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arXiv · 1712.09551

On the K-theory of C*-algebras for substitution tilings (a pedestrian version)

Daniel Gonçalves·Maria Ramirez-Solano
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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 with $S$, $U$, and $A$ their corresponding $C^*$-algebras in the sense of Renault. In this article 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.

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Daniel Gonçalves, Maria Ramirez-Solano. 2017-12-27. On the K-theory of C*-algebras for substitution tilings (a pedestrian version). https://arxiv.org/abs/1712.09551

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