Stone–Weierstrass theorems for Riesz ideals of continuous functions

• Schötz, Matthias ^[1]
  1. [1] Université Libre de Bruxelles

     Université Libre de Bruxelles

     Arrondissement Brussel-Hoofdstad, Bélgica

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 72, Fasc. 3, 2021, págs. 587-603
• Idioma: inglés
• DOI: 10.1007/s13348-020-00301-6
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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.

• Referencias bibliográficas
  • 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),...
  • Sb, Ng, Warner, S.: Continuity of positive and multiplicative functionals. Duke Math. J. 39(2), 281–284 (1972). https://doi-org.sire.ub.edu/10.1215/S0012-7094-72-03933-6
  • Schmüdgen, K.: The Moment Problem. Springer, Berlin (2017)