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Toeplitz-composition $ C^{*}$-algebras for certain finite Blaschke products


Authors: Hiroyasu Hamada and Yasuo Watatani
Journal: Proc. Amer. Math. Soc. 138 (2010), 2113-2123
MSC (2010): Primary 46L55, 47B33; Secondary 37F10, 46L08
DOI: https://doi.org/10.1090/S0002-9939-10-10270-6
Published electronically: February 9, 2010
MathSciNet review: 2596050
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Abstract: Let $ R$ be a finite Blaschke product of degree at least two with $ R(0)=0$. Then there exists a relation between the associated composition operator $ C_R$ on the Hardy space and the $ C^*$-algebra $ \mathcal{O}_R (J_R)$ associated with the complex dynamical system $ (R^{\circ n})_n$ on the Julia set $ J_R$. We study the $ C^*$-algebra $ \mathcal{TC}_R$ generated by both the composition operator $ C_R$ and the Toeplitz operator $ T_z$ to show that the quotient algebra by the ideal of the compact operators is isomorphic to the $ C^*$-algebra $ \mathcal{O}_R (J_R)$, which is simple and purely infinite.


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Additional Information

Hiroyasu Hamada
Affiliation: Graduate School of Mathematics, Kyushu University, Hakozaki, Fukuoka, 812-8581, Japan
Email: h-hamada@math.kyushu-u.ac.jp

Yasuo Watatani
Affiliation: Department of Mathematical Sciences, Kyushu University, Hakozaki, Fukuoka, 812-8581, Japan
Email: watatani@math.kyushu-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-10-10270-6
Keywords: Composition operator, Blaschke product, Toeplitz operator, $C*$-algebra, complex dynamical system
Received by editor(s): October 23, 2008
Received by editor(s) in revised form: October 8, 2009
Published electronically: February 9, 2010
Communicated by: Marius Junge
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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