Weak-star closed algebras containing R(K)
type: Item
instance of: scholarly article
title: Weak-star closed algebras containing R(K)
zbMATH Open document ID: 0677.46041
DOI: 10.1007/BF01197975
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: Let \(K\subset {\mathbb{C}}\) be a compact subset of the complex plane, and let R(K) be the function algebra consisting of the uniform limits on K of rational functions with poles off K. Let \(\nu\) be a finite positive measure on the boundary of K, and let \(R^{\infty}(\nu)\) be the weak- star closure of R(K) in \(L^{\infty}(\nu)\). This paper deals with the problem of classifying the weak-star closed subalgebras of \(L^{\infty}(\nu)\) containing \(R^{\infty}(\nu)\). When K is the unit disk and \(\nu\) is the arc-length measure on the boundary of K, then \(R^{\infty}(\nu)=H^{\infty}(d\theta)\) is a maximal weak-star closed subalgebra of \(L^{\infty}(\nu)\). There is a similar weak-star maximality theorem in the case K is finitely connected and has connected interior and \(\nu\) is a harmonic measure on the boundary of K. The author deals with the case of infinitely connected sets K such that (1) the interior of K is connected and dense in K, (2) there is a free boundary arc in the boundary of K, (3) R(K) is pointwise boundedly dense in \(H^{\infty}(K^ 0).\) The main result, which describes the proper weak-star closed subalgebras of \(L^{\infty}(\nu)\) containing \(R^{\infty}(\nu)\), is then applied to the roadrunner set and the champagne bubble set.
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 46J30
Mathematics Subject Classification ID: 46J10
zbMATH DE Number: 4109385
zbMATH Keywords: function algebra
zbMATH Keywords: classifying the weak-star closed subalgebras of \(L^{\infty }(\nu )\) containing \(R^{\infty }(\nu )\)
zbMATH Keywords: weak-star maximality theorem
zbMATH Keywords: harmonic measure
zbMATH Keywords: roadrunner set
zbMATH Keywords: champagne bubble set
author: T. W. Gamelin
reviewed by: João B. Prolla
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01197975
OpenAlex ID: W2086415531
description: scientific article; zbMATH DE number 4109385