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

TL;DR

This paper explores the relationship between composition operators, Toeplitz operators, and complex dynamical systems for finite Blaschke products, revealing a deep algebraic structure linking operator theory and complex dynamics.

Contribution

It establishes an isomorphism between a quotient C*-algebra generated by composition and Toeplitz operators and the C*-algebra of the associated dynamical system, showing it is simple and purely infinite.

Findings

The quotient algebra is isomorphic to the C*-algebra of the dynamical system.

The resulting algebra is simple and purely infinite.

The study connects operator theory with complex dynamical systems.

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 associated with the complex dynamical system on the Julia set of R. We study the C*-algebra 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 associated with the complex dynamical system, which is simple and purely infinite.