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.