A characterization of a class of locally compact abelian groups via Korovkin theory
type: Item
instance of: scholarly article
title: A characterization of a class of locally compact abelian groups via Korovkin theory
zbMATH Open document ID: 0682.22005
DOI: 10.1007/BF02570885
author: Michael Pannenberg
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183782
review text: Let G be a locally compact abelian (LCA) group. We characterize the fact that the associated group algebra \(L^ 1(G)\) possesses a finite universal Korovkin system by a property of G as well as by a dual property of the character group \({\mathbb{X}}=\hat G:\) It turns out that the first assertion is equivalent to the fact that \({\mathbb{X}}\) is a finite- dimensional second countable group, or equivalently, that G is a second countable group such that a certain discrete factor group has finite torsion-free rank. The latter class of LCA groups is characterized by the existence of a finite family of functions in \(L^ 1(G)\) whose Fourier transforms strongly separate the points of \({\mathbb{X}}=\hat G\). We also discuss the relation of the above results to the Menger-Nöbeling theorem and give a vector-valued version of the above characterization.
Mathematics Subject Classification ID: 22B10
Mathematics Subject Classification ID: 43A20
Mathematics Subject Classification ID: 43A40
zbMATH DE Number: 4116837
zbMATH Keywords: group algebra
zbMATH Keywords: finite universal Korovkin system
zbMATH Keywords: character group
zbMATH Keywords: finite- dimensional second countable group
zbMATH Keywords: finite torsion-free rank
zbMATH Keywords: LCA groups
zbMATH Keywords: Fourier transforms
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4116837