Totally disconnected semigroup compactifications of topological groups
Alexander Stephens, Ross Stokke

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 of closed-and-open subsets of a topological group , show that the associated Stone space is a totally disconnected semigroup compactification of , and show that every totally disconnected semigroup compactification of 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 . Our main results are obtained independently of Gelfand theory and well-known properties of the (typically non-totally disconnected) universal compactifications , and , though we do employ Gelfand theory to clarify the relationship between these familiar universal compactifications and their…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Homotopy and Cohomology in Algebraic Topology · Topological and Geometric Data Analysis
