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

Abstract

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 PerronFrobenius theory, and is self-similar with scaling factor given by the Perron-Frobenius eigenvalue. We show that each spectral triple is θ-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.

Classification of tiling $C^*$-algebras

“…Since T is aperiodic and repetitive, the induced representation extends to a faithful nondegenerate representation of A punc on ℓ 2 (T ), as shown in [32,Section 4]. Restricting (6.5) to delta functions δ t ′′ ∈ ℓ 2 (T ) gives (6.6)…”

Index Theory and Topological Phases of Aperiodic Lattices

“…Spectral triple constructions for C * r (G) building from the Pearson-Bellissard framework have already appeared in the tiling literature [57,64]. While the setting of each construction is quite different, it would be interesting to better understand the relationship between these spectral triples and our unbounded Fredholm module.…”

“…As previously mentioned, it would be interesting to compare the K-homology class of the ε-unbounded Fredholm module from Theorem 5.14 with similar constructions in the tiling literature [64,57].…”