### 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

**Representations of cuntz algebras on fractal sets.** / Mori, Makoto; Suzuki, Osamu; Watatani, Yasuo.

Research output: Contribution to journal › Article

*Kyushu Journal of Mathematics*, vol. 61, no. 2, pp. 443-456. https://doi.org/10.2206/kyushujm.61.443

}

TY - JOUR

T1 - Representations of cuntz algebras on fractal sets

AU - Mori, Makoto

AU - Suzuki, Osamu

AU - Watatani, Yasuo

PY - 2007/1/1

Y1 - 2007/1/1

N2 - 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 ℝn. We completely classify the Hausdorff representations up to unitary equivalence. The complete invariant is the list (λ 1D,..., λND), where λj 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 ℝn.

AB - 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 ℝn. We completely classify the Hausdorff representations up to unitary equivalence. The complete invariant is the list (λ 1D,..., λND), where λj 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 ℝn.

UR - http://www.scopus.com/inward/record.url?scp=43049120020&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43049120020&partnerID=8YFLogxK

U2 - 10.2206/kyushujm.61.443

DO - 10.2206/kyushujm.61.443

M3 - Article

VL - 61

SP - 443

EP - 456

JO - Kyushu Journal of Mathematics

JF - Kyushu Journal of Mathematics

SN - 1340-6116

IS - 2

ER -