Representations of cuntz algebras on fractal sets

Makoto Mori, Osamu Suzuki, Yasuo Watatani

    Research output: Contribution to journalArticlepeer-review

    4 Citations (Scopus)


    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.

    Original languageEnglish
    Pages (from-to)443-456
    Number of pages14
    JournalKyushu Journal of Mathematics
    Issue number2
    Publication statusPublished - 2007

    All Science Journal Classification (ASJC) codes

    • General Mathematics


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