Abstract
We give several characterizations of order continuous vector lattice homomorphisms between Archimedean vector lattices. We reduce the proofs of some of the equivalences to the case of composition operators between vector lattices of continuous functions, and we obtain a characterization of order continuity of such operators. Motivated by this, we investigate various properties of the sublattices of the space \(\mathcal {C}\left( X\right) \), where X is a Tychonoff topological space. We also obtain several characterizations of a regular sublattice of a vector lattice, and show that the closure of a regular sublattice of a Banach lattice is also regular.
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Acknowledgements
The author wants to thank Vladimir Troitsky for many valuable discussions on the topic of this paper, and Taras Banakh who contributed an idea for Example 7.6 and the service MathOverflow which made it possible.
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Bilokopytov, E. Order continuity and regularity on vector lattices and on lattices of continuous functions. Positivity 27, 52 (2023). https://doi.org/10.1007/s11117-023-01002-7
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DOI: https://doi.org/10.1007/s11117-023-01002-7