2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/172815Let $ϕ$ be a linear-fractional self map of the open unit disk, not an automorphism, such that $ϕ(ζ)=η$ for distinct points $ζ,η$ in the unit circle. We consider the question of which composition operators lie in the unital C*-algebra generated by the composition operator $C_ϕ$ and the ideal of compact operators, acting on the Hardy space $H^2$.21 pages; minor revision of earlier versionFunctional AnalysisOperator Algebras47B33text