dorsal/arxiv
View SchemaRelative uniform completion of a vector lattice
| Authors | Eugene Bilokopytov, Vladimir G. Troitsky |
|---|---|
| Categories | |
| ArXiv ID | 2601.09015vv1 |
| URL | https://arxiv.org/abs/2601.09015 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
In the paper, we revisit several approaches to the concept of uniform completion $X^{\mathrm{ru}}$ of a vector lattice $X$. We show that many of this approaches yield the same result. In particular, if $X$ is a sublattice of a uniformly complete vector lattice $Z$ then $X^{\mathrm{ru}}$ may be viewed as the intersection of all uniformly complete sublattices of $Z$ containing $X$. $X^{\mathrm{ru}}$ may also be constructed via a transfinite process of taking uniform adherences in $Z$ with regulators coming from the previous adherences. If, in addition, $X$ is majorizing in $Z$ then $X^{\mathrm{ru}}$ may be viewed as the uniform closure of $X$ in $Z$. We show that $X^{\mathrm{ru}}$ may also be characterized via a universal property: every positive operator from $X$ to a uniformly complete vector lattice extends uniquely to $X^{\mathrm{ru}}$. Moreover, the class of positive operators here may be replaced with several other important classes of operators (e.g., lattice homomorphisms). We also discuss conditions when the uniform adherence of a sublattice equals its uniform closure, and present an example (based on a construction by R.N. Ball and A.W. Hager) where this fails.
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"abstract": "In the paper, we revisit several approaches to the concept of uniform completion $X^{\\mathrm{ru}}$ of a vector lattice $X$. We show that many of this approaches yield the same result. In particular, if $X$ is a sublattice of a uniformly complete vector lattice $Z$ then $X^{\\mathrm{ru}}$ may be viewed as the intersection of all uniformly complete sublattices of $Z$ containing $X$. $X^{\\mathrm{ru}}$ may also be constructed via a transfinite process of taking uniform adherences in $Z$ with regulators coming from the previous adherences. If, in addition, $X$ is majorizing in $Z$ then $X^{\\mathrm{ru}}$ may be viewed as the uniform closure of $X$ in $Z$. We show that $X^{\\mathrm{ru}}$ may also be characterized via a universal property: every positive operator from $X$ to a uniformly complete vector lattice extends uniquely to $X^{\\mathrm{ru}}$. Moreover, the class of positive operators here may be replaced with several other important classes of operators (e.g., lattice homomorphisms). We also discuss conditions when the uniform adherence of a sublattice equals its uniform closure, and present an example (based on a construction by R.N. Ball and A.W. Hager) where this fails.",
"arxiv_id": "2601.09015",
"authors": [
"Eugene Bilokopytov",
"Vladimir G. Troitsky"
],
"categories": [
"math.FA"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Relative uniform completion of a vector lattice",
"url": "https://arxiv.org/abs/2601.09015",
"version": "v1"
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