Toeplitz-Composition C*-Algebras

Let ζ and η be distinct points on the unit circle and suppose that ϕ is a linear-fractional self-map of the unit disk D, not an automorphism, with ϕ(ζ)=η. We describe the C*-algebra generated by the associated composition operator C_ϕ and the shift operator, acting on the Hardy space on D.