Finer closed left ideal decompositions of

Let G be a countably infinite discrete group and let βG be the Stone–Čech compactification of G. Let I denote the finest decomposition of G ⁎ = β G ∖ G into closed left ideals of βG with the property that the corresponding quotient space of G ⁎ is Hausdorff, and I ′ the finest decomposition of G ⁎ into closed left ideals of βG. We show that there is a dense subset of points p ∈ G ⁎ such that ( β G ) p ∈ I ′ ∖ I . In particular, I ′ is finer than I .