Relative uniform completion of a vector lattice
Eugene Bilokopytov, Vladimir G. Troitsky
TL;DR
This work investigates relative uniform (ru) completion of vector lattices, clarifying how multiple natural constructions—intersection of uniformly complete sublattices, uniform closure within an ambient space, transfinite uniform adherences, and universal properties for operator extensions—align under suitable assumptions while exposing ambient-space dependencies. It proves that the ru-completion $X^{\mathrm{ru}}$ is the least uniformly complete lattice containing $X$, and, when possible, coincides with the uniform closure of $X$ in the order completion $X^{\delta}$ or in a majorizing ambient lattice $Y$; it also frames ru-completion via a universal property for several operator classes, including lattice homomorphisms and positive operators. The paper provides structural results on the adherence of $X$ with regulators, and identifies conditions (e.g., $\sigma$-PP, PP) under which the first ru-adherence equals the uniform closure in $X^{\delta}$, linking to representations in $C(K)$ spaces and norm completions. It also presents Ball–Hager type examples demonstrating the necessity of certain hypotheses, showing that ru-adherence can fail to be ru-closed in general, with concrete constructions in $C(P)$ and related spaces, and discusses implications for ambient-space independence and universal completions.
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.