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

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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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Correspondence to Pavol Zlatoš.

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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

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