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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Nebenhülle and the Gleason problem
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by Linus Carlsson PDF
Proc. Amer. Math. Soc. 138 (2010), 267-273 Request permission

Abstract:

This article concerns the Gleason property as a local phenomenon. We prove that there always exists an open set where the domain $D\Subset \mathbb {C}^{2}$ has the Gleason $\mathcal {B}$ property whenever the boundary of the Nebenhülle of $D$ coincides with a $C^{2}$ smooth part of the boundary $bD$; here $\mathcal {B}$ is either one of the Banach algebras, $H^{\infty }$ or $A$. As an easy consequence of this, we see that if the extremal boundary points are $C^{2}$-smooth, then $D$ has the Gleason $\mathcal {B}$ property close to those points. Also a $\overline {\partial }$-problem for locally supported forms is solved.
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Additional Information
  • Linus Carlsson
  • Affiliation: Department of Mathematics and Mathematical Statistics, Umeå University, S-901 87 Umeå, Sweden
  • Email: linus.carlsson@math.umu.se
  • Received by editor(s): December 9, 2008
  • Received by editor(s) in revised form: May 26, 2009
  • Published electronically: August 24, 2009
  • Communicated by: Mei-Chi Shaw
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 138 (2010), 267-273
  • MSC (2000): Primary 32A65, 32W05, 46J20
  • DOI: https://doi.org/10.1090/S0002-9939-09-10064-3
  • MathSciNet review: 2550192