Abstract
Notions of convergence and continuity specifically adapted to Riesz ideals \(\mathscr {I}\) of the space of continuous real-valued functions on a Lindelöf locally compact Hausdorff space are given, and used to prove Stone–Weierstrass-type theorems for \(\mathscr {I}\). As applications, sufficient conditions are discussed that guarantee that various types of positive linear maps on \(\mathscr {I}\) are uniquely determined by their restriction to various point-separating subsets of \(\mathscr {I}\). A very special case of this is the characterization of the strong determinacy of moment problems, which is rederived here in a rather general setting and without making use of spectral theory.
Similar content being viewed by others
References
Bishop, E.: A generalization of the Stone–Weierstrass theorem. Pac. J. Math. 11(3), 777–783 (1961)
Dugundji, J.: Topology. Allyn and Bacon, inc, Boston (1978)
Jurzak, J.P.: Dominated convergence and Stone–Weierstrass theorem. J. Appl. Anal. 11(2), 207–223 (2010)
Kelley, J.L.: General Topology. D. van Nostrand Company Inc, New York (1955)
Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North-Holland Publishing Company, Amsterdam (1971)
Nachbin, L.: Weighted approximation for algebras and modules of continuous functions: real and self-adjoint complex cases. Ann. Math. 81(2), 289–302 (1965)
Sb, Ng, Warner, S.: Continuity of positive and multiplicative functionals. Duke Math. J. 39(2), 281–284 (1972). https://doi.org/10.1215/S0012-7094-72-03933-6
Schmüdgen, K.: The Moment Problem. Springer, Berlin (2017)
Acknowledgements
I would like to thank Prof. K. Schmüdgen for some valuable hints and remarks. The author is “Boursier de l’ULB” (stipendiary of the Université libre de Bruxelles). This work was supported by the Fonds de la Recherche Scientifique (FNRS) and the Fonds Wetenschappelijk Onderzoek - Vlaaderen (FWO) under EOS Project no. 30950721.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Schötz, M. Stone–Weierstrass theorems for Riesz ideals of continuous functions. Collect. Math. 72, 587–603 (2021). https://doi.org/10.1007/s13348-020-00301-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13348-020-00301-6