Totally disconnected semigroup compactifications of topological groups

TL;DR

This paper develops a framework for totally disconnected semigroup compactifications of topological groups using introverted Boolean algebras, providing a new perspective independent of Gelfand theory.

Contribution

It introduces a novel approach to totally disconnected compactifications via Boolean algebras and characterizes universal such compactifications, expanding understanding beyond classical Gelfand theory.

Findings

Constructs totally disconnected semigroup compactifications from Boolean algebras.

Identifies universal totally disconnected compactifications and their properties.

Clarifies relationships with classical universal compactifications like G^{LUC}, G^{WAP}, G^{AP}.

Abstract

We introduce the notion of an introverted Boolean algebra $\cal B$ of closed-and-open subsets of a topological group $G$, show that the associated Stone space $(\nu_{\cal B} G, \nu_{\cal B})$ is a totally disconnected semigroup compactification of $G$, and show that every totally disconnected semigroup compactification of $G$ takes this form. We identify and study the universal totally disconnected semigroup compactification, the universal totally disconnected semitopological semigroup compactification and the universal totally disconnected group compactification of $G$. Our main results are obtained independently of Gelfand theory and well-known properties of the (typically non-totally disconnected) universal compactifications $G^{LUC}$, $G^{WAP}$ and $G^{AP}$, though we do employ Gelfand theory to clarify the relationship between these familiar universal compactifications and their totally disconnected counterparts.