Toeplitz-Composition C*-Algebras
  Let $\zeta$ and $\eta$ be distinct points on the unit circle and suppose that $\phi$ is a linear-fractional self-map of the unit disk D, not an automorphism, with $\phi(\zeta)=\eta$. We describe the C*-algebra generated by the associated composition operator $C_{\phi}$ and the shift operator, acting on the Hardy space on D. 
Thomas L. Kriete, Barbara D. MacCluer, Jennifer L. Moorhouse