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arXiv · 1604.03085

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

Sara E. Arklint·James Gabe·Efren Ruiz
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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.

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Sara E. Arklint, James Gabe, Efren Ruiz. 2016-04-11. Hereditary $C^*$-subalgebras of graph $C^*$-algebras. https://arxiv.org/abs/1604.03085

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