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arXiv · 1908.00770

Classification of tiling $C^*$-algebras

Luke J. Ito·Michael F. Whittaker·Joachim Zacharias
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Abstract

We prove that Kellendonk's $C^*$-algebra of an aperiodic and repetitive tiling with finite local complexity is classifiable by the Elliott invariant. Our result follows from showing that tiling $C^*$-algebras are $\mathcal{Z}$-stable, and hence have finite nuclear dimension. To prove $\mathcal{Z}$-stability, we extend Matui's notion of almost finiteness to the setting of \'etale groupoid actions following the footsteps of Kerr. To use some of Kerr's techniques we have developed a version of the Ornstein-Weiss quasitiling theorem for general \'etale groupoids.

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Luke J. Ito, Michael F. Whittaker, Joachim Zacharias. 2019-08-02. Classification of tiling $C^*$-algebras. https://arxiv.org/abs/1908.00770

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