Fredholm operators in the Toeplitz Algebra $\mathcal{I}(QC)$

TL;DR

This paper characterizes the set of invertible quasicontinuous functions and Fredholm operators in the Toeplitz algebra, revealing their complex topological structure with uncountably many path-connected components.

Contribution

It provides a complete classification of invertible quasicontinuous functions and Fredholm operators in the Toeplitz algebra, including their path-connected components.

Findings

Uncountably many path-connected components in the set of invertible quasicontinuous functions.

Uncountably many path-connected components in the set of Fredholm operators.

Complete description of invertible quasicontinuous functions on the unit circle.

Abstract

We will give a complete description of $\mathcal{I}$, the set of invertible quasicontinuous functions on the unit circle. After doing this, we will then classify the path-connected components of $\mathcal{I}$ and show that $\mathcal{I}$ has uncountably many path-connected components. We will then use the above classifications to characterize $\mathcal{F}$, the set of Fredholm operators of the C$^*$-algebra generated by the Toeplitz operators $T_\phi$ with quasicontinuous symbols $\phi$. Then we will classify the path-connected components of $\mathcal{F}$ and show that $\mathcal{F}$ also has uncountably many path-connected components.