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# Closed Left Ideal Decompositions of $U(G)$

Published:2011-08-31
Printed: Jun 2013
• Yevhen Zelenyuk,
School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
 Format: LaTeX MathJax PDF

## Abstract

Let $G$ be an infinite discrete group and let $\beta G$ be the Stone--Čech compactification of $G$. We take the points of $ėta G$ to be the ultrafilters on $G$, identifying the principal ultrafilters with the points of $G$. The set $U(G)$ of uniform ultrafilters on $G$ is a closed two-sided ideal of $\beta G$. For every $p\in U(G)$, define $I_p\subseteq\beta G$ by $I_p=\bigcap_{A\in p}\operatorname{cl} (GU(A))$, where $U(A)=\{p\in U(G):A\in p\}$. We show that if $|G|$ is a regular cardinal, then $\{I_p:p\in U(G)\}$ is the finest decomposition of $U(G)$ into closed left ideals of $\beta G$ such that the corresponding quotient space of $U(G)$ is Hausdorff.
 Keywords: Stone--Čech compactification, uniform ultrafilter, closed left ideal, decomposition
 MSC Classifications: 22A15 - Structure of topological semigroups 54H20 - Topological dynamics [See also 28Dxx, 37Bxx] 22A30 - Other topological algebraic systems and their representations 54D80 - Special constructions of spaces (spaces of ultrafilters, etc.)

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