Composition operators within singly generated composition C*-algebras
  Let $\phi$ be a linear-fractional self map of the open unit disk, not an automorphism, such that $\phi(\zeta)=\eta$ for distinct points $\zeta,\eta$ in the unit circle. We consider the question of which composition operators lie in the unital C*-algebra generated by the composition operator $C_{\phi}$ and the ideal of compact operators, acting on the Hardy space $H^2$. 
Thomas Kriete, Barbara MacCluer and Jennifer Moorhouse