$C^*$-algebra extensions associated to continued fraction expansions of rational numbers

TL;DR

This paper solves the isomorphism problem for specific $C^*$-algebra extensions related to continued fraction expansions of rational numbers, connecting them to Effros Shen AF algebras and path categories.

Contribution

It provides a classification of certain $C^*$-algebra extensions associated with continued fractions of rationals, linking them to known AF algebra structures.

Findings

Solved the isomorphism problem for the specified $C^*$-algebra extensions.

Established a connection between these extensions and Effros Shen AF algebras.

Used categories of paths from continued fraction expansions to analyze the algebras.

Abstract

We solve the isomorphism problem for essential unital $C^*$-algebra extensions of the form $0 \to \mathcal{K} \oplus \mathcal{K} \to E \xrightarrow{\pi} M_n \otimes C(\mathbb{T}) \to 0$. We then relate these to analogs of the Effros Shen AF algebras for rational numbers. This involves $C^*$-algebras constructed from categories of paths built from certain nonsimple continued fraction expansions.