Band projections and order idempotents in Banach lattice algebras

TL;DR

This paper investigates the relationship between band projections and order idempotents in Banach lattice algebras, extending the understanding of their algebraic and order-theoretic properties.

Contribution

It introduces a new perspective on band projections in Banach lattice algebras and explores their connection with order idempotents, building on recent developments.

Findings

Characterization of band projections via multiplication operators

Relationship between band projections and order idempotents

Properties of the ideal generated by the identity element

Abstract

Motivated by recent work about band projections on spaces of regular operators over a Banach lattice, given a Banach lattice algebra $A$, we will say an element $a \in A_+$ is a band projection if the multiplication operator $L_aR_a\in \mathcal L_r(A)$ is a band projection. Our aim in this note is to explore the relations between this and the notion of order idempotent (those elements $a$ in a Banach lattice algebra $A$ with identity $e$ such that $0\leq a\leq e$ and $a^2=a$). We also revisit the properties of the ideal generated by the identity on a Banach lattice algebra, motivated by those of the centre of a Banach lattice.