## 抄録

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.

本文言語 | English |
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ページ（範囲） | 3968-3996 |

ページ数 | 29 |

ジャーナル | Journal of Functional Analysis |

巻 | 266 |

号 | 6 |

DOI | |

出版ステータス | Published - 2014 3月 15 |

## ASJC Scopus subject areas

- 分析

## フィンガープリント

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