## Abstract

Let Λ be a strongly connected, finite higher-rank graph. In this paper, we construct representations of C^{*}(Λ) on certain separable Hilbert spaces of the form L^{2}(X, μ), by introducing the notion of a Λ-semibranching function system (a generalization of the semibranching function systems studied by Marcolli and Paolucci). In particular, if Λ is aperiodic, we obtain a faithful representation of C^{*}(Λ) on L^{2}(Λ^{∞}, M), where M is the Perron-Frobenius probability measure on the infinite path space Λ^{∞} recently studied by an Huef, Laca, Raeburn, and Sims. We also show how a Λ-semibranching function system gives rise to KMS states for C^{*}(Λ). For the higher-rank graphs of Robertson and Steger, we also obtain a representation of C^{*}(Λ) on L^{2}(X, μ), where X is a fractal subspace of [0, 1] by embedding Λ^{∞} into [0, 1] as a fractal subset X of [0, 1]. Moreover, when the Radon-Nikodym derivatives of a Λ-semibranching function system are constant, we show that we can associate to it a KMS state for C^{*}(Λ). Finally, we construct a wavelet system for L^{2}(Λ^{∞}, M) by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.

Original language | English |
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Article number | 19767 |

Pages (from-to) | 241-270 |

Number of pages | 30 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 434 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2016 |

### Funding

This work was partially supported by a grant from the Simons Foundation (# 316981 to Judith Packer). The third author would like to thank the Department of Mathematics of University of Colorado, Boulder for hosting her during the initial portion of this research.

Funders | Funder number |
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Simons Foundation | 316981 |

## Keywords

- Cantor-type fractal subspaces of [0, 1]
- Cuntz-Krieger C-algebras of k-graphs
- Separable representations
- Λ-semibranching function systems