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

Tsuyoshi Kajiwara, Yasuo Watatani

Research output: Contribution to journalArticle

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 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).

Original languageEnglish
JournalJournal of Mathematical Analysis and Applications
DOIs
Publication statusAccepted/In press - 2017

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Complex Dynamical Systems
(0, 1)-matrices
Algebra
C*-algebra
Subalgebra
Dynamical systems
Rational functions
Rational function
Cuntz-Krieger Algebra
Open Set Condition
Path Space
Julia set
Compact Metric Space
Analogy
Permutation

Keywords

  • Complex dynamical systems
  • Maximal abelian subalgebras
  • Self-similar maps

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

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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 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).",
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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).

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