Geodesic
On Closed Ideals in a Certain Class of Algebras of Holomorphic Functions
Canadian mathematical bulletin, Tome 58 (2015) no. 2, pp. 350-355

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-2015-003-6
Mots-clés : 46J15, 46J20, 30H50, Banach algebra, disc algebra, holomorphic spaces, standard ideal 
Merino-Cruz, Héctor; Wawrzyńczyk, Antoni. On Closed Ideals in a Certain Class of Algebras of Holomorphic Functions. Canadian mathematical bulletin, Tome 58 (2015) no. 2, pp. 350-355. doi: 10.4153/CMB-2015-003-6
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[1] [1] Gale, J. E., Miana, P. J., and Royo, J. J., Estimates of the Laplace transform on convolution Sobolev algebras. J. Approx. Theory 164 (2012), no. 1,162–178. http://dx.doi.Org/10.1016/j.jat.2O11.09.011 Google Scholar

[2] [2] Gale, J. E. and Wawrzynczyk, A., Standard ideals in weighted algebras of Korenblyum and Wiener types. Math. Scand. 108 (2011), no. 2, 291–319. Google Scholar

[3] [3] Hoffman, K., Banach spaces of analytic functions. Prentice-Hall Series in Modern Analysis, Prentice Hall, Englewood Cliffs, NJ, 1962. Google Scholar

[4] [4] Korenblyum, B. I., Closed ideals in the ring A”. Funkcional. Anal, i Prilozen. 6 (1972), no. 3, 38–52. Google Scholar

[5] [5] Merino-Cruz, H. and Wawrzynczyk, A., Closed ideals in a new class of algebras of holomorphic functions on the disc. Comment. Math. 54 (2014), no. 1, 29–37. Google Scholar

[6] [6] Soltysiak, A. and Wawrzynczyk, A., Ditkin's condition and ideals with at most countable hull in algebras of functions analytic in the unit disc. Comment. Math. 52 (2012), no. 1,101–112. Google Scholar

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