A relationship between two almost \(f\)-algebra products
type: Item
instance of: scholarly article
title: A relationship between two almost \(f\)-algebra products
zbMATH Open document ID: 1012.06022
author: Karim Boulabiar
published in: Algebra Universalis
publication date: 2003-04-23T00:00:00Z
review text: The theory of almost \(f\)-algebras was established in 1967 by Birkhoff and it was launched in 1981 by the fundamental work of \textit{E. Scheffold} [Math. Z. 177, 193-205 (1981; Zbl 0439.46037)], in which a representation theorem was given. The starting point of the paper was the observation that if \(*\) is an almost \(f\)-algebra multiplication on \(C(X)\) (the Banach algebra of real continuous functions on a compact Hausdorff space \(X\)) then there exists a positive linear operator \(T\) such that \[ f*g=T(fg)\text{ for all }f,g\in C(X). \] More precisely, if \(A\) is a uniformly complete almost \(f\)-algebra and \(T\) a positive linear operator, necessary and sufficient conditions on \(T\) are given so that the multiplication \(*\) defined by the formula above becomes an almost \(f\)-algebra multiplication. Also the converse problem is studied. If \(A\) is a uniformly complete almost \(f\)-algebra equipped with another multiplication \(*,\) conditions on the initial multiplication are given so that there exists a positive linear operator \(T\) checking the formula above. The proofs are purely order-theoretical and do not involve any analytical means.
reviewed by: Radomír Halaš
Mathematics Subject Classification ID: 06F25
Mathematics Subject Classification ID: 46A40
zbMATH DE Number: 1899944
zbMATH Keywords: vector lattice
zbMATH Keywords: uniformly complete vector lattice
zbMATH Keywords: lattice-ordered algebra
zbMATH Keywords: almost \(f\)-algebra
zbMATH Keywords: \(d\)-algebra
zbMATH Keywords: \(f\)-algebra
MaRDI profile type: Publication
full work available at URL: s000120050164
OpenAlex ID: W1982764607
DOI: 10.1007/S000120050164
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description: scientific article; zbMATH DE number 1899944