AF-Embeddings of Graph Algebras

TL;DR

This paper characterizes when graph C*-algebras are AF-embeddable based on the absence of entrances to loops in the graph, providing a constructive proof similar to desingularization.

Contribution

It establishes a precise criterion for AF-embeddability of graph C*-algebras related to graph loops and introduces a constructive proof approach.

Findings

AF-embeddability occurs iff no loop has an entrance

Provides a constructive proof method

Connects graph properties with C*-algebra embeddability

Abstract

Let $E$ be a countable directed graph. We show that $C^*(E)$ is AF-embeddable if and only if no loop in $E$ has an entrance. The proof is constructive and is in the same spirit as the Drinen-Tomforde desingularization.