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arXiv 1908.007702019-08math.OA

Classification of tiling $C^*$-algebras

Luke J. Ito, Michael F. Whittaker, Joachim Zacharias

备注：: Small updates based on various comments and suggestions

英文摘要：

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 étale 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 étale groupoids.

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