Fractal spectral triples on Kellendonk's C*-algebra of a substitution tiling

Mampusti, M. and Whittaker, M. F. (2017) Fractal spectral triples on Kellendonk's C*-algebra of a substitution tiling. Journal of Geometry and Physics, 112, pp. 224-239. (doi: 10.1016/j.geomphys.2016.11.010)

Abstract

We introduce a new class of noncommutative spectral triples on Kellendonk's C*-algebra associated with a nonperiodic substitution tiling. These spectral triples are constructed from fractal trees on tilings, which define a geodesic distance between any two tiles in the tiling. Since fractals typically have infinite Euclidean length, the geodesic distance is defined using Perron-Frobenius theory, and is self-similar with scaling factor given by the Perron-Frobenius eigenvalue. We show that each spectral triple is $\theta$-summable, and respects the hierarchy of the substitution system. To elucidate our results, we construct a fractal tree on the Penrose tiling, and explicitly show how it gives rise to a collection of spectral triples.

• Item Type: Articles
• Additional Information: This research was partially supported by the Australian Research Council (DP130100490).
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Whittaker, Professor Mike
• Authors: Mampusti, M., and Whittaker, M. F.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Geometry and Physics
• Publisher: Elsevier
• ISSN: 0393-0440
• Published Online: 22 November 2016
• Copyright Holders: Copyright © 2016 Elsevier B.V.
• First Published: First published in Journal of Geometry and Physics 112: 224-239
• Publisher Policy: Reproduced in accordance with the publisher copyright policy