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Dimension groups for self-similar maps and matrix representations of the cores of the associated C*-algebras

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  12 May 2020

Tsuyoshi Kajiwara  and
Yasuo Watatani
Tsuyoshi Kajiwara*
Affiliation:
Department of Environmental and Mathematical Sciences, Okayama University, Tsushima, Okayama, 700-8530, Japan
Yasuo Watatani
Affiliation:
Department of Mathematical Sciences, Kyushu University, Motooka, Fukuoka, 819-0395, Japan e-mail: watatani@math.kyushu-u.ac.jp
*
e-mail: kajiwara@okayama-u.ac.jp
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Abstract

We introduce a dimension group for a self-similar map as the $\mathrm {K}_0$ -group of the core of the C*-algebra associated with the self-similar map together with the canonical endomorphism. The key step for the computation is an explicit description of the core as the inductive limit using their matrix representations over the coefficient algebra, which can be described explicitly by the singularity structure of branched points. We compute that the dimension group for the tent map is isomorphic to the countably generated free abelian group ${\mathbb Z}^{\infty }\cong {\mathbb Z}[t]$ together with the unilateral shift, i.e. the multiplication map by t as an abstract group. Thus the canonical endomorphisms on the $\mathrm {K}_0$ -groups are not automorphisms in general. This is a different point compared with dimension groups for topological Markov shifts. We can count the singularity structure in the dimension groups.

Keywords

Self-similar mapcoredimension group

MSC classification

Primary: 46L80: $K$-theory and operator algebras (including cyclic theory)
Secondary: 46L08: $C^*$-modules 46L35: Classifications of $C^*$-algebras 46L55: Noncommutative dynamical systems
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 4 , August 2021 , pp. 1013 - 1056
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Copyright
© Canadian Mathematical Society 2020

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