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Topologically simple Banach algebras with derivation

Published online by Cambridge University Press:  17 April 2009

El Hossein Illoussamen  and
Volker Runde
El Hossein Illoussamen
Affiliation:
Ecole Normale Supérieure TakaddoumDépartement de MathématiquesB.P. 511810105 RabatMorocco
Volker Runde
Affiliation:
Fachbereich 9 MathematikUniversitát des SaarlandesPostfach 15115066041 SaarbrückenGermany e-mail: runde@math.uni-sb.de
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Abstract

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It is not known if a commutative, topologically simple, radical Banach algebra exists. If, however, every derivation on such an algebra is continuous, this yields the automatic continuity of all derivations on commutative, semiprime Banach algebras. Utilising techniques used by Thomas in his proof of the Singer-Wermer conjecture, we show that, if A is a commutative, topologically simple Banach algebra with a non-zero derivation on it, then a quotient of a certain localisation of A has a power series structure. A pivotal role is played by what we call ample sets of denominators.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 60 , Issue 1 , August 1999 , pp. 153 - 161
Copyright
Copyright © Australian Mathematical Society 1999

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