Finer closed left ideal decompositions of G⁎

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⁎=βG∖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 (βG)p∈I′∖I. In particular, I′ is finer than I.