$D$-domains and the corona
HTML articles powered by AMS MathViewer
- by W. M. Deeb and D. R. Wilken PDF
- Trans. Amer. Math. Soc. 231 (1977), 107-115 Request permission
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
- Michael Frederick Behrens, The maximal ideal space of algebras of bounded analytic functions on infinitely connected domains, Trans. Amer. Math. Soc. 161 (1971), 359–379. MR 435420, DOI 10.1090/S0002-9947-1971-0435420-9
- Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559. MR 141789, DOI 10.2307/1970375
- W. M. Deeb, A class of infinitely connected domains and the corona, Trans. Amer. Math. Soc. 231 (1977), no. 1, 101–106. MR 477784, DOI 10.1090/S0002-9947-1977-0477784-1
- T. W. Gamelin, Localization of the corona problem, Pacific J. Math. 34 (1970), 73–81. MR 276742, DOI 10.2140/pjm.1970.34.73
- T. W. Gamelin and John Garnett, Distinguished homomorphisms and fiber algebras, Amer. J. Math. 92 (1970), 455–474. MR 303296, DOI 10.2307/2373334
- Gordon M. Petersen, Regular matrix transformations, McGraw-Hill Publishing Co., Ltd., London-New York-Toronto, Ont., 1966. MR 0225045
- E. L. Stout, Two theorems concerning functions holomorphic on multiply connected domains, Bull. Amer. Math. Soc. 69 (1963), 527–530. MR 150274, DOI 10.1090/S0002-9904-1963-10981-7
- Lawrence Zalcman, Bounded analytic functions on domains of infinite connectivity, Trans. Amer. Math. Soc. 144 (1969), 241–269. MR 252665, DOI 10.1090/S0002-9947-1969-0252665-2
Additional 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