Classifying $C^*$-algebras with both finite and infinite subquotients

TL;DR

This paper develops a classification framework for a specific class of $C^*$-algebras over finite topological spaces, especially those with a separating open set, and applies it to graph $C^*$-algebras.

Contribution

It introduces a new classification method for $C^*$-algebras with finite and infinite subquotients over finite spaces, including applications to graph algebras.

Findings

Classification result for $C^*$-algebras over finite spaces with separating open sets

Application of the classification to graph $C^*$-algebras

Identification of structural properties distinguishing finite and infinite subquotients

Abstract

We give a classification result for a certain class of $C^{*}$-algebras $\mathfrak{A}$ over a finite topological space $X$ in which there exists an open set $U$ of $X$ such that $U$ separates the finite and infinite subquotients of $\mathfrak{A}$. We will apply our results to $C^{*}$-algebras arising from graphs.