Toeplitz-Composition $\hbox {C}^*$-Algebras Induced by Linear-Fractional Non-automorphism Self-Maps of the Disk

Abstract

Let ${\mathcal {M}}$ be an arbitrary collection of linear-fractional non-automorphism self-maps of the unit disk $\mathbb {D}$. We consider the unital $\hbox {C}^*$-algebras $C^*(T_z, \{C_{\varphi } : \varphi \in {\mathcal {M}}\})$ and $C^*(\{C_{\varphi } : \varphi \in {\mathcal {M}}\}, {\mathcal {K}})$ generated by the composition operators induced by the maps in ${\mathcal {M}}$ and either the unilateral shift $T_z$ or the ideal of compact operators ${\mathcal {K}}$ on the Hardy space $H^2(\mathbb {D})$. We describe the structures of these $\hbox {C}^*$-algebras, modulo the ideal of compact operators, for all finite collections ${\mathcal {M}}$ as well as all collections ${\mathcal {M}}$ that have finite boundary behavior. This work completes a line of research investigating the structures, modulo the ideal of compact operators, of Toeplitz-composition and composition $\hbox {C}^*$-algebras induced by linear-fractional non-automorphism self-maps of $\mathbb {D}$ that has unfolded over the last decade and half. While all results in this paper are stated for $H^2(\mathbb {D})$, the descriptions of the structures of these $\hbox {C}^*$-algebras, modulo the ideal of compact operators, also apply to the weighted Bergman spaces $A^2_{\alpha }(\mathbb {D})$ for $\alpha > -1$.