### 抜粋

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.

元の言語 | 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 |

発行部数 | 6 |

DOI | |

出版物ステータス | Published - 2014 3 15 |

### ASJC Scopus subject areas

- Analysis

## フィンガープリント Identifying AF-algebras that are graph C<sup>*</sup>-algebras' の研究トピックを掘り下げます。これらはともに一意のフィンガープリントを構成します。

## これを引用

^{*}-algebras.

*Journal of Functional Analysis*,

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