Gelfand spectra and Wallman compactifications

TL;DR

This paper explores Wallman compactifications and their connections with Gelfand spectra and Stone-Cech compactifications, providing a categorical framework that unifies various dualities in topology and algebra.

Contribution

It introduces a functorial framework for representing topological spaces via Wallman bases, establishing categorical equivalences with distributive lattices and C*-algebras.

Findings

Categorical equivalence between commutative C*-algebras and distributive lattices.

General theorem representing Stone-Cech compactification as a Wallman compactification.

Unified framework for Wallman and Gelfand spectra relations.

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.