Fractal spectral triples on Kellendonk’s -algebra of a substitution tiling

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• 作者：Michael Mampusti ^{mm554@uowmail.edu.au} ; Michael F. Whittaker ; ^{Mike.Whittaker@glasgow.ac.uk}
• 关键词：primary ; 46L55 ; secondary ; 47B25 ; 37D40
• 刊名：Journal of Geometry and Physics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：112
• 期：Complete
• 页码：224-239
• 全文大小：1350 K
• 卷排序：112

文摘

We introduce a new class of noncommutative spectral triples on Kellendonk’s C∗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 θθ-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.