Band projections in spaces of regular operators

TL;DR

This paper introduces inner band projections in regular operator spaces on Dedekind complete Banach lattices, providing new characterizations of atomic order continuous lattices and multiplication operators as band projections.

Contribution

It offers a novel characterization of atomic order continuous Banach lattices and characterizes multiplication operators that are band projections.

Findings

Characterization of atomic order continuous Banach lattices via band projections

Identification of multiplication operators that are band projections

Introduction of inner band projections in regular operator spaces

Abstract

We introduce inner band projections in the space of regular operators on a Dedekind complete Banach lattice and study some structural properties of this class. In particular, we provide a new characterization of atomic order continuous Banach lattices as those for which all band projections in the corresponding space of regular operators are inner. We also characterize the multiplication operators $L_AR_B$ which are band projections precisely as those with $A,B$ being band projections up to a scalar multiple.