Approximate regularity of commutative Beurling algebras and Korovkin approximation
type: Item
instance of: scholarly article
title: Approximate regularity of commutative Beurling algebras and Korovkin approximation
zbMATH Open document ID: 0765.46038
DOI: 10.1007/BF01301072
author: S. Singh
published in: Monatshefte für Mathematik
publication date: 1993-05-12T00:00:00Z
full work available at URL: https://eudml.org/doc/178567
review text: We study generalizations of the classical Korovkin theorem to the context of commutative Beurling algebras associated to a weight \(\omega\) on a locally compact abelian group \(G\). On the way, we discuss the parametrization of the possible spectra of commutative Beurling algebras by the rate of growth \(\Omega\) of the weight \(\omega\) and deduce an analogue of Domar's famous characterization of the regularity of a Beurling algebra via the norm-quasianalyticity of \(\omega\): It turns out that a commutative Beurling algebra is approximately regular iff \(\omega\) has rate of growth identically one. We also calculate diverse stable ranks of commutative Beurling algebras in terms of the dimension of the dual group. We finally use our results to characterize those weighted locally compact abelian groups \((G,\omega)\) whose associated Beurling algebra contains a finite universal Korovkin system: These are exactly the second countable groups \(G\) whose dual group has finite covering dimension, endowed with a weight function having rate of growth identically one. Stated in terms of the Beurling algebra, there exists a finite universal Korovkin system if and only if the Beurling algebra is separable as well as approximately regular and has finite stable rank.
Mathematics Subject Classification ID: 46J05
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 41A36
zbMATH DE Number: 164929
zbMATH Keywords: commutative Beurling algebras associated to a weight
zbMATH Keywords: norm- quasianalyticity
zbMATH Keywords: approximately regular
zbMATH Keywords: dimension of the dual group
zbMATH Keywords: universal Korovkin system
zbMATH Keywords: finite stable rank
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 164929