A characterization of the disk algebra
type: Item
instance of: scholarly article
title: A characterization of the disk algebra
zbMATH Open document ID: 1029.46069
author: Nazim Sadik
author: Evgeny A. Poletsky
author: Brian J. Cole
published in: Illinois Journal of Mathematics
publication date: 2003-01-01T00:00:00Z
review text: The authors show that a complex unital uniform algebra is isomorphic to the disk algebra \(A(\mathbb D)\) if and only if every closed unital subalgebra with one generator is isomorphic to the whole algebra. Moreover, every such subalgebra of \(A(\mathbb D)\) is isometrically isomorphic to \(A(\mathbb D)\). Along the way, they prove the following interesting result: Let \(f\in A(\mathbb D)\) be non-constant. Then every component of the complement of \(f(\overline {\mathbb D})\) is a Jordan domain. The proof uses the theory of prime ends from the theory of conformal mappings.
reviewed by: Raymond Mortini
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 30C35
zbMATH DE Number: 1839961
zbMATH Keywords: uniform algebras
zbMATH Keywords: disk-algebra
zbMATH Keywords: generators of subalgebras
zbMATH Keywords: polynomial convex hull
zbMATH Keywords: prime ends
zbMATH Keywords: Jordan domains
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1839961