  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.
