### Abstract

We prove that the classes of graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence. This result answers the long-standing open question of whether every Exel-Laca algebra is Morita equivalent to a graph algebra. Given an ultragraph script G sign we construct a directed graph E such that C*(script G sign) is isomorphic to a full corner of C*(E). As applications, we characterize real rank zero for ultragraph algebras and describe quotients of ultragraph algebras by gauge-invariant ideals.

Original language | English |
---|---|

Pages (from-to) | 135-165 |

Number of pages | 31 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Issue number | 640 |

DOIs | |

Publication status | Published - 2010 Mar |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal fur die Reine und Angewandte Mathematik*, (640), 135-165. https://doi.org/10.1515/CRELLE.2010.023

**Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence.** / Katsura, Takeshi; Muhly, Paul S.; Sims, Aidan; Tomforde, Mark.

Research output: Contribution to journal › Article

*Journal fur die Reine und Angewandte Mathematik*, no. 640, pp. 135-165. https://doi.org/10.1515/CRELLE.2010.023

}

TY - JOUR

T1 - Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence

AU - Katsura, Takeshi

AU - Muhly, Paul S.

AU - Sims, Aidan

AU - Tomforde, Mark

PY - 2010/3

Y1 - 2010/3

N2 - We prove that the classes of graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence. This result answers the long-standing open question of whether every Exel-Laca algebra is Morita equivalent to a graph algebra. Given an ultragraph script G sign we construct a directed graph E such that C*(script G sign) is isomorphic to a full corner of C*(E). As applications, we characterize real rank zero for ultragraph algebras and describe quotients of ultragraph algebras by gauge-invariant ideals.

AB - We prove that the classes of graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence. This result answers the long-standing open question of whether every Exel-Laca algebra is Morita equivalent to a graph algebra. Given an ultragraph script G sign we construct a directed graph E such that C*(script G sign) is isomorphic to a full corner of C*(E). As applications, we characterize real rank zero for ultragraph algebras and describe quotients of ultragraph algebras by gauge-invariant ideals.

UR - http://www.scopus.com/inward/record.url?scp=77951537410&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951537410&partnerID=8YFLogxK

U2 - 10.1515/CRELLE.2010.023

DO - 10.1515/CRELLE.2010.023

M3 - Article

AN - SCOPUS:77951537410

SP - 135

EP - 165

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

SN - 0075-4102

IS - 640

ER -