Let A be an Archimedean f -algebra and let N (A) be the set of all nilpotent elements of A. Colville et al. [6] proved that a positive linear map D : A → A is a derivation if and only if D(A) ⊂ N (A) and D(A 2 ) = {0} , where A 2 is the set of all products ab in A. In this paper, we establish a result corresponding to the ColvilleDavis-Keimel theorem for an order bounded derivation D on an Archimedean almost f -algebra, which generalizes the results of Boulabiar [3].

The paper presents simple proofs of the Cauchy-Schwartz inequality and the negative discriminant property in archimedean almost f -algebras [5] , based on a sequence approximation.