Faithful representations of graph algebras via branching systems

TL;DR

This paper explores how branching systems of directed graphs can produce faithful representations of graph algebras, establishing conditions for faithfulness and proving a converse to the Cuntz-Krieger uniqueness theorem.

Contribution

It provides a sufficient condition for faithfulness of representations from branching systems and constructs a broad class of such systems, also proving a converse to a key theorem in graph algebra theory.

Findings

Identified a sufficient condition for faithful representations

Constructed a large class of branching systems satisfying this condition

Proved the converse of the Cuntz-Krieger uniqueness theorem using branching systems

Abstract

We continue to investigate branching systems of directed graphs and their connections with graph algebras. We give a sufficient condition under which the representation induced from a branching system of a directed graph is faithful and construct a large class of branching systems that satisfy this condition. We finish the paper by providing a proof of the converse of the Cuntz-Krieger uniqueness theorem for graph algebras by means of branching systems.