# Dedekind complete and order continuous Banach $C(K)$-modules

## TL;DR

This paper extends the concepts of Dedekind completeness and order continuity from Banach lattices to Banach C(K)-modules, providing an analogue of Lozanovsky's characterization for these modules.

## Contribution

It introduces and develops the theory of Dedekind complete and sigma-Dedekind complete Banach C(K)-modules, extending classical lattice results.

## Key findings

- Established an analogue of Lozanovsky's characterization for Banach C(K)-modules.
- Extended Dedekind completeness notions to a broader class of modules.
- Provided foundational results for the structure of Banach C(K)-modules.

## Abstract

We extend the notions of Dedekind complete and sigma-Dedekind complete Banach lattices to Banach C(K)-modules. As our main result we prove for these modules an analogue of Lozanovsky's well known characterization of Banach lattices with order continuous norm.