### Abstract

We consider the problem of identifying exactly which AF-algebras are isomorphic to a graph C^{*}-algebra. We prove that any separable, unital, Type I C^{*}-algebra with finitely many ideals is isomorphic to a graph C^{*}-algebra. This result allows us to prove that a unital AF-algebra is isomorphic to a graph C^{*}-algebra if and only if it is a Type I C^{*}-algebra with finitely many ideals. We also consider nonunital AF-algebras that have a largest ideal with the property that the quotient by this ideal is the only unital quotient of the AF-algebra. We show that such an AF-algebra is isomorphic to a graph C^{*}-algebra if and only if its unital quotient is Type I, which occurs if and only if its unital quotient is isomorphic to M_{k} for some natural number k. All of these results provide vast supporting evidence for the conjecture that an AF-algebra is isomorphic to a graph C^{*}-algebra if and only if each unital quotient of the AF-algebra is Type I with finitely many ideals, and bear relevance for the extension problem for graph C^{*}-algebras.

Original language | English |
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Pages (from-to) | 3968-3996 |

Number of pages | 29 |

Journal | Journal of Functional Analysis |

Volume | 266 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2014 Mar 15 |

### Keywords

- AF-algebras
- Bratteli diagrams
- Graph C-algebras
- Type I C-algebras

### ASJC Scopus subject areas

- Analysis

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## Cite this

^{*}-algebras.

*Journal of Functional Analysis*,

*266*(6), 3968-3996. https://doi.org/10.1016/j.jfa.2014.01.009