Convergence structures and locally solid topologies on vector lattices of operators

TL;DR

This paper develops new locally solid topologies on the space of order bounded operators between vector lattices, establishing convergence properties and relationships, especially for orthomorphisms, with implications for operator theory.

Contribution

Introduces a locally solid absolute strong operator topology on order bounded operators and analyzes convergence structures, extending known results for orthomorphisms on vector lattices.

Findings

The space of operators admits a Hausdorff uo-Lebesgue topology under certain conditions.

Five of six possible inclusions of convergence structures are valid for orthomorphisms.

Orthomorphisms preserve order, unbounded order, and uo-Lebesgue convergence, unlike general operators.

Abstract

For vector lattices $E$ and $F$, where $F$ is Dedekind complete and supplied with a locally solid topology, we introduce the corresponding locally solid absolute strong operator topology on the order bounded operators $\mathcal L_{\mathrm{ob}}(E,F)$ from $E$ into $F$. Using this, it follows that $\mathcal L_{\mathrm{ob}}(E,F)$ admits a Hausdorff uo-Lebesgue topology whenever $F$ does. For each of order convergence, unbounded order convergence, and-when applicable-convergence in the Hausdorff uo-Lebesgue topology, there are both a uniform and a strong convergence structure on $\mathcal L_{\mathrm {ob}}(E,F)$. Of the six conceivable inclusions within these three pairs, only one is generally valid. On the orthomorphisms of a Dedekind complete vector lattice, however, five are generally valid, and the sixth is valid for order bounded nets. The latter condition is redundant in the case of sequences of orthomorphisms on a Banach lattice, as a consequence of a uniform order boundedness principle for orthomorphisms that we establish. We also show that, in contrast to general order bounded operators, the orthomorphisms preserve not only order convergence of nets, but unbounded order convergence and -- when applicable -- convergence in the Hausdorff uo-Lebesgue topology as well.