Parabolic Non-Automorphism Induced Toeplitz-Composition C*-Algebras with Piece-wise Quasi-continuous Symbols

In this paper we consider the C*-algebra $C^{\ast }(\{C_{\phi }\}\cup \mathcal{T}(PQC(\mathbb{T})))\mathrm{/}K(H^2)$ generated by Toeplitz operators with piece-wise quasi-continuous symbols and a composition operator induced by a parabolic linear fractional non-automorphism symbol modulo compact operators on the Hilbert-Hardy space $H^2$. This C*-algebra is commutative. We characterize its maximal ideal space. We apply our results to the question of determining the essential spectra of linear combinations of a class of composition operators and Toeplitz operators.