Analytic properties in the spectrum of certain Banach algebras
type: Item
instance of: scholarly article
title: Analytic properties in the spectrum of certain Banach algebras
zbMATH Open document ID: 1176.32004
author: Linus Carlsson
published in: Mathematische Zeitschrift
publication date: 2008-12-01T00:00:00Z
review text: Let \(D\subset \mathbb{C}^n\) be a bounded pseudoconvex domain and let \(S(bD)\) denote the set of strictly pseudoconvex boundary points. By \(\mathcal{B}(D)\), the author denotes one of the Banach algebras \(H^\infty (D)\) or \(A(D)\) or \(A^k(D).\) The domain \(D\) is said to have the Gleason \(\mathcal{B}\) property at \(p\in D\) if for every \(f\in \mathcal{B}(D)\) such that \(f(p)=0\), there exist functions \(f_1,\dots , f_n\in \mathcal{B}(D)\) such that \(f(z)=\sum (z_j-p_j)f_j(z)\) for all \(z\in D.\) Now let \(V \subset \subset S(bD)\cap C^2\) be a non-empty open set in the intersection of \(bD\) and the convex hull of \(D.\) It is shown that for every point \(\xi \in V\), there exists an open neighborhood \(U_\xi\) of \(\xi\) such that \(D\) has the Gleason \(\mathcal{B}\) property at \(p\) for all \(p\in U_\xi \cap D.\) In the sequel, the author discusses some properties concerning spectrum schlichtness following the approach of [\textit{U. Backlund} and \textit{A. Fällström}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 26, No.~3, 595--603 (1998; Zbl 0933.32002)].
reviewed by: Fritz Haslinger
Mathematics Subject Classification ID: 32A65
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 46J20
zbMATH DE Number: 5376028
zbMATH Keywords: holomorphic functions
zbMATH Keywords: Banach algebras
zbMATH Keywords: Nebenhülle
zbMATH Keywords: \({\overline\partial}\) -problems
zbMATH Keywords: generalized Shilov boundary
MaRDI profile type: Publication
full work available at URL: s00209-008-0322-9
OpenAlex ID: W2054368053
cites work: Q4230944
cites work: A Generalized Shilov Boundary and Analytic Structure
cites work: Nebenhülle and the Gleason problem
cites work: Trivial generators for nontrivial fibres
cites work: Spectrum of certain Banach algebras and ∂-problems
cites work: Q4318838
cites work: Q5727911
cites work: Extension of holomorphic functions
DOI: 10.1007/S00209-008-0322-9
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description: scientific article; zbMATH DE number 5376028