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Completion of normed algebras of polunomials

Published online by Cambridge University Press:  09 April 2009

H. G. Dales  and
J. P. McClure
H. G. Dales
Affiliation:
School of Mathematics, University of LeedsLeeds LS2 9JT, England.
J. P. McClure
Affiliation:
Department of Mathematics and Astronomy University of ManitobaWinnipeg, Canada.
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Let P be the algebra of polynomials in one inderminate x over the complex field C. Suppose ∥ · ∥ is a norm on P such that the coefficient functionals cj: ∑αix1 → αj (j = 0,1,2,…) are all continuous with respect to ∥·∥, and Let K ⊂ C be the set of characters on P which are ∥·∥-continuous. then K is compact, C\K is connected, and 0∈K. K. Let A be the completion of P with respect to ∥·∥. Then A is a singly generated Banach algebra, with space of characters (homeomorphic with) K. The functionals cj have unique extensions to bounded linear functionals on A, and the map a →∑Ci(a)xi (a ∈ A) is a homomorphism from A onto an algebra of formal power series with coefficients in C. We say that A is an algebra of power series if this homomorphism is one-to-one, that is if a ∈ A and a≠O imply cj(a)≠ 0 for some j.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 20 , Issue 4 , November 1975 , pp. 504 - 510
Copyright
Copyright © Australian Mathematical Society 1975

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