Nebenhülle and the Gleason problem
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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.References
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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