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 $$\varvec{\beta }G\rightarrow \mathfrak {b}G$$ from the Stone–Cech compactification $$\varvec{\beta }G$$ of G to its Bohr compactification $$\mathfrak {b}G$$ is a homomorphism with respect to the semigroup operation on $$\varvec{\beta }G$$ , 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 $$\varvec{\beta }G$$ with respect to the least closed congruence relation on $$\varvec{\beta }G$$ merging all the Schur ultrafilters on G into the unit of G.