2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/127826This is a companion to recent papers of the authors; here we construct the `noncommutative Shilov boundary' of a (possibly nonunital) selfadjoint ordered space of Hilbert space operators. The morphisms in the universal property of the boundary preserve order. As an application, we consider `maximal' and `minimal' unitizations of such ordered operator spaces.11 pages. To appear, PositivityOperator AlgebrasFunctional Analysistext