Relative uniform completion of a vector lattice

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 these 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.