Joincompact spaces, continuous lattices, and C*-algebras

Abstract.

Recent work in the ideal theory of commutative rings and that of C*-algebra is unified and generalized by first noting that these spaces are Lawson-closed subspaces of continuous lattices, equipped with the restriction of the lower topology. These topologies were first studied by Nachbin in the late 1940's (in [32]), as the topologies of those open sets in a compact Hausdorff space which are upper sets with respect to a partial order closed in its square. Such spaces have an intrinsic duality, obtained by reversing the order.¶ We use a different 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 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 here takes the form, $(X, \tau , \tau^*) \rightarrow (X, \tau^*, \tau)$, but duality is absent in general continuous lattices. Duality permits efficient proofs of very general results on compactness, the Baire property, and coincidence of topologies, for maximal and minimal elements in Lawson-closed subspaces of continuous lattices.