### Abstract

We consider an analogy among Markov shifts, complex dynamical systems and self-similar maps. Their dynamics are given by 0–1 matrices A, rational functions R and self-similar maps γ on a compact metric space K, respectively. If the 0–1 matrix A is irreducible and not a permutation, then the Cuntz–Krieger algebra O_{A} is simple and purely infinite. Similarly, if the rational function R is restricted to the Julia set J_{R} and the self-similar map γ satisfies the open set condition respectively, then the associated C^{⁎}-algebras O_{R}(J_{R}) and O_{γ}(K) are simple and purely infinite. Let Σ_{A} be the associated infinite path space for the 0–1 matrix A, then C(Σ_{A}) is known to be a maximal abelian subalgebra of O_{A}. In this paper we shall show that C(J_{R}) is a maximal abelian subalgebra of O_{R}(J_{R}) and C(K) is a maximal abelian subalgebra of O_{γ}(K).

Original language | English |
---|---|

Pages (from-to) | 1383-1400 |

Number of pages | 18 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 455 |

Issue number | 2 |

DOIs | |

Publication status | Published - Nov 15 2017 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

^{⁎}-algebras associated with complex dynamical systems and self-similar maps.

*Journal of Mathematical Analysis and Applications*,

*455*(2), 1383-1400. https://doi.org/10.1016/j.jmaa.2017.06.044

**Maximal abelian subalgebras of C ^{⁎}-algebras associated with complex dynamical systems and self-similar maps.** / Kajiwara, Tsuyoshi; Watatani, Yasuo.

Research output: Contribution to journal › Article

^{⁎}-algebras associated with complex dynamical systems and self-similar maps',

*Journal of Mathematical Analysis and Applications*, vol. 455, no. 2, pp. 1383-1400. https://doi.org/10.1016/j.jmaa.2017.06.044

^{⁎}-algebras associated with complex dynamical systems and self-similar maps. Journal of Mathematical Analysis and Applications. 2017 Nov 15;455(2):1383-1400. https://doi.org/10.1016/j.jmaa.2017.06.044

}

TY - JOUR

T1 - Maximal abelian subalgebras of C⁎-algebras associated with complex dynamical systems and self-similar maps

AU - Kajiwara, Tsuyoshi

AU - Watatani, Yasuo

PY - 2017/11/15

Y1 - 2017/11/15

N2 - We consider an analogy among Markov shifts, complex dynamical systems and self-similar maps. Their dynamics are given by 0–1 matrices A, rational functions R and self-similar maps γ on a compact metric space K, respectively. If the 0–1 matrix A is irreducible and not a permutation, then the Cuntz–Krieger algebra OA is simple and purely infinite. Similarly, if the rational function R is restricted to the Julia set JR and the self-similar map γ satisfies the open set condition respectively, then the associated C⁎-algebras OR(JR) and Oγ(K) are simple and purely infinite. Let ΣA be the associated infinite path space for the 0–1 matrix A, then C(ΣA) is known to be a maximal abelian subalgebra of OA. In this paper we shall show that C(JR) is a maximal abelian subalgebra of OR(JR) and C(K) is a maximal abelian subalgebra of Oγ(K).

AB - We consider an analogy among Markov shifts, complex dynamical systems and self-similar maps. Their dynamics are given by 0–1 matrices A, rational functions R and self-similar maps γ on a compact metric space K, respectively. If the 0–1 matrix A is irreducible and not a permutation, then the Cuntz–Krieger algebra OA is simple and purely infinite. Similarly, if the rational function R is restricted to the Julia set JR and the self-similar map γ satisfies the open set condition respectively, then the associated C⁎-algebras OR(JR) and Oγ(K) are simple and purely infinite. Let ΣA be the associated infinite path space for the 0–1 matrix A, then C(ΣA) is known to be a maximal abelian subalgebra of OA. In this paper we shall show that C(JR) is a maximal abelian subalgebra of OR(JR) and C(K) is a maximal abelian subalgebra of Oγ(K).

UR - http://www.scopus.com/inward/record.url?scp=85021830354&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021830354&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2017.06.044

DO - 10.1016/j.jmaa.2017.06.044

M3 - Article

AN - SCOPUS:85021830354

VL - 455

SP - 1383

EP - 1400

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -