The Bohr compactification of an abelian group as a quotient of its Stone-\v{C}ech compactification

TL;DR

This paper demonstrates that for any abelian group, the Bohr compactification can be obtained as a quotient of its Stone-cech compactification, with the canonical map being a semigroup homomorphism.

Contribution

It establishes a canonical isomorphism between the Bohr compactification and a quotient of the Stone-cech compactification by a specific congruence relation.

Findings

The canonical map from cech to Bohr compactification is a semigroup homomorphism.

The Bohr compactification is isomorphic to a quotient of the Stone-cech compactification.

The quotient is formed by merging all Schur ultrafilters into the identity element.

Abstract

We will prove that, for any abelian group $G$, the canonical (surjective and continuous) mapping $\boldsymbol{\beta}G \to {\frak b}G$ from the Stone-\v{C}ech compactification $\boldsymbol{\beta}G$ of $G$ to its Bohr compactfication ${\frak b}G$ is a homomorphism with respect to the semigroup operation on $\boldsymbol{\beta}G$, 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 $\boldsymbol{\beta}G$ with respect to the least closed congruence relation on $\boldsymbol{\beta}G$ merging all the Schur ultrafilters on $G$ into the unit of $G$.