Joincompact Spaces, Continuous Lattices, and C*-algebras
Document Type
Article
Department
Mathematics (HMC)
Publication Date
5-19-1997
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, τ, τ*) is joincompact if τ∨τ* is quasicompact and T0, whenever x is in the τ*-closure of y then y is the τ-closure of x, and for any x, y ε X, if x is not in the τ-closure of y, there
exist disjoint Τ ε τ, T* ε τ* such that x ε T and y ε 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, τ, τ*) → (X, τ*, τ), 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.
Rights Information
© 1997 Birkhauser Verlag
DOI
10.1007/s000120050056
Recommended Citation
Henriksen, M., Kopperman, R., Mack, J., and Somerset, D. W. B. 1997. Joincompact spaces, continuous lattices, and C*-algebras. Algebra Universalis. 38(3):289-323.
