# Journal of Operator Theory

Volume 84, Issue 1, Summer 2020 pp. 99-126.

Hereditary $C^*$-subalgebras of graph $C^*$-algebras**Authors**: Sara E.Arklint (1), James Gabe (2), Efren Ruiz (3)

**Author institution:**(1) Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark

(2) School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW 2522, Australia

(3) Department of Mathematics, University of Hawaii, Hilo, 200 W. Kawili St., Hilo, Hawaii, 96720-4091 U.S.A.

**Summary:**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.

**DOI:**http://dx.doi.org/10.7900/jot.2019jan21.2230

**Keywords:**graph $C^*$-algebras, hereditary $C^*$-subalgebras

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