On closed left ideal decompositions of G⁎

Let G be a countably infinite discrete group, let βG be the Stone–Čech compactification of G, and let G⁎=βG∖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 2c maximal principal left ideals of βG.