On closed left ideal decompositions of

Let G be a countably infinite discrete group, let βG be the Stone–Čech compactification of G, and let G ⁎ = β G ∖ G . Let I ( G ) denote the finest decomposition of G ⁎ into closed left ideals of βG with the property that the corresponding quotient space of G ⁎ is Hausdorff. We show that it is consistent with ZFC that, if G can be embedded algebraically into a compact group, then every I ∈ I ( G ) contains 2 c maximal principal left ideals of βG.