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Transactions of the American Mathematical Society

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

 

 

A class of infinitely connected domains and the corona


Author: W. M. Deeb
Journal: Trans. Amer. Math. Soc. 231 (1977), 101-106
MSC: Primary 46J15
MathSciNet review: 0477784
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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 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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DOI: https://doi.org/10.1090/S0002-9947-1977-0477784-1
Article copyright: © Copyright 1977 American Mathematical Society