Toeplitz-composition 𝐶*-algebras for certain finite Blaschke products

Hamada, Hiroyasu; Watatani, Yasuo

doi:10.1090/S0002-9939-10-10270-6

Toeplitz-composition $C^{*}$-algebras for certain finite Blaschke products
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by Hiroyasu Hamada and Yasuo Watatani

Proc. Amer. Math. Soc. 138 (2010), 2113-2123

DOI: https://doi.org/10.1090/S0002-9939-10-10270-6

Published electronically: February 9, 2010

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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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