The theory of almost \(f\)-algebras was established in 1967 by Birkhoff and it was launched in 1981 by the fundamental work of E. Scheffold [Math. Z. 177, 193-205 (1981; Zbl 0439.46037)], in which a representation theorem was given.
The starting point of the paper was the observation that if \(*\) is an almost \(f\)-algebra multiplication on \(C(X)\) (the Banach algebra of real continuous functions on a compact Hausdorff space \(X\)) then there exists a positive linear operator \(T\) such that \[ f*g=T(fg)\text{ for all }f,g\in C(X). \] More precisely, if \(A\) is a uniformly complete almost \(f\)-algebra and \(T\) a positive linear operator, necessary and sufficient conditions on \(T\) are given so that the multiplication \(*\) defined by the formula above becomes an almost \(f\)-algebra multiplication.
Also the converse problem is studied. If \(A\) is a uniformly complete almost \(f\)-algebra equipped with another multiplication \(*,\) conditions on the initial multiplication are given so that there exists a positive linear operator \(T\) checking the formula above.
The proofs are purely order-theoretical and do not involve any analytical means.