Hereditary $C^{\ast }$-subalgebras of graph $C^{\ast }$-algebras

Abstract

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

23 pages. Ver 2: 24 pages, minor changes to introduction, bibliography updated