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The stability of derivations on Banach algebras. (English) Zbl 0965.46036

This paper is devoted to the concept of derivation on Banach algebras and its generalizations. A linear operator \(D\) on a Banach algebra \(A\) is an \(\epsilon\)-approximate derivation if \(\|D(ab)-(Da)b-a(Db)\|\leq\epsilon \|a\|\|b\|\) for all \(a, b\in A\), and a continuous linear operator \(T\) on \(A\) is an \(\epsilon\)-near derivation if there exists a continuous derivation \(d\) such that \(\|T-d\|\leq\epsilon \). It is well known that every derivation on a semisimple Banach algebra is continuous. The author shows that every \(\epsilon\)-approximate derivation on a semisimple Banach algebra is continuous even \(\epsilon\) is large. He also investigates relations between approximate and near derivations and proves the following theorem:
Let \(A\) be a finite dimensional Banach algebra. Then for each \(\epsilon >0\) there exists a \(\delta >0\) such that every \(\delta\)-approximate derivation on \(A\) is an \(\epsilon\)-near derivation.

MSC:

46H40 Automatic continuity
46J05 General theory of commutative topological algebras
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