Convergence structures and Hausdorff uo-Lebesgue topologies on vector lattice algebras of operators

TL;DR

This paper explores various convergence structures on vector sublattices of operators on Dedekind complete vector lattices, analyzing their relationships, continuity properties, and implications for representation theory.

Contribution

It establishes the validity of implications between six convergence types and investigates continuity and adherence properties, extending results to general Dedekind complete vector lattices.

Findings

Validity of implications between convergence structures determined.

Continuity of multiplication operations analyzed.

Adherence of sublattices characterized.

Abstract

A vector sublattice of the order bounded operators on a Dedekind complete vector lattice can be supplied with the convergence structures of order convergence, strong order convergence, unbounded order convergence, strong unbounded order convergence, and, when applicable, convergence with respect to a Hausdorff uo-Lebesgue topology and strong convergence with respect to such a topology. We determine the general validity of the implications between these six convergences on the order bounded operator and on the orthomorphisms. Furthermore, the continuity of left and right multiplications with respect to these convergence structures on the order bounded operators, on the order continuous operators, and on the orthomorphisms is investigated, as is their simultaneous continuity. A number of results are included on the equality of adherences of vector sublattices of the order bounded operators and of the orthomorphisms with respect to these convergence structures. These are consequences of more general results for vector sublattices of arbitrary Dedekind complete vector lattices. The special attention that is paid to vector sublattices of the orthomorphisms is motivated by explaining their relevance for representation theory on vector lattices.