A short note on band projections in partially ordered vector spaces

TL;DR

This paper extends the concept of band projections from vector lattices to more general Archimedean partially ordered vector spaces, showing that positive projections with positive complements have ranges that are bands.

Contribution

It demonstrates that band projections in vector lattices can be generalized to Archimedean partially ordered vector spaces, establishing conditions under which their ranges are bands.

Findings

Range of positive projections with positive complements are bands

Extension of band projection concept to pre-Riesz spaces

Bridges the gap between vector lattice theory and ordered vector spaces

Abstract

Consider an Archimedean partially ordered vector space $X$ with generating cone (or, more generally, a pre-Riesz space $X$). Let $P$ be a linear projection on $X$ such that both $P$ and its complementary projection $I - P$ are positive; we prove that the range of $P$ is a band. This shows that the well-known concept of band projections on vector lattices can, to a certain extent, be transferred to the framework of ordered vector spaces.