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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

References

Browder, A. (1969), Introduction to function algebras (Benjamin, New York, 1969).Google Scholar
Dales, H. G. and Davie, A. M. (1973), ‘Quasi-analytic Babach function algebras’, J. Functional Analysis 13, 28–50.CrossRefGoogle Scholar
Gamelin, T. W. (1969), Uniform algebras (Prentice-Hall, Englewood Cliffs, N. J., 1969).Google Scholar
Gelbaum, B. R. (1962), ‘Tensor products and related questions’, Trans. Amer. Math. Soc. 103, 525–548.CrossRefGoogle Scholar
Korenbljum, B. I. (1965), ‘Quasianalytic classes of functions in a circle’, Dokl. Akad. nauk. SSSR 164, 36–39 and Soviet Math. Dokl. 6, 1155–1158.Google Scholar
Loy, R. J. (1974), ‘Babach algebras of power series’, J. Austral. Math. Soc. 17, 263–273.CrossRefGoogle Scholar
Loy, R. J. (1974a), ‘Commutative Banach algebras with non-unique complete norm topology’, Bull. Austral. Math. Soc. 10, 409–420.CrossRefGoogle Scholar
Rickart, C. E. (1960), General theory of Banach algebras (Van Nostrand, Princeton, 1960).Google Scholar