Ext classes and embeddings for C*-algebras of graphs with sinks

TL;DR

This paper introduces a new Ext class invariant for C*-algebras of graphs with sinks and uses it to classify when one such algebra can embed into another, extending existing classification results.

Contribution

It defines an Ext class for graphs with sinks and establishes a criterion for algebra embeddings, generalizing previous classification theorems.

Findings

Ext classes classify graph C*-algebra embeddings

Embedding criteria depend on Ext class equality

Generalizes classification theorems for graphs with sinks

Abstract

We consider directed graphs E which have been obtained by adding a sink to a fixed graph G. We associate an element of Ext(C*(G)) to each such E, and show that the classes of two such graphs are equal in Ext(C*(G)) if and only if the associated C*-algebra of one can be embedded as a full corner in the C*-algebra of the other in a particular way. If every loop in G has an exit, then we are able to use this result to generalize some of the classification theorems of Raeburn, Tomforde, and Williams for C*-algebras of graphs with sinks.