Completion of normed algebras of polunomials

Dales, H.G. and McClure, J. P. (1975) Completion of normed algebras of polunomials. Journal of the Australian Mathematical Society, 20 (4). pp. 504-510. ISSN 1446-7887

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Let P be the algebra of polynomials in one inderminate x over the complex field C. Suppose xs2225 · xs2225 is a norm on P such that the coefficient functionals cj: ∑αix1 → αj (j = 0,1,2,…) are all continuous with respect to xs2225·xs2225, and Let K xs2282 C be the set of characters on P which are xs2225·xs2225-continuous. then K is compact, C\K is connected, and 0xs2208K. K. Let A be the completion of P with respect to xs2225·xs2225. 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 xs2208 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 xs2208 A and a≠O imply cj(a)≠ 0 for some j.

Item Type:
Journal Article
Journal or Publication Title:
Journal of the Australian Mathematical Society
Additional Information: The final, definitive version of this article has been published in the Journal, Journal of the Australian Mathematical Society, 20 (4), pp 504-510 1975, © 1975 Cambridge University Press.
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26 Nov 2013 12:04
Last Modified:
19 Dec 2023 01:22