On generalized limits and ultrafilters
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915312440442880 |
|---|---|
| author | Leonetti, Paolo Orhan, Cihan |
| author_facet | Leonetti, Paolo Orhan, Cihan |
| contents | Given an ideal $\mathcal{I}$ on $ω$, we denote by $\mathrm{SL}(\mathcal{I})$ the family of positive normalized linear functionals on $\ell_\infty$ which assign value $0$ to all characteristic sequences of sets in $\mathcal{I}$. We show that every element of $\mathrm{SL}(\mathcal{I})$ is a Choquet average of certain ultrafilter limit functionals. Also, we prove that the diameter of $\mathrm{SL}(\mathcal{I})$ is $2$ if and only if $\mathcal{I}$ is not maximal, and that the latter claim can be considerably strengthened if $\mathcal{I}$ is meager. Lastly, we provide several applications: for instance, recovering a result of Freedman in [Bull. Lond. Math. Soc. 13 (1981), 224--228], we show that the family of bounded sequences for which all functionals in $\mathrm{SL}(\mathcal{I})$ assign the same value coincides with the closed vector space of bounded $\mathcal{I}$-convergent sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_23263 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On generalized limits and ultrafilters Leonetti, Paolo Orhan, Cihan Functional Analysis General Topology Given an ideal $\mathcal{I}$ on $ω$, we denote by $\mathrm{SL}(\mathcal{I})$ the family of positive normalized linear functionals on $\ell_\infty$ which assign value $0$ to all characteristic sequences of sets in $\mathcal{I}$. We show that every element of $\mathrm{SL}(\mathcal{I})$ is a Choquet average of certain ultrafilter limit functionals. Also, we prove that the diameter of $\mathrm{SL}(\mathcal{I})$ is $2$ if and only if $\mathcal{I}$ is not maximal, and that the latter claim can be considerably strengthened if $\mathcal{I}$ is meager. Lastly, we provide several applications: for instance, recovering a result of Freedman in [Bull. Lond. Math. Soc. 13 (1981), 224--228], we show that the family of bounded sequences for which all functionals in $\mathrm{SL}(\mathcal{I})$ assign the same value coincides with the closed vector space of bounded $\mathcal{I}$-convergent sequences. |
| title | On generalized limits and ultrafilters |
| topic | Functional Analysis General Topology |
| url | https://arxiv.org/abs/2505.23263 |