
16W10Associative rings with involution, etc.
46H70Nonassociative topological algebras