Hereditary $C^*$-subalgebras of graph $C^*$-algebras

TL;DR

This paper characterizes when certain hereditary subalgebras of graph $C^*$-algebras are themselves graph $C^*$-algebras, focusing on properties like stable isomorphism and approximate units of projections.

Contribution

It establishes new criteria for hereditary subalgebras of graph $C^*$-algebras to be isomorphic to graph $C^*$-algebras, especially involving approximate units of projections.

Findings

Hereditary subalgebras of unital real rank zero graph $C^*$-algebras are isomorphic to graph $C^*$-algebras.

A $C^*$-algebra with an approximate unit of projections is isomorphic to a graph $C^*$-algebra iff it is stably isomorphic to a unital graph $C^*$-algebra.

Minimal unitization of such $C^*$-algebras is isomorphic to a graph $C^*$-algebra under certain conditions.

Abstract

We show that a $C^*$-algebra $\mathfrak{A}$ which is stably isomorphic to a unital graph $C^*$-algebra, is isomorphic to a graph $C^*$-algebra if and only if it admits an approximate unit of projections. As a consequence, a hereditary $C^*$-subalgebra of a unital real rank zero graph $C^*$-algebra is isomorphic to a graph $C^*$-algebra. Furthermore, if a $C^*$-algebra $\mathfrak{A}$ admits an approximate unit of projections, then its minimal unitization is isomorphic to a graph $C^*$-algebra if and only if $\mathfrak{A}$ is stably isomorphic to a unital graph $C^*$-algebra.