Quotient algebras of Toeplitz-composition C*-algebras for finite Blaschke products

TL;DR

This paper investigates the structure of quotient C*-algebras generated by composition and Toeplitz operators associated with finite Blaschke products, revealing their simplicity criteria and classification invariants based on complex dynamics.

Contribution

It characterizes the simplicity of quotient algebras via dynamics near the Denjoy-Wolff point and establishes the degree of Blaschke products as a complete isomorphism invariant.

Findings

Simplicity of OC_R linked to dynamics near the Denjoy-Wolff point.

Degree of Blaschke product is a complete invariant for certain quotient algebras.

Classification achieved using Kirchberg-Phillips theorem.

Abstract

Let R be a finite Blaschke product. We study the C*-algebra 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 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 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.