Characterizations of the projection bands and some order properties of the lattices of continuous functions
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911855431122944 |
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| author | Bilokopytov, Eugene |
| author_facet | Bilokopytov, Eugene |
| contents | We show that for an ideal $H$ in an Archimedean vector lattice $F$ the following conditions are equivalent:
$\bullet$ $H$ is a projection band;
$\bullet$ Any collection of mutually disjoint vectors in $H$, which is order bounded in $F$, is order bounded in $H$;
$\bullet$ $H$ is an infinite meet-distributive element of the lattice $\mathcal{I}_{F}$ of all ideals in $F$ in the sense that $\bigcap\limits_{J\in \mathcal{J}}\left(H+ J\right)=H+ \bigcap \mathcal{J}$, for any $\mathcal{J}\subset \mathcal{I}_{F}$.
Additionally, we show that if $F$ is uniformly complete and $H$ is a uniformly closed principal ideal, then $H$ is a projection band. In the process we investigate some order properties of lattices of continuous functions on Tychonoff topological spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_11192 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Characterizations of the projection bands and some order properties of the lattices of continuous functions Bilokopytov, Eugene Functional Analysis General Topology 06E15, 46A40, 46E25 We show that for an ideal $H$ in an Archimedean vector lattice $F$ the following conditions are equivalent: $\bullet$ $H$ is a projection band; $\bullet$ Any collection of mutually disjoint vectors in $H$, which is order bounded in $F$, is order bounded in $H$; $\bullet$ $H$ is an infinite meet-distributive element of the lattice $\mathcal{I}_{F}$ of all ideals in $F$ in the sense that $\bigcap\limits_{J\in \mathcal{J}}\left(H+ J\right)=H+ \bigcap \mathcal{J}$, for any $\mathcal{J}\subset \mathcal{I}_{F}$. Additionally, we show that if $F$ is uniformly complete and $H$ is a uniformly closed principal ideal, then $H$ is a projection band. In the process we investigate some order properties of lattices of continuous functions on Tychonoff topological spaces. |
| title | Characterizations of the projection bands and some order properties of the lattices of continuous functions |
| topic | Functional Analysis General Topology 06E15, 46A40, 46E25 |
| url | https://arxiv.org/abs/2211.11192 |