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arXiv · 1411.7269

Gabor Frames for Quasicrystals, $K$-theory, and Twisted Gap Labeling

Michael Kreisel
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Abstract

We study the connection between Gabor frames for quasicrystals, the topology of the hull of a quasicrystal $\Lambda,$ and the $K$-theory of the twisted groupoid $C^*$-algebra $\mathcal{A}_\sigma$ arising from a quasicrystal. In particular, we construct a finitely generated projective module $\mathcal{H}_\L$ over $\mathcal{A}_\sigma$ related to time-frequency analysis, and any multiwindow Gabor frame for $\Lambda$ can be used to construct an idempotent in $M_N(\mathcal{A}_\sigma)$ representing $\mathcal{H}_\L$ in $K_0(\mathcal{A}_\sigma).$ We show for lattice subsets in dimension two, this element corresponds to the Bott element in $K_0(\mathcal{A}_\sigma),$ allowing us to prove a twisted version of Bellissard's gap labeling theorem.

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Michael Kreisel. 2014-11-26. Gabor Frames for Quasicrystals, $K$-theory, and Twisted Gap Labeling. https://arxiv.org/abs/1411.7269

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