## $D$-domains and the corona

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- by W. M. Deeb and D. R. Wilken
- Trans. Amer. Math. Soc.
**231**(1977), 107-115 - DOI: https://doi.org/10.1090/S0002-9947-1977-0477785-3
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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 $\text {weak}^\ast$ 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.

## References

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## Bibliographic Information

- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**231**(1977), 107-115 - MSC: Primary 46J15
- DOI: https://doi.org/10.1090/S0002-9947-1977-0477785-3
- MathSciNet review: 0477785