REPRESENTATIONS OF CUNTZ ALGEBRAS ON FRACTAL SETS REPRESENTATIONS OF CUNTZ ALGEBRAS ON FRACTAL SETS
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Abstract
We consider representations of Cuntz algebras on selfsimilar fractal sets for proper/improper systems of contractions. Natural representations, called Hausdorff representations, are associated with selfsimilar 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 nontrivial 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 selfsimilar 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 selfsimilar fractal sets for proper/improper systems of contractions. Natural representations, called Hausdorff representations, are associated with selfsimilar 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 nontrivial 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 selfsimilar 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), 443456, 2007
Kyushu University