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

TL;DR

This paper connects Gabor frames for quasicrystals with the topology of their hulls and the $K$-theory of associated twisted groupoid $C^*$-algebras, leading to a twisted gap labeling theorem.

Contribution

It constructs a finitely generated projective module over the twisted algebra related to Gabor frames and links it to $K$-theory, extending gap labeling results.

Findings

Constructed a projective module related to Gabor frames.

Linked the module to the $K$-theory of the algebra.

Proved a twisted gap labeling theorem for quasicrystals.

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.