### Abstract

We consider representations of Cuntz algebras on self-similar fractal sets for proper/improper systems of contractions. Natural representations, called Hausdorff representations, are associated with self-similar sets and Hausdorff measures in the case of similitudes in ℝ^{n}. We completely classify the Hausdorff representations up to unitary equivalence. The complete invariant is the list (λ _{1}^{D},..., λ_{N}^{D}), where λ_{j} is the Lipschitz constant of the jth contraction and D is the Hausdorff dimension of the fractal set. Any non-trivial list can be realized by similitudes on the unit interval. There exists an improper system of contractions such that its representation of a Cuntz algebra on the self-similar fractal set is not unitarily equivalent to any Hausdorff representation for a proper system of similitudes in ℝ^{n}.

Original language | English |
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Pages (from-to) | 443-456 |

Number of pages | 14 |

Journal | Kyushu Journal of Mathematics |

Volume | 61 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2007 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Kyushu Journal of Mathematics*,

*61*(2), 443-456. https://doi.org/10.2206/kyushujm.61.443