Geometric classification of simple graph algebras

TL;DR

This paper introduces a geometric classification method for simple graph algebras, using graph moves that preserve stable isomorphism, enabling transformation between graphs with stably isomorphic simple unital algebras.

Contribution

It provides a set of allowed graph moves that classify simple graph algebras up to stable isomorphism, extending ideas from symbolic dynamics.

Findings

Identified a list of graph moves preserving stable isomorphism.

Proved that graphs with stably isomorphic simple unital algebras can be transformed into each other.

Established a classification framework inspired by Franks' work on shifts of finite type.

Abstract

Inspired by Franks' classification of irreducible shifts of finite type we provide a short list of allowed moves on graphs that preserves the stable isomorphism class of the associated C*-algebras. We show that if two graphs have stably isomorphic and simple unital algebras then we can use these moves to transform one into the other.