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.
Categories: math.OA