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 ∈ A + is a band projection if the multiplication operator LaRa ∈Lr(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 ≤ a ≤ e and a2 = 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