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Continuity of derivations on topological algebras of power series

Published online by Cambridge University Press:  17 April 2009

R.J. Loy
Affiliation:
Department of Mathematics, Carleton University, Ottawa, Canada.
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Abstract

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Let A be an algebra of formal power series in one indeterminate over the complex field, D a derivation on A. It is shown that if A has a Fréchet space topology under which it is a topological algebra, then D is necessarily continuous provided the coordinate projections satisfy a certain equicontinuity condition. This condition is always satisfied if A is a Banach algebra and the projections are continuous. A second result is given, with weaker hypothesis on the projections and correspondingly weaker conclusion.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Arens, Richard, “Linear topological division algebras”, Bull. Amer. Math. Soc. 53 (1947), 623630.CrossRefGoogle Scholar
[2]Hochschild, G., “Semisimple algebras and generalized derivations”, Amer. J. Math. 64 (1942), 677694.CrossRefGoogle Scholar
[3]Johnson, B.E., “Continuity of derivations on commutative algebras”, Amer. J. Math. 91 (1969), 110.CrossRefGoogle Scholar
[4]Loy, R.J., “A class of topological algebras of power series”, Carleton Mathematical Series 11 (to appear).Google Scholar
[5] Donald Newman, J., “A radical algebra without derivations”, Proc. Amer. Math. Soc. 10 (1959), 584586.CrossRefGoogle Scholar