Closed Left Ideal Decompositions of U(G)

Zelenyuk, Yevhen

Canadian mathematical bulletin, Tome 56 (2013) no. 2, pp. 442-448

Résumé

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.

Détail

DOI : 10.4153/CMB-2011-175-8

Mots-clés : 22A15, 54H20, 22A30, 54D80, Stone–Čech compactification, uniform ultrafilter, closed left ideal, decomposition.

Bibliographie

[1] [1] Davenport, D. and Hindman, N., A proof of van Douwen's right ideal theorem. Proc. Amer. Math. Soc. 113(1991), no. 2, 573–580. Google Scholar

[2] [2] Engelking, R., General topology. Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989. Google Scholar

[3] [3] Hindman, N. and Strauss, D., Algebra in the Stone- ˇ Cech compactification. Theory and applications. de Gruyter Expositions in Mathematics, 27, Walter de Gruyter, Berlin, 1998. Google Scholar

[4] [4] Filali, M. and Salmi, P., Slowly oscillating functions in semigroup compactifications and convolution algebras. J. Funct. Anal. 250(2007), no. 1, 144–166. Google Scholar | DOI

[5] [5] Protasov, I. V., Coronas of balleans. Topology Appl. 149(2005), no. 1–3. 149–160. Google Scholar | DOI