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

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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