Representations of cuntz algebras on fractal sets

Makoto Mori, Osamu Suzuki, Yasuo Watatani

Research output: Contribution to journalArticle

2 Citations (Scopus)

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 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
Volume61
Issue number2
DOIs
Publication statusPublished - Jan 1 2007

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Cuntz Algebra
Self-similar Set
Fractal Set
Contraction
Hausdorff Measure
Hausdorff Dimension
Lipschitz
Classify
Equivalence
Interval
Unit
Invariant

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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

In: Kyushu Journal of Mathematics, Vol. 61, No. 2, 01.01.2007, p. 443-456.

Research output: Contribution to journalArticle

Mori, Makoto ; Suzuki, Osamu ; Watatani, Yasuo. / Representations of cuntz algebras on fractal sets. In: Kyushu Journal of Mathematics. 2007 ; Vol. 61, No. 2. pp. 443-456.
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