On generalized limits and ultrafilters

Leonetti P., ORHAN C.

Journal of Mathematical Analysis and Applications, vol.556, no.1P1, 2026 (SCI-Expanded, Scopus) identifier identifier

  • Publication Type: Article / Article
  • Volume: 556 Issue: 1P1
  • Publication Date: 2026
  • Doi Number: 10.1016/j.jmaa.2025.130090
  • Journal Name: Journal of Mathematical Analysis and Applications
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Keywords: Generalized limits, Ideals on ω, Positive extensions of limit functional, Positive linear functional on ℓ∞
  • Ankara University Affiliated: Yes

Abstract

Given an ideal I on ω, we denote by SL(I) the family of positive normalized linear functionals on ℓ∞ which assign value 0 to all characteristic sequences of sets in I. We show that every element of SL(I) is a Choquet average of certain ultrafilter limit functionals. Also, we prove that the diameter of SL(I) is 2 if and only if I is not maximal, and that the latter claim can be considerably strengthened if I is meager. Lastly, we provide several applications: for instance, recovering a result of Freedman (1981) [19], we show that the family of bounded sequences for which all functionals in SL(I) assign the same value coincides with the closed vector space of bounded I-convergent sequences.