### 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 1 2006 |

### Keywords

- Hilbert bimodule
- Purely infinite C*-algebra
- Self-similar set

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Kajiwara, T., & Watatani, Y. (2006). C*-algebras associated with self-similar sets.

*Journal of Operator Theory*,*56*(2), 225-247.