Quotient Algebras of Toeplitz-Composition $$C^{*}$$ C ∗ -Algebras for Finite Blaschke Products

Abstract

Let \(R\) be a finite Blaschke product. We study the \(C^*\)-algebra \(\mathcal TC _R\) generated by both the composition operator \(C_R\) and the Toeplitz operator \(T_z\) on the Hardy space. We show that the simplicity of the quotient algebra \(\mathcal OC _R\) by the ideal of the compact operators can be characterized by the dynamics near the Denjoy–Wolff point of \(R\) if the degree of \(R\) is at least two. Moreover we prove that the degree of finite Blaschke products is a complete isomorphism invariant for the class of \(\mathcal OC _R\) such that \(R\) is a finite Blaschke product of degree at least two and the Julia set of \(R\) is the unit circle, using the Kirchberg–Phillips classification theorem.