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On the $K$-theory of $C^*$-algebras associated to substitution tilings

Volume 551 / 2020

Daniel Gonçalves, Maria Ramirez-Solano Dissertationes Mathematicae 551 (2020), 1-133 MSC: 52C23, 46L80, 37D15. DOI: 10.4064/dm800-4-2020 Published online: 28 July 2020

Abstract

Under suitable conditions, a substitution tiling gives rise to a Smale space, from which three equivalence relations can be constructed, namely the stable, unstable, and asymptotic equivalence relations. We denote by $S$, $U$, and $A$ their corresponding $C^*$-algebras in the sense of Renault. We show that the $K$-theories of $S$ and $U$ can be computed from the cohomology and homology of a single cochain complex with connecting maps for tilings of the line and of the plane. Moreover, we provide formulas to compute the $K$-theory for these three $C^*$-algebras. Furthermore, we show that the $K$-theory groups for tilings of dimension 1 are always torsion free. For tilings of dimension 2, only $K_0(U)$ and $K_1(S)$ can contain torsion.

Authors

  • Daniel GonçalvesDepartamento de Matemática
    Universidade Federal de Santa Catarina
    Florianópolis, 88040-900 SC, Brazil
    e-mail
  • Maria Ramirez-SolanoDepartment of Mathematics
    University of Southern Denmark
    Odense, Denmark
    e-mail

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