Fixed point composition and Toeplitz-composition C⁎-algebras

Let φ be a linear-fractional, non-automorphism self-map of D that fixes ζ∈T and satisfies φ′(ζ)≠1 and consider the composition operator Cφ acting on the Hardy space H2(D). We determine which linear-fractionally-induced composition operators are contained in the unital C⁎-algebra generated by Cφ and the ideal K of compact operators. We apply these results to show that C⁎(Cφ,K) and C⁎(Fζ), the unital C⁎-algebra generated by all composition operators induced by linear-fractional, non-automorphism self-maps of D that fix ζ, are each isomorphic, modulo the ideal of compact operators, to a unitization of a crossed product of C0([0,1]). We compute the K-theory of C⁎(Cφ,K) and calculate the essential spectra of a class of operators in this C⁎-algebra. We also obtain a full description of the structures, modulo the ideal of compact operators, of the C⁎-algebras generated by the unilateral shift Tz and a single linear-fractionally-induced composition operator.