Abstract
We show that for an ideal H in an Archimedean vector lattice F the following conditions are equivalent:
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H is a projection band;
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Any collection of mutually disjoint vectors in H, which is order bounded in F, is order bounded in H;
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H is an infinite meet-distributive element of the lattice \({\mathcal {I}}_{F}\) of all ideals in F in the sense that \(\bigcap \nolimits _{J\in {\mathcal {J}}}\left( H+ J\right) =H+ \bigcap {\mathcal {J}}\), 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.
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Acknowledgements
The author wants to express gratitude to Vladimir Troitsky for many valuable discussions on the topic of this paper, and in particular for contributing to the proof of Proposition 2.1. Marten Wortel gets credit for bringing the author’s attention to the concept of a self-majorizing element. The author also thanks Joseph van Name for contributing an idea to the proof of Theorem 3.10, as well as Remy van Dobben de Bruyn and KP Hart who contributed Proposition 2.8, and the service MathOverflow which made it possible. Finally, the anonymous reviewer provided feedback that helped to improve the exposition.
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Bilokopytov, E. Characterizations of the projection bands and some order properties of the lattices of continuous functions. Positivity 28, 35 (2024). https://doi.org/10.1007/s11117-024-01050-7
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DOI: https://doi.org/10.1007/s11117-024-01050-7