Order topologies on \(l\)-algebras.
type: Item
instance of: scholarly article
title: Order topologies on \(l\)-algebras.
zbMATH Open document ID: 1053.06012
DOI: 10.1016/S0166-8641(03)00212-8
author: D. Massart
published in: Topology and its Applications
publication date: 2004-03-14T00:00:00Z
review text: The order topology and abstractions of the order topology are studied in the setting of lattice-ordered algebras. The order topology is characterized as the topology defined by all lattice seminorms. A seminorm is said be a lattice seminorm if \(q(x)\leq q(y)\) whenever \(| x| \leq| y| \). It is important in the context of algebras to consider m-convex sets or m-seminorms (\(q\) is an m-seminorm if it satisfies \(q(fg)\leq q(f)q(g) \) for \(f\) and \(g\) in the lattice-ordered algebra). Thus m-convex topologies can be considered. This paper provides an instructive comparison of different topologies on lattice-ordered algebras as well as conditions when the order topology is m-convex. In the special case of uniformly closed Archimedean lattice-ordered algebras where \(1\) is a weak order unit, a number of the topologies under consideration coincide. Further, if these algebras are also closed under inversion, then the order topology coincides with an appropriate topology of compact convergence (thus, m-convex). This analysis leads to a characterization of certain spaces \( C_k(X) \), all continuous real-valued functions on \(X\) with the topology of compact convergence . In particular, it is shown that for an \(f\)-algebra \(A\) there exists a topology that is locally m-convex, bornological, complete, and generated by lattice seminorms if and only if \(A\) with its order topology is isomorphic and homeomorphic to \(C_k(X)\) for a realcompact \( k_r \)-space.
Mathematics Subject Classification ID: 06F30
Mathematics Subject Classification ID: 06F25
Mathematics Subject Classification ID: 54H12
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 2056856
zbMATH Keywords: vector lattice
zbMATH Keywords: lattice-ordered algebra
zbMATH Keywords: order topology
zbMATH Keywords: topology of compact convergence
zbMATH Keywords: m-convex sets
zbMATH Keywords: m-seminorms
reviewed by: William A. Feldman
MaRDI profile type: Publication
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full work available at URL: s0166-8641(03)00212-8
OpenAlex ID: W2092562747
description: scientific article; zbMATH DE number 2056856