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

  • Katie S. QuertermousEmail author
Article
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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 \(\alpha > -1\).

Keywords

Composition operator Toeplitz operator \(\hbox {C}^*\)-algebra Hardy space Crossed product \(\hbox {C}^*\)-algebra 

Mathematics Subject Classification

Primary 47B33 Secondary 47B35 47L80 47L65 

Notes

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsJames Madison UniversityHarrisonburgUSA

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