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

Abstract

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