Joincompact spaces, continuous lattices and \(C^*\)-algebras
type: Item
instance of: scholarly article
title: Joincompact spaces, continuous lattices and \(C^*\)-algebras
zbMATH Open document ID: 0933.54002
author: Ralph Kopperman
author: John Mack
author: M. Henriksen
author: Douglas W. B. Somerset
published in: Algebra Universalis
publication date: 2000-02-27T00:00:00Z
review text: The paper explores the similarity between two areas of mathematics: the theory of minimal primes in algebraic lattices and the functional analytic theory of minimal closed primal ideals in \(C^*\)-algebras. The basic theorems on both the algebraic side and the functional analytic side look very similar, and the aim of this paper is to explain why this is so. The authors use a characterization of these spaces, which allows convenient topological proofs: a bitopological space \((X,\tau ,\tau ^*)\) is joincompact if \(\tau \vee \tau ^*\) is quasicompact and \(T_0\), whenever \(x\) is in the \(\tau ^*\)-closure of \(y\) then \(y\) is in the \(\tau \)-closure of \(x\), and for any \(x,y\in X\), if \(x\) is not in the \(\tau \)-closure of \(y\), there exist disjoint \(T\in \tau \), \(T^* \in \tau ^*\) such that \(x\in T\) and \(y\in T^*\). Joincompact spaces are shown to be precisely the Lawson-closed subsets of continuous lattices, with the restrictions of the lower and Scott topologies. The natural duality of Nachbin's spaces permits efficient proofs of very general results on compactness, the Baire property, and coincidence topologies, for maximal and minimal elements in Lawson-closed subspaces of continuous lattices.
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 06B35
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 54C05
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54E55
zbMATH DE Number: 1407078
zbMATH Keywords: bitopological and join space
zbMATH Keywords: continuous lattice
zbMATH Keywords: \(C^*\)-algebra
MaRDI profile type: Publication
full work available at URL: s000120050056
OpenAlex ID: W2098451700
DOI: 10.1007/S000120050056
description: scientific article; zbMATH DE number 1407078