Composition operators within singly generated composition <Emphasis Type="Italic">C</Emphasis>*-algebras

Let ϕ be a linear-fractional self-map of the open unit disk D, not an automorphism, such that ϕ(ζ) = η for two distinct points ζ,η in the unit circle ∂D. We consider the problem of determining which composition operators, acting on the Hardy space H ², lie in C*(C _ϕ ,K), the unital C*-algebra generated by the composition operator C _ϕ and the ideal K of compact operators. This necessitates a companion study of the unital C*-algebra generated by the composition operators induced by all parabolic non-automorphisms with common fixed point on the unit circle.