On the Structure of Almost F-Algebras

Abstract

Abstract. This paper is mainly concerned with a representation theorem for almost /-algebras.

On orthosymmetric bilinear maps

“…From then on the interest in orthosymmetric bilinear maps has steadily grown and a series of papers devoted to the subject appeared in the literature. See [4,5,9,15]. The following result due to Buskes-van Rooij in [10, Corollary 2] for positive orthosymmetric bilinear maps and to Buskes-Kusraev in [9, 3.4] for order bounded orthosymmetric bilinear maps play a key role in the context of this paper.…”

“…, f p ∈ L where L is an -algebra and p ≥ 2 is natural number. If L is a uniformly complete almost f -algebra and p ≥ 3 then L p is a uniformly complete semiprime f -algebra under the ordering and multiplication inherited from L, see [3,Theorems 4,5,and 6]. Contrary to L p ( p ≥ 3) , L 2 need not be a vector lattice under the ordering inherited from L as is proved in [7, Example 6.6].…”

The Dedekind Completion of d-Algebras

“…The bilinear map A×A → B is said to be positive if f g ∈ B + for all f g ∈ A + The positive bilinear map A×A → B is called orthosymmetrical map if f ∧g = 0 implies f g = 0 It is shown in [8] that every orthosymmetrical map is symmetrical and that extends to an othosymmetrical map from A ×A into B (for representation-free proofs of these facts, see [5,7]). …”

“…Recently, Buskes and van Rooij gave a positive answer to the Huijsmans's question for the class of almost f -algebras (see [8]). The d-algebra case was partially solved by Boulabiar and the author, who proved that there exists a d-algebra multiplication on the Dedekind completion of an Archimedean commutative d-algebra (see [7]). The principal purpose of this paper is to improve this latter result, namely that the multiplication in an Archimedean (not necessarily commutative) d-algebra A can be extended to a multiplication in the Dedekind completion A of A in such a fashion that A becomes a d-algebra under this extended multiplication.…”

On Riesz homomorphisms in unital $f$-algebras

“…This conjecture was proved by Buskes and van Rooij in [7,Theorem 4.1] for the class of almost f -algebras. In contrast to the original approach of Buskes and van Rooij, Boulabiar and the author gave in [6] an order theoretical and algebraic approach of the Buskes and van Rooij theorem. As an application, Boulabiar and the author gave in [6] an affirmative answer to Huijsmans's question for the class of commutative d-algebras.…”