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

  Published:2015-03-04
 Printed: Jun 2015
  • Héctor Merino-Cruz,
    Universidad Autónoma de Guerrero , Unidad Académica de Matemáticas , Av. Lázaro Cárdenas S/N, Col. La Haciendita , 39127 Chilpancingo, Gro., México
  • Antoni Wawrzynczyk,
    Departamento de Matemáticas, Universidad Autónoma Metropolitana-Izta\-palapa AP 55-534, 09340 México D. F., México
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Abstract

We recently introduced a weighted Banach algebra $\mathfrak{A}_G^n$ of functions which are holomorphic on the unit disc $\mathbb{D}$, continuous up to the boundary and of the class $C^{(n)}$ 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 $\mathfrak{A}_G^n$ with at most countable hull are standard. In the present paper, on the assumption that $G$ is an outer function in $C^{(n)}(\overline{\mathbb{D}})$ having infinite roots in $\mathfrak{A}_G^n$ and countable zero set $h(G)$, we show that all the closed ideals $I$ with hull containing $h(G)$ are standard.
Keywords: Banach algebra, disc algebra, holomorphic spaces, standard ideal Banach algebra, disc algebra, holomorphic spaces, standard ideal
MSC Classifications: 46J15, 46J20, 30H50 show english descriptions Banach algebras of differentiable or analytic functions, $H^p$-spaces [See also 30H10, 32A35, 32A37, 32A38, 42B30]
Ideals, maximal ideals, boundaries
Algebras of analytic functions
46J15 - Banach algebras of differentiable or analytic functions, $H^p$-spaces [See also 30H10, 32A35, 32A37, 32A38, 42B30]
46J20 - Ideals, maximal ideals, boundaries
30H50 - Algebras of analytic functions
 

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