Runde, Volker Automatic continuity of derivations and epimorphisms. (English) Zbl 0666.46052 Pac. J. Math. 147, No. 2, 365-374 (1991). 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 Cited in 1 ReviewCited in 4 Documents MSC: 46J05 General theory of commutative topological algebras 46H05 General theory of topological algebras 47B47 Commutators, derivations, elementary operators, etc. Keywords:automatic continuity problems for derivations on commutative Banach algebras; derivation on a commutative Banach algebra; semiprime Banach algebra; nilpotent separating space; accessible prime ideals; epimorphisms Citations:Zbl 0518.46038; Zbl 0617.46056 PDFBibTeX XMLCite \textit{V. Runde}, Pac. J. Math. 147, No. 2, 365--374 (1991; Zbl 0666.46052) Full Text: DOI