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
Pages (from-to)1383-1400
Number of pages18
JournalJournal of Mathematical Analysis and Applications
Volume455
Issue number2
DOIs
Publication statusPublished - Nov 15 2017

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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