A characterization of a class of locally compact abelian groups via Korovkin theory
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.
finite- dimensional second countable group
scientific article; zbMATH DE number 4116837
scientific article