Pervasiveness of ${\mathcal{L}}^r(E,F)$ in ${\mathcal{L}}^r(E,F^{\delta })$

Abstract

Let $E,F$ be Archimedean Riesz spaces, and let $F^{\delta }$ denote an order completion of $F$. In this note, we provide necessary conditions under which the space of regular operators ${\mathcal{L}}^r(E,F)$ is pervasive in ${\mathcal{L}}^r(E,F^{\delta })$. Pervasiveness of ${\mathcal{L}}^r(E,F)$ in ${\mathcal{L}}^r(E,F^{\delta })$ implies that the Riesz completion of ${\mathcal{L}}^r(E,F)$ can be realized as a Riesz subspace of ${\mathcal{L}}^r(E,F^{\delta }$. It also ensures that the regular part of the space of order continuous operators ${\mathcal{L}}^{oc}(E,F)$ forms a band of ${\mathcal{L}}^r(E,F)$. Furthermore, the positive part $T^{+}$ of any operator $T\in {\mathcal{L}}^r(E,F)$, provided it exists, is given by the Riesz-Kantorovich formula. The results apply in particular to cases where $E={\ell }_0^{\infty }$, $E=c$, or $F$ is atomic, and they provide solutions to some problems posed in [3] and [16].