An analytic algebra without analytic structure in the spectrum
type: Item
instance of: scholarly article
title: An analytic algebra without analytic structure in the spectrum
zbMATH Open document ID: 0678.32010
DOI: 10.1007/BF02386361
published in: Arkiv för Matematik
publication date: 1989-01-01T00:00:00Z
review text: This paper deals with the following example due to J. Wermer: There is a compact subset Y of \({\mathbb{C}}^ 2\) such that \(X\setminus Y\) contains no analytic variety of positive dimension, where X stands for the polynomially convex hull of Y [\textit{J. Wermer}, Ark. Mat. 20, 129-135 (1982; Zbl 0491.32013)]. Here the author proves that, nonetheless, the identity principle holds for the algebra \({\mathcal P}(X)\) of continuous complex-valued functions on X which are uniform limits of polynomials. The proof is quite delicate and uses techniques of Banach algebras and complex analysis. The author also includes an example showing that for an arbitrary Y the conclusion above may fail.
Mathematics Subject Classification ID: 32E20
Mathematics Subject Classification ID: 32B05
zbMATH DE Number: 4111044
zbMATH Keywords: polynomial convexity
zbMATH Keywords: analytic algebras
zbMATH Keywords: identity principle
author: Helmut Goldmann
reviewed by: Jesús M. Ruiz
MaRDI profile type: Publication
cites work: Polynomially convex hulls and analyticity
cites work: Q4104417
cites work: Q3767598
description: scientific article; zbMATH DE number 4111044