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)

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)

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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, Dr Michael
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

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