TY - JOUR

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

AU - Kajiwara, Tsuyoshi

AU - Watatani, Yasuo

N1 - Funding Information:
This work was supported by JSPS KAKENHI Grant Number 23540242, 19340040 and 25287019.
Publisher Copyright:
© 2017 Elsevier Inc.

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).

KW - Complex dynamical systems

KW - Maximal abelian subalgebras

KW - Self-similar maps

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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 -