Band projections and order idempotents in Banach lattice algebras

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.