On Products in Lattice-ordered Algebras

Abstract

AbstractLet A be a uniformly complete vector sublattice of an Archimedean semiprime f-algebra B and p ∈ {1, 2,…}. It is shown that the set ΠBp (A) = {f1 … fp: fk ∈ A, k = 1, …, p } is a uniformly complete vector sublattice of B. Moreover, if A is provided with an almost f-algebra multiplication * then there exists a positive operator Tp, from ΠBp(A) into A such that fi *…* fp = Tp(f1 …fp) for all f1…fp ∈ A.As application, being given a uniformly complete almost f-algebra (A, *) and a natural number p ≧ 3, the set Π*p(A) = {f1 *… *fp: fk ∈ A, k = 1…p} is a uniformly complete semiprime f-algebra under the ordering and the multiplication inherited from A.