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Nebenhülle and the Gleason problem

Author: Linus Carlsson
Journal: Proc. Amer. Math. Soc. 138 (2010), 267-273
MSC (2000): Primary 32A65, 32W05, 46J20
Published electronically: August 24, 2009
MathSciNet review: 2550192
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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

Keywords: Holomorphic functions, Banach algebras, Nebenh\"{u}lle, $\overline {\partial }$-problems
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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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