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On Closed Ideals in a Certain Class of Algebras of Holomorphic Functions

Published online by Cambridge University Press:  20 November 2018

Héctor Merino-Cruz  and
Antoni Wawrzyńczyk
Héctor Merino-Cruz
Affiliation:
Universidad Autónoma de Guerrero, Unidad Académica deMatemáticas, Av. Lázaro Cárdenas S/N, Col. La Haciendita, 39127 Chilpancingo, Gro., México. e-mail: [email protected]
Antoni Wawrzyńczyk
Affiliation:
Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa AP 55-534 México D. F., México. e-mail: [email protected]
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Abstract

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We recently introduced a weighted Banach algebra $\mathcal{A}_{G}^{n}$ of functions that are holomorphic on the unit disc $\mathbb{D}$, continuous up to the boundary, and of the class ${{C}^{\left( n \right)}}$ at all points where the function $G$ does not vanish. Here, $G$ refers to a function of the disc algebra without zeros on $\mathbb{D}$. Then we proved that all closed ideals in $\mathcal{A}_{G}^{n}$ with at most countable hull are standard. In this paper, on the assumption that $G$ is an outer function in ${{C}^{\left( n \right)}}\,\left( {\bar{\mathbb{D}}} \right)$ having infinite roots in $\mathcal{A}_{G}^{n}$ and countable zero set ${{h}_{o}}\left( G \right)$, we show that all the closed ideals $I$ with hull containing ${{h}_{o}}\left( G \right)$ are standard.

Keywords

46J1546J2030H50Banach algebradisc algebraholomorphic spacesstandard ideal
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 2 , 01 June 2015 , pp. 350 - 355
Copyright
Copyright © Canadian Mathematical Society 2015

References

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