A relationship between two almost f -algebra products

Abstract

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.

Characterization of hyper-archimedean vector lattices via disjointness preserving bilinear maps

“…Moreover, he showed in [4] that if (A, * ) is, in addition, an almost f -algebra, then there exists a positive operator T from Π(A) into A such that T (ab) = a * b, for all a, b ∈ A. In fact, by using the same argument as in ( [4], Theorem 5.4) we deduce the following result. Ψ(x, y) = T (x * y).…”

Lattice bimorphisms on f -algebras

“…In other words, the multiplications by positive elements in the d-algebra A are lattice homomorphisms. Contrary to the Archimedean f -algebras, Archimedean d-algebras need not be commutative nor have positive squares (see [4,Example 2.4]). For elementary theory of d-algebras we refer to [2,16].…”

“…Some years after, Bernau and Huijsmans in [3] generalized (4) to the case where A is an arbitrary Archimedean f -algebra. Later in [5], Buskes and van Rooij established the Cauchy-Schwarz inequality for Archimedean almost f -algebras (we say that an -algebra A is an almost f -algebra whenever f ∧ g = 0 implies f g = 0) and therefore for Archimedean commutative d-algebras (note that any commutative d-algebra is an almost f -algebra).…”

On the Structure of Almost F-Algebras

“…On the other hand, it follows from Proposition 1 that a (A & ) p is a vector sublattice of B with a as positive cone. Therefore, there exists 0< he A 6 such that a (a) a (f3) -a (/) a (g) = a (h) 2 . Hence, by (4.2), we get and the theorem follows.…”

“…The closure of A in A 6 with respect to the (relatively) uniform topology is a uniformly complete vector lattice denoted by A ru (for definition and basic properties of the (relatively) uniform topology we refer to [10, §16] 6 and (A ru ) u = A u . On the other hand, it is well-known that if A is an ¿-algebra then the multiplication in A extends uniquely to a multiplication in A TU in such a fashion that A ru is again a (uniformly complete) ¿-algebra.…”