Let A be an uniformly complete almost f -algebra. Then (A) = {ab : a, b ∈ A} is a positively generated ordered vector subspace of A with (A) = {a 2 : a ∈ A} as a positive cone. If T : (A) → A is a positive linear operator, we put ρ : A → L b (A) the linear operator defined by ρ(a) = T a with T a (b) = T (ab) for all b ∈ A(L b (A) is the algebra of all order bounded linear operators of A). Let L T b (A) denote the range of ρ and let's define a new product * by putting athis shows that if it happens that the product * is associative then A is an almost falgebra with respect to this new product. It turns out that a necessarily and sufficient condition in order that * be an associative product is that L T b (A) is a commutative subalgebra of L b (A). We find necessarily and sufficient conditions on T in order that * is an almost f -algebra (respect.; d-algebra, f -algebra) product. Such conditions are described in terms of the algebraic and order structure of the algebra L T b (A). The converse problem is also studied. More precisely, let A be an uniformly complete almost f -algebra and assume that * is another almost f -algebra product on A. The aim is to find sufficient conditions in order that there exist T : (A) → A such that a * b = T (ab) for all a, b ∈ A. It will be showed that a sufficient condition is that A is a d-algebra with respect to the initial product. An example is produced which shows that the condition "A is a d-algebra with respect to the initial product" can not be weakened.
In this paper we study properties of bilinear maps of order bounded variation. Theorems of preservation of properties in passage to the triadjoint and the tensor product are presented.
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