### Abstract

Let γ = (γ_{1},..., γ_{N}), N ≥ 2, be a system of proper contractions on a complete metric space. Then there exists a unique self-similar non-empty compact subset K. We consider the union G = ∪_{i=1}^{N} {(x,y) ∈ K^{2};x = γ_{i}(y)} of the cographs of γ_{i}. Then X = C(G) is a Hilbert bimodule over A = C(K). We associate a C*-algebra script O sign_{γ}(K) with them as a Cuntz-Pimsner algebra script O sign_{X}. We show that if a system of proper contractions satisfies the open set condition in K, then the C*-algebra script O sign _{γ}(K) is simple, purely infinite and, in general, not isomorphic to a Cuntz algebra.

Original language | English |
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Pages (from-to) | 225-247 |

Number of pages | 23 |

Journal | Journal of Operator Theory |

Volume | 56 |

Issue number | 2 |

Publication status | Published - Sep 2006 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Journal of Operator Theory*,

*56*(2), 225-247.

**C*-algebras associated with self-similar sets.** / Kajiwara, Tsuyoshi; Watatani, Yasuo.

Research output: Contribution to journal › Article

*Journal of Operator Theory*, vol. 56, no. 2, pp. 225-247.

}

TY - JOUR

T1 - C*-algebras associated with self-similar sets

AU - Kajiwara, Tsuyoshi

AU - Watatani, Yasuo

PY - 2006/9

Y1 - 2006/9

N2 - Let γ = (γ1,..., γN), N ≥ 2, be a system of proper contractions on a complete metric space. Then there exists a unique self-similar non-empty compact subset K. We consider the union G = ∪i=1N {(x,y) ∈ K2;x = γi(y)} of the cographs of γi. Then X = C(G) is a Hilbert bimodule over A = C(K). We associate a C*-algebra script O signγ(K) with them as a Cuntz-Pimsner algebra script O signX. We show that if a system of proper contractions satisfies the open set condition in K, then the C*-algebra script O sign γ(K) is simple, purely infinite and, in general, not isomorphic to a Cuntz algebra.

AB - Let γ = (γ1,..., γN), N ≥ 2, be a system of proper contractions on a complete metric space. Then there exists a unique self-similar non-empty compact subset K. We consider the union G = ∪i=1N {(x,y) ∈ K2;x = γi(y)} of the cographs of γi. Then X = C(G) is a Hilbert bimodule over A = C(K). We associate a C*-algebra script O signγ(K) with them as a Cuntz-Pimsner algebra script O signX. We show that if a system of proper contractions satisfies the open set condition in K, then the C*-algebra script O sign γ(K) is simple, purely infinite and, in general, not isomorphic to a Cuntz algebra.

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M3 - Article

VL - 56

SP - 225

EP - 247

JO - Journal of Operator Theory

JF - Journal of Operator Theory

SN - 0379-4024

IS - 2

ER -