**Rights statement:**http://journals.cambridge.org/action/displayJournal?jid=JAZ 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.Final published version, 299 KB, PDF-document

- 10.1017/S1446788700016207
Final published version

Research output: Contribution to journal › Journal article

Published

<mark>Journal publication date</mark> | 11/1975 |
---|---|

<mark>Journal</mark> | Journal of the Australian Mathematical Society |

Issue number | 4 |

Volume | 20 |

Number of pages | 7 |

Pages (from-to) | 504-510 |

<mark>State</mark> | Published |

<mark>Original language</mark> | English |

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.

http://journals.cambridge.org/action/displayJournal?jid=JAZ 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.