REPRESENTATIONS OF CUNTZ ALGEBRAS ON FRACTAL SETS REPRESENTATIONS OF CUNTZ ALGEBRAS ON FRACTAL SETS

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Author(s)

    • MORI Makoto
    • Department of Mathematics College of Humanities and Sciences Nihon University
    • SUZUKI Osamu
    • Department of Mathematics College of Humanities and Sciences Nihon University
    • WATATANI Yasuo
    • Department of Mathematical Sciences Faculty of Mathematics Kyushu University

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 $. We completely classify the Hausdorff representations up to unitary equivalence. The complete invariant is the list$ (\lambda_1 ^D, \ldots ,\lambda_N ^D) $, where $ \lambda_j $ is the Lipschitz constant of the $ j $th 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 $.

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 <B>R</B><SUP><I>n</I></SUP>. We completely classify the Hausdorff representations up to unitary equivalence. The complete invariant is the list(λ<SUB>1</SUB><SUP>D</SUP>, . . . ,λ<SUB>N</SUB><SUP>D</SUP>), where λ<SUB>j</SUB> is the Lipschitz constant of the <I>j</I> th contraction and <I>D</I> 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 <B>R</B><SUP><I>n</I></SUP>.

Journal

  • Kyushu Journal of Mathematics

    Kyushu Journal of Mathematics 61(2), 443-456, 2007

    Kyushu University

Codes

  • NII Article ID (NAID)
    110006377556
  • NII NACSIS-CAT ID (NCID)
    AA10994346
  • Text Lang
    ENG
  • Article Type
    Departmental Bulletin Paper
  • ISSN
    1340-6116
  • Data Source
    NII-ELS  IR  J-STAGE 
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