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Extensions of operators between ordered vector spaces

  • Eugene Bilokopytov
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Abstract

We prove analogues of several classical theorems about operators between vector lattices in the context of directed Archimedean ordered vector spaces. In particular, we show that every order continuous operator between pervasive Archimedean ordered vector spaces can be extended to an order continuous operator between their Dedekind completions. Additionally, we introduce the universal completion of a directed Archimedean ordered vector space, and show that every supremum-preserving operator between directed Archimedean ordered vector spaces can be extended to an order continuous homomorphism between their universal completions. We also establish some universal properties for the relative uniform completion.

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Cite

BibTeXRIS
BibTeX
@misc{bilokopytov2026extensions,
  title = {{Extensions of operators between ordered vector spaces}},
  author = {Eugene Bilokopytov},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08735},
  eprint = {2611.08735},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08735},
}
RIS
TY  - UNPB
TI  - Extensions of operators between ordered vector spaces
AU  - Eugene Bilokopytov
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.08735
UR  - https://arxiv.org/abs/2611.08735
AB  - We prove analogues of several classical theorems about operators between vector lattices in the context of directed Archimedean ordered vector spaces. In particular, we show that every order continuous operator between pervasive Archimedean ordered vector spaces can be extended to an order continuous operator between their Dedekind completions. Additionally, we introduce the universal completion of a directed Archimedean ordered vector space, and show that every supremum-preserving operator between directed Archimedean ordered vector spaces can be extended to an order continuous homomorphism between their universal completions. We also establish some universal properties for the relative uniform completion.
ER  -