  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 $H^2$.
