Convergence structures and locally solid topologies on vector lattices of 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 {mathscr{L}}_{mathrm{ob}}(E,F) from E into F. Using this, it follows that {mathscr{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 {mathscr{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, as a consequence of a uniform order boundedness principle for orthomorphisms that we establish. We furthermore show that, in contrast to general order bounded operators, orthomorphisms preserve not only order convergence of nets, but unbounded order convergence and—when applicable—convergence in the Hausdorff uo-Lebesgue topology as well.