2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/121553Let $ζ$ 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.20 pages To appear in Journal of Operator TheoryOperator Algebras47B33 47B35, 47L80text