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Some characterizations of $C(\mathcal M)$

Author(s): Christopher J. Bishop
Journal: Proc. Amer. Math. Soc. 124 (1996), 2695-2701.
MSC (1991): Primary 46J10
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Abstract: We show that a function $f$ on the unit disk extends continuously to $\mathcal M$ , the maximal ideal space of $H^\infty (\mathbb D)$ iff it is uniformly continuous (in the hyperbolic metric) and close to constant on the complementary components of some Carleson contour.

References:

1.: S. Axler and A. Shields Extensions of harmonic and analytic functions, Pacific J. Math. 145:1--15, 1990. MR 91h:30056
2.: C.J. Bishop A distance formula for algebras on the disk, Pacific J. Math. (to appear).
3.: J.B. Garnett Bounded analytic functions, Academic Press 1981. MR 83g:30037
4.: K. Hoffman Bounded analytic functions and Gleason parts, Ann. Math. 86:74--111, 1967. MR 35:5945
5.: O.V. Ivanov Nontangential limits and Shilov boundary of the algebra $H^\infty$ , Dokl. Akad. Nauk. Ukraine SSR Ser. A 5--8, 1991. MR 93c:46096

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Additional Information:

Christopher J. Bishop
Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651
Email: bishop@math.sunysb.edu

DOI: 10.1090/S0002-9939-96-03328-X
PII: S 0002-9939(96)03328-X
Keywords: Harmonic functions, holomorphic functions, function algebras, maximal ideal space, Carleson measures, uniform approximation
Received by editor(s): November 23, 1994
Received by editor(s) in revised form: February 24, 1995
Additional Notes: The author is partially supported by NSF Grant DMS 92-04092 and an Alfred P. Sloan research fellowship.
Communicated by: T. W. Gamelin
Copyright of article: Copyright 1996, American Mathematical Society


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