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–Čech 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.
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Acknowledgements
The author acknowledges with thanks the support of his research by the Grant No. 1/0333/17 of the Slovak Grant agency VEGA.
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Communicated by Mikhail Volkov.
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Zlatoš, P. The Bohr compactification of an abelian group as a quotient of its Stone–Čech compactification. Semigroup Forum 101, 497–506 (2020). https://doi.org/10.1007/s00233-020-10121-6
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DOI: https://doi.org/10.1007/s00233-020-10121-6