Toeplitz-Composition C*-Algebras

TL;DR

This paper characterizes the C*-algebra generated by a specific composition operator and the shift operator on the Hardy space, focusing on non-automorphic linear-fractional maps with prescribed boundary behavior.

Contribution

It provides a detailed description of the C*-algebra generated by composition and shift operators for non-automorphic linear-fractional maps with boundary points mapped to each other.

Findings

Explicit description of the generated C*-algebra

Analysis of the algebra's structure based on boundary points

Connection between boundary behavior and algebra properties

Abstract

Let $\zeta$ and $\eta$ be distinct points on the unit circle and suppose that $\phi$ is a linear-fractional self-map of the unit disk D, not an automorphism, with $\phi(\zeta)=\eta$. We describe the C*-algebra generated by the associated composition operator $C_{\phi}$ and the shift operator, acting on the Hardy space on D.