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

Sara E. Arklint; James Gabe; Efren Ruiz

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

Operator Algebras 2017-02-01 v2

Authors: Sara E. Arklint , James Gabe , Efren Ruiz

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.

Keywords

graph C*-algebra hereditary subalgebra stable isomorphism approximate unit projection real rank zero minimal unitization

Cite

@article{arxiv.1604.03085,
  title  = {Hereditary $C^*$-subalgebras of graph $C^*$-algebras},
  author = {Sara E. Arklint and James Gabe and Efren Ruiz},
  journal= {arXiv preprint arXiv:1604.03085},
  year   = {2017}
}

Comments

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