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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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A class of infinitely connected domains and the corona
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by W. M. Deeb
Trans. Amer. Math. Soc. 231 (1977), 101-106
DOI: https://doi.org/10.1090/S0002-9947-1977-0477784-1

Abstract:

Let D be a bounded domain in the complex plane. Let ${H^\infty }(D)$ be the Banach algebra of bounded analytic functions on D. The corona problem asks whether D is $\text {weak}^\ast$ dense in the space $\mathfrak {M}(D)$ of maximal ideals of ${H^\infty }(D)$. Carleson [3] proved that the open unit disc ${\Delta _0}$ is dense in $\mathfrak {M}({\Delta _0})$. Stout [9] extended Carleson’s result to finitely connected domains. Behrens [2] found a class of infinitely connected domains for which the corona problem has an affirmative answer. In this paper we will use Behrens’ idea to extend the results to more general domains. See [11] for further extensions and applications of these techniques.
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Bibliographic Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 231 (1977), 101-106
  • MSC: Primary 46J15
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0477784-1
  • MathSciNet review: 0477784