Joincompact spaces, continuous lattices and \(C^*\)-algebras
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.
scientific article; zbMATH DE number 1407078