On Almost f-Algebras

Abstract

For an Archimedian vector lattice A and a Dedekind complete vector lattice B, it is shown that if a positive bimorphism ψ₀ from A × A to B satisfies the (AF) property (the image of each pair of disjoint elements is zero), then there exists a positive extension of ψ₀ to the cartesian product of the Dedekind completion of A with itself that also satisfies the (AF) property. As a consequence, we demonstrate that the multiplication in an Archimedean almost f-algebra can be extended to an almost f-algebra multiplication in the Dedekind completion, which gives a postive answer to the problem posed by C.B. Huijsmans [9, last paragraph of Section 7].