Interpolation and Gleason parts in $L$-domains
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- by Michael Frederick Behrens PDF
- Trans. Amer. Math. Soc. 286 (1984), 203-225 Request permission
Abstract:
We describe the closure of $[ - 1/2,0)$ in the maximal ideal space $\mathcal {M}(\mathcal {D})$ of ${H^\infty }(\mathcal {D})$) for an arbitrary $L$-domain $\mathcal {D}$. For $L$-domains satisfying $\sup ({c_{n + 1}}/{c_n}) < 1$ and $\Sigma {({r_n}/{c_n})^p} < \infty$, some $p \geqslant 1$, we describe all interpolation sequences for ${H^\infty }(\mathcal {D})$, we show that a homomorphism (except the distinguished homomorphism, when it exists) lies in a nontrivial Gleason part if and only if it is contained in the closure of an interpolating sequence, and we describe all the analytic structure occurring in $\mathcal {M}(\mathcal {D})$.References
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M. Behrens, Analytic sets in $\mathcal {M}(\mathcal {D})$, Victoria Symposium on Nonstandard Analysis (Hurd and Loeb, eds.), Lecture Notes in Math., vol. 369, Springer-Verlag, Berlin, 1974.
—, Geometry of analytic sets in maximal ideal spaces (to appear).
- M. Behrens, The corona conjecture for a class of infinitely connected domains, Bull. Amer. Math. Soc. 76 (1970), 387–391. MR 256166, DOI 10.1090/S0002-9904-1970-12487-9
- 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
- John Garnett, Interpolating sequences for bounded harmonic functions, Indiana Univ. Math. J. 21 (1971/72), 187–192. MR 284589, DOI 10.1512/iumj.1971.21.21016
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
- Kenneth Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86 (1967), 74–111. MR 215102, DOI 10.2307/1970361
- M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894
- 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 1984 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 286 (1984), 203-225
- MSC: Primary 46J15; Secondary 03D55, 03H05, 30H05
- DOI: https://doi.org/10.1090/S0002-9947-1984-0756036-X
- MathSciNet review: 756036