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On Products in Lattice-ordered Algebras

Part of: Ordered structures Topological linear spaces and related structures

Published online by Cambridge University Press:  09 April 2009

Karim Boulabiar
Karim Boulabiar
Affiliation:
Département des Classes Préparatoires Institut Préparatoire aux Etudes Scientifiques et Techniques Université7 Novembre á Carthage BP 51, 2070-La MarsaTunisia e-mail: karim.boulabiar@ipest.rnu.tn
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Abstract

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Let 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.

Keywords

vector latticeuniformly complete vector latticepositive operatorlattice-ordered algebraalmost f-algebrad-algebraf-algebra.

MSC classification

Secondary: 06F25: Ordered rings, algebras, modules 46A40: Ordered topological linear spaces, vector lattices
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 75 , Issue 1 , August 2003 , pp. 23 - 40
Copyright
Copyright © Australian Mathematical Society 2003

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