Classification of tiling $C^{\ast }$-algebras

Abstract

We prove that Kellendonk's $C^{\ast }$-algebra of an aperiodic and repetitive tiling with finite local complexity is classifiable by the Elliott invariant. Our result follows from showing that tiling $C^{\ast }$-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.

arXiv commentSmall updates based on various comments and suggestions

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Cited by in corpus (2)

• Almost elementary étale groupoids
• Fiberwise amenability of ample étale groupoids