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Transactions of the American Mathematical Society

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



$ D$-domains and the corona

Authors: W. M. Deeb and D. R. Wilken
Journal: Trans. Amer. Math. Soc. 231 (1977), 107-115
MSC: Primary 46J15
MathSciNet review: 0477785
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Abstract: Let D be a bounded domain in the complex plane C. Let $ {H^\infty }(D)$ denote the usual Banach algebra of bounded analytic functions on D. The Corona Conjecture asserts that D is weak$ ^\ast$ dense in the space $ \mathfrak{M}(D)$ of maximal ideals of $ {H^\infty }(D)$. In [2] Carleson proved that the unit disk $ {\Delta _0}$ is dense in $ \mathfrak{M}({\Delta _0})$. In [7] Stout extended Carleson's result to finitely connected domains. In [4] Gamelin showed that the problem is local. In [1] Behrens reduced the problem to very special types of infinitely connected domains and established the conjecture for a large class of such domains.

In this paper we extract some of the crucial ingredients of Behrens' methods and extend his results to a broader class of infinitely connected domains.

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Article copyright: © Copyright 1977 American Mathematical Society