Pervasiveness of Lr(E,F) in Lr(E,Fδ)

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 all regular operators $L^r(E,F)$ is pervasive in $L^r(E,F^{\delta })$. Pervasiveness of $L^r(E,F)$ in $L^r(E,F^{\delta })$ implies that the Riesz completion of $L^r(E,F)$ can be realized as a Riesz subspace of $L^r(E,F^{\delta })$. It also ensures that the regular part of the space of order continuous operators $L^{oc}(E,F)$ forms a band of $L^r(E,F)$. Furthermore, the positive part $T^{+}$ of any operator $T\in L^r(E,F)$, 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 Abramovich and Wickstead (1991) and Wickstead (2024).