The Bohr compactification of an abelian group as a quotient of its Stone-Čech compactification

Pavol Zlatoš

Published 2018-08-14Version 1

We will prove that, for any abelian group $G$, the canonical (surjective and continuous) mapping $\btG \to \fbG$ from the Stone-\v{C}ech compactification $\btG$ of $G$ to its Bohr compactfication $\fbG$ is a homomorphism with respect to the semigroup operation on $\btG$, extending the multiplication on $G$, and the group operation on $\fbG$. Moreover, the Bohr compactification $\fbG$ is canonically isomorphic (both in algebraic and topological sense) to the quotient of $\btG$ with respect to the least closed congruence relation on $\btG$ merging all the \textit{Schur ultrafilters} on $G$ into the unit of \,$G$.