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 $$\alpha > -1$$.

中文翻译：

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Toeplitz-Composition $$\hbox {C}^*$$C∗-由磁盘的线性分数非自同构自映射引起的代数

令 $${\mathcal {M}}$$ 是单位盘 $$\mathbb {D}$$ 的线性分数非自同构自映射的任意集合。我们考虑单位 $$\hbox {C}^*$$-代数 $$C^*(T_z, \{C_{\varphi } : \varphi \in {\mathcal {M}}\})$$ 和$$C^*(\{C_{\varphi } : \varphi \in {\mathcal {M}}\}, {\mathcal {K}})$$ 由 $$ 中的映射诱导的复合算子生成{\mathcal {M}}$$ 和哈代空间 $$H^2(\mathbb {D })$$。我们描述了这些 $$\hbox {C}^*$$-代数的结构，以紧凑运算符的理想为模，对于所有有限集合 $${\mathcal {M}}$$ 以及所有集合 $${ \mathcal {M}}$$ 具有有限边界行为。这项工作完成了一系列研究结构的研究，以已展开的 $$\mathbb {D}$$ 的线性分数非自同构自映射诱导的 Toeplitz 组合和组合 $$\hbox {C}^*$$-代数的紧凑算子的理想为模在过去的十五年里。虽然本文中的所有结果都是针对 $$H^2(\mathbb {D})$$ 陈述的，但这些 $$\hbox {C}^*$$-代数的结构描述，以紧致理想为模运算符，也适用于加权伯格曼空间 $$A^2_{\alpha }(\mathbb {D})$$ $$\alpha > -1$$。