Non-Abelian factors for actions of Z and other non-C^⁎-simple groups

Let Γ be a countable group and (X,Γ) a compact topological dynamical system. We study the question of the existence of an intermediate C^⁎-subalgebra A C_r^⁎(Γ)<A<C(X)⋊_rΓ, which is not of the form A=C(Y)⋊_rΓ, corresponding to a factor map (X,Γ)→(Y,Γ). Here C_r^⁎(Γ) is the reduced C^⁎-algebra of Γ and C(X)⋊_rΓ is the reduced C^⁎-crossed-product of (X,Γ). Our main results are: (1) For Γ which is not C^⁎-simple, when (X,Γ) admits a Γ-invariant probability measure, then such a sub-algebra always exists. (2) For Γ=Z and (X,Γ) an irrational rotation of the circle X=R/Z, we give a full description of all these non-crossed-product subalgebras.