### Abstract

We study Kubo-Martin-Schwinger (KMS) states on finite-graph C*-algebras with sinks and sources. We compare finite-graph C*-algebras with C*-algebras associated with complex dynamical systems of rational functions. We show that if the inverse temperature β is large, then the set of extreme β-KMS states is parametrized by the set of sinks of the graph. This means that the sinks of a graph correspond to the branched points of a rational function from the point of KMS states. Since we consider graphs with sinks and sources, left actions of the associated bimodules are not injective. Then the associated graph C*-algebras are realized as (relative) Cuntz-Pimsner algebras studied by Katsura. We need to generalize Laca-Neshvyev's theorem of the construction of KMS states on Cuntz-Pimsner algebras to the case that left actions of bimodules are not injective.

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

Pages (from-to) | 83-104 |

Number of pages | 22 |

Journal | Kyushu Journal of Mathematics |

Volume | 67 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 25 2013 |

### Fingerprint

### Keywords

- C*-correspondences
- Graph C*-algebras
- KMS states

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Kyushu Journal of Mathematics*,

*67*(1), 83-104. https://doi.org/10.2206/kyushujm.67.83

**KMS states on finite-graph C*-algebras.** / Kajiwara, Tsuyoshi; Watatani, Yasuo.

Research output: Contribution to journal › Article

*Kyushu Journal of Mathematics*, vol. 67, no. 1, pp. 83-104. https://doi.org/10.2206/kyushujm.67.83

}

TY - JOUR

T1 - KMS states on finite-graph C*-algebras

AU - Kajiwara, Tsuyoshi

AU - Watatani, Yasuo

PY - 2013/7/25

Y1 - 2013/7/25

N2 - We study Kubo-Martin-Schwinger (KMS) states on finite-graph C*-algebras with sinks and sources. We compare finite-graph C*-algebras with C*-algebras associated with complex dynamical systems of rational functions. We show that if the inverse temperature β is large, then the set of extreme β-KMS states is parametrized by the set of sinks of the graph. This means that the sinks of a graph correspond to the branched points of a rational function from the point of KMS states. Since we consider graphs with sinks and sources, left actions of the associated bimodules are not injective. Then the associated graph C*-algebras are realized as (relative) Cuntz-Pimsner algebras studied by Katsura. We need to generalize Laca-Neshvyev's theorem of the construction of KMS states on Cuntz-Pimsner algebras to the case that left actions of bimodules are not injective.

AB - We study Kubo-Martin-Schwinger (KMS) states on finite-graph C*-algebras with sinks and sources. We compare finite-graph C*-algebras with C*-algebras associated with complex dynamical systems of rational functions. We show that if the inverse temperature β is large, then the set of extreme β-KMS states is parametrized by the set of sinks of the graph. This means that the sinks of a graph correspond to the branched points of a rational function from the point of KMS states. Since we consider graphs with sinks and sources, left actions of the associated bimodules are not injective. Then the associated graph C*-algebras are realized as (relative) Cuntz-Pimsner algebras studied by Katsura. We need to generalize Laca-Neshvyev's theorem of the construction of KMS states on Cuntz-Pimsner algebras to the case that left actions of bimodules are not injective.

KW - C-correspondences

KW - Graph C-algebras

KW - KMS states

UR - http://www.scopus.com/inward/record.url?scp=84880768062&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84880768062&partnerID=8YFLogxK

U2 - 10.2206/kyushujm.67.83

DO - 10.2206/kyushujm.67.83

M3 - Article

AN - SCOPUS:84880768062

VL - 67

SP - 83

EP - 104

JO - Kyushu Journal of Mathematics

JF - Kyushu Journal of Mathematics

SN - 1340-6116

IS - 1

ER -