Gelfand spectra and Wallman compactifications

Abstract

We carry out a systematic, topos-theoretically inspired, investigation of Wallman compactifications with a particular emphasis on their relations with Gelfand spectra and Stone-Cech compactifications. In addition to proving several specific results about Wallman bases and maximal spectra of distributive lattices, we establish a general framework for functorializing the representation of a topological space as the maximal spectrum of a Wallman base for it, which allows to generate different dualities between categories of topological spaces and subcategories of the category of distributive lattices; in particular, this leads to a categorical equivalence between the category of commutative C*-algebras and a natural category of distributive lattices. We also establish a general theorem concerning the representation of the Stone-Cech compactification of a locale as a Wallman compactification, which subsumes all the previous results obtained on this problem.