Totally disconnected semigroup compactifications of topological groups.

The article develops a comprehensive framework for totally disconnected (TD) semigroup compactifications of a topological group \( G \) by identifying them with Stone compactifications associated with introverted Boolean algebras of clopen subsets of \( G \). It establishes that every TD semigroup compactification arises from a left introverted Boolean subalgebra of the Boolean algebra of clopen subsets, and characterizes universal TD compactifications corresponding to well-known function spaces: the largest left introverted Boolean algebra \(\mathcal{B}^{LMC}\) yields the universal TD semigroup compactification; \(\mathcal{B}^{LUC}\) corresponds to the universal TD semigroup compactification with joint continuity; \(\mathcal{B}^{WAP}\) to the universal TD semitopological semigroup compactification; and \(\mathcal{B}^{AP}\) to the universal TD topological group compactification, which is identified with the profinite completion of \( G \). The paper further relates these Stone compactifications to Gelfand compactifications arising from m-introverted \( C^* \)-subalgebras of bounded continuous functions on \( G \), showing that the Stone approach provides an alternative, function-space-independent construction of these universal compactifications.