Totally disconnected semigroup compactifications of topological groups

Abstract

Abstract We introduce the notion of an introverted Boolean algebra $\mathcal{B}$ of closed-and-open subsets of a topological group G, show that the associated Stone space $(\nu_{\mathcal{B}} G, \nu_{\mathcal{B}})$ 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 GLUC, GW AP and GAP, although we do employ Gelfand theory to clarify the relationship between these familiar universal compactifications and their totally disconnected counterparts.