The RFD property for graph $C^*$-algebras

TL;DR

This paper characterizes when the C*-algebra of a graph is residually finite dimensional, based on specific graph properties such as absence of infinite receivers and cycles with exits.

Contribution

It provides a complete characterization of the RFD property for graph C*-algebras in terms of graph-theoretic conditions.

Findings

C*-algebra of a graph is RFD iff the graph has no infinite receiver.

The graph must have no cycle with an exit.

Every vertex must connect via a finite path to a sink, cycle, or infinite emitter.

Abstract

It is proved that the C*-algebra of a graph is residually finite dimensional (RFD) if and only if the graph has no infinite receiver, no cycle with an exit, no infinite ackward chain and from each vertex, there is a finite path to a sink or a cycle or an infinite emitter.