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Discontinuity of multiplication and left translations in $\beta G$


Author: Yevhen Zelenyuk
Journal: Proc. Amer. Math. Soc. 143 (2015), 877-884
MSC (2010): Primary 03E35, 22A15; Secondary 22A05, 54D35
DOI: https://doi.org/10.1090/S0002-9939-2014-12267-2
Published electronically: October 6, 2014
MathSciNet review: 3283674
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Abstract: The operation of a discrete group $G$ naturally extends to the Stone-Čech compactification $\beta G$ of $G$ so that for each $a\in G$, the left translation $\beta G\ni x\mapsto ax\in \beta G$ is continuous, and for each $q\in \beta G$, the right translation $\beta G\ni x\mapsto xq\in \beta G$ is continuous. We show that for every Abelian group $G$ with finitely many elements of order 2 such that $|G|$ is not Ulam-measurable and for every $p,q\in G^*=\beta G\setminus G$, the multiplication $\beta G\times \beta G\ni (x,y)\mapsto xy\in \beta G$ is discontinuous at $(p,q)$. We also show that it is consistent with ZFC, the system of usual axioms of set theory, that for every Abelian group $G$ and for every $p,q\in G^*$, the left translation $G^*\ni x\mapsto px\in G^*$ is discontinuous at $q$.


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Additional Information

Yevhen Zelenyuk
Affiliation: School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
Email: yevhen.zelenyuk@wits.ac.za

Keywords: Stone-Čech compactification, ultrafilter, $P$-point, Ulam-measurable cardinal, discontinuity, weak $(p,q)$-homomorphism.
Received by editor(s): February 19, 2013
Received by editor(s) in revised form: May 18, 2013
Published electronically: October 6, 2014
Additional Notes: The author was supported by NRF grant IFR2011033100072.
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2014 American Mathematical Society