Closed Left Ideal Decompositions of U(G)

Abstract

Let $G$ be an infinite discrete group and let $\beta G$ be the Stone-Čech compactification of $G$. We take the points of $\beta G$ to be the ultrafilters on $G$, identifying the principal ultrafilters with the points of $G$. The set $U\left( G \right)$ of uniform ultrafilters on $G$ is a closed two-sided ideal of $\beta G$. For every $p\,\in \,U\left( G \right)$, define ${{I}_{p}}\,\subseteq \,\beta G$ by ${{I}_{p}}\,=\,{{\bigcap }_{A\in p}}\text{cl}\left( GU\left( A \right) \right)$, where $U\left( A \right)\,=\,\left\{ p\,\in \,U\left( G \right)\,:\,A\in \,p \right\}$. We show that if $\left| G \right|$ is a regular cardinal, then $\left\{ {{I}_{p}}\,:\,p\,\in \,U\left( G \right) \right\}$ is the finest decomposition of $U\left( G \right)$ into closed left ideals of $\beta G$ such that the corresponding quotient space of $U\left( G \right)$ is Hausdorff.