TY - JOUR

T1 - Dimension groups for self-similar maps and matrix representations of the cores of the associated C∗-Algebras

AU - Kajiwara, Tsuyoshi

AU - Watatani, Yasuo

N1 - Publisher Copyright:
Copyright © 2018, The Authors. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2018/3/23

Y1 - 2018/3/23

N2 - We introduce a dimension group for a self-similar map as the K0-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 ℤ∞ ≅= ℤ[t] together with the unilatral shift, i.e. the multiplication map by t as an abstract group. Thus the canonical endomorphisms on the K0-groups are not automorphisms in geneal. This is a different point compared with dimension groups for topological Markov shifts. We can count the singularity structure in the dimension groups.46L35, 46L80, 46L55

AB - We introduce a dimension group for a self-similar map as the K0-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 ℤ∞ ≅= ℤ[t] together with the unilatral shift, i.e. the multiplication map by t as an abstract group. Thus the canonical endomorphisms on the K0-groups are not automorphisms in geneal. This is a different point compared with dimension groups for topological Markov shifts. We can count the singularity structure in the dimension groups.46L35, 46L80, 46L55

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M3 - Article

AN - SCOPUS:85093029624

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SN - 0402-1215

ER -