Pervasiveness of $\mathcal{L}^r(E,F)$ in $\mathcal{L}^r(E,F^δ)$
Quinn Kiervin Starkey, Foivos Xanthos
TL;DR
This work identifies conditions under which the regular operator space $\mathcal{L}^r(E,F)$ is pervasive in its completion $\mathcal{L}^r(E,F^{\delta})$, allowing results typically requiring order completeness to be derived from pervasiveness. The authors prove a central result: when $\mathcal{L}^r(E,F)$ is pervasive in $\mathcal{L}^r(E,F^{\delta})$, the Riesz completion $\mathcal{L}^r(E,F)^{\rho}$ can be realized as the Riesz subspace of $\mathcal{L}^r(E,F^{\delta})$ generated by $\mathcal{L}^r(E,i(F))$, and $\mathcal{L}^r(E,F)$ enjoys the Riesz-Kantorovich property; in addition, $\mathcal{L}^{oc}(E,F)\cap \mathcal{L}^r(E,F)$ forms a band in $\mathcal{L}^r(E,F)$. The paper further identifies sufficient pervasiveness conditions, notably when $F$ is atomic, and derives explicit band-projection results and convergence characterizations for domains like $E=\ell_0^{\infty}$ or $E=c$. It also shows that $\mathcal{L}^r(E,\ell_0^{\infty})$ always has the Riesz-Kantorovich property, and discusses intricate nuances such as $\mathcal{L}^{oc}(\ell_0^{\infty},F)$ being a band but not necessarily directed, thereby addressing several open problems from prior literature.
Abstract
Let $E, F$ be Archimedean Riesz spaces, and let $F^δ$ 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 $\mathcal{L}^r(E, F^δ)$. Pervasiveness of $\mathcal{L}^r(E, F)$ in $\mathcal{L}^r(E, F^δ)$ implies that the Riesz completion of $ \mathcal{L}^r(E, F)$ can be realized as a Riesz subspace of $ \mathcal{L}^r(E, F^δ$. 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].