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.