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Automatic continuity of derivations and epimorphisms. (English) Zbl 0666.46052
Two automatic continuity problems for derivations on commutative Banach algebras are discussed:
(a) Every derivation on a commutative Banach algebra maps into the radical, and
(b) Every derivation on a semiprime Banach algebra is continuous.
We show that (b) implies (a). Furthermore, we prove that (b) and the following statements are equivalent:
(c) Every derivation on a commutative Banach algebra has a nilpotent separating space,
(d) Every derivation on an integral domain is continuous, and
(e) Every derivation on a commutative, topologically simple Banach algebra other than \({\mathbb{C}}\) is continuous.
Using accessible prime ideals as introduced by P. C. Curtis [Lect. Notes Math. 975, 328-333 (1983; Zbl 0518.46038)], we show that (b) holds in some special cases, thus improving older results by R. J. Loy [Bull. Austr. Math. Soc. 1, 419-424 (1969)] and R. V. Garimella [Proc. Am. Math. Soc. 99, 289-292 (1987; Zbl 0617.46056)]; related results for epimorphisms are given.
Reviewer: V.Runde

46J05 General theory of commutative topological algebras
46H05 General theory of topological algebras
47B47 Commutators, derivations, elementary operators, etc.
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