The Spectral Theory of Distributive Continuous Lattices

In this paper various properties of the spectrum (i.e. the set of prime elements endowed with the hull-kernel topology) of a distributive continuous lattice are developed. It is shown that the spectrum is always a locally quasicompact sober space and conversely that the lattice of open sets of a locally quasicompact sober space is a continuous lattice. Algebraic lattices are a special subclass of continuous lattices and the special properties of their spectra are treated. The concept of the patch topology is extended from algebraic lattices to continuous lattices, and necessary and sufficient conditions for its compactness are given. The spectral theory of lattices serves the purpose of representing a lattice L as a lattice of open sets of a topological space X. The spectral theory of rings and algebras practically reduces to this situation in view of the fact that for the most part one considers the lattice of ring (or algebra) ideals and then develops the spectral theory of that lattice. (The occasional complications due to the fact that ideal products are not intersections have been dealt with elsewhere, e.g. [4].) The lattice of all ring (or algebra) ideals forms a particular kind of continuous lattice, namely an algebraic lattice. It should be the case, however, that more general continuous lattices arise in the study of certain objects endowed with both an algebraic and a topological structure. Indeed the first author has shown in a seminar report using the concept of Pedersen's ideal that the closed ideals of a C*-algebra always form a distributive continuous lattice with respect to intersection. How widely continuous lattices occur in such contexts is, at this point, a largely uncharted sea. We show that the spectrum of a distributive continuous lattice is a locally quasicompact sober space (see 2.6 for the definition of sobriety). This implies, e.g., that the space of closed two sided prime ideals of a C*-algebra is locally quasicompact in the hull-kernel topology. (This is usually proved for primitive ideals by different methods.) On the other hand, the question of what topological consequences follow Received by the editors November 4, 1977. AMS (MOS) subject classifications (1970). Primary 54H10; Secondary 22A30, 06A35.