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 $\mathbf{\beta }G\to \mathfrak{b}G$ from the Stone-Čech compactification $\mathbf{\beta }G$ of $G$ to its Bohr compactfication $\mathfrak{b}G$ is a homomorphism with respect to the semigroup operation on $\mathbf{\beta }G$, extending the multiplication on $G$, and the group operation on $\mathfrak{b}G$. Moreover, the Bohr compactification $\mathfrak{b}G$ is canonically isomorphic (both in algebraic and topological sense) to the quotient of $\mathbf{\beta }G$ with respect to the least closed congruence relation on $\mathbf{\beta }G$ merging all the Schur ultrafilters on $G$ into the unit of $G$.