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arXiv · 1310.5549

K-theory of crossed products of tiling C*-algebras by rotation groups

Charles Starling
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Abstract

Let $Ω$ be a tiling space and let $G$ be the maximal group of rotations which fixes $Ω$. Then the cohomology of $Ω$ and $Ω/G$ are both invariants which give useful geometric information about the tilings in $Ω$. The noncommutative analog of the cohomology of $Ω$ is the K-theory of a C*-algebra associated to $Ω$, and for translationally finite tilings of dimension 2 or less the K-theory is isomorphic to the direct sum of cohomology groups. In this paper we give a prescription for calculating the noncommutative analog of the cohomology of $Ω/G$, that is, the K-theory of the crossed product of the tiling C*-algebra by $G$. We also provide a table with some calculated K-groups for many common examples, including the Penrose and pinwheel tilings.

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Charles Starling. 2013-10-21. K-theory of crossed products of tiling C*-algebras by rotation groups. https://doi.org/10.1007/s00220-014-2070-5

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