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

We study the connection between Gabor frames for quasicrystals, the topology of the hull of a quasicrystal $\mathrm{\Lambda },$ and the $K$-theory of the twisted groupoid $C^{\ast }$-algebra ${\mathcal{A}}_{\sigma }$ arising from a quasicrystal. In particular, we construct a finitely generated projective module ${\mathcal{H}}_{\text{L}}$ over ${\mathcal{A}}_{\sigma }$ related to time-frequency analysis, and any multiwindow Gabor frame for $\mathrm{\Lambda }$ can be used to construct an idempotent in $M_N({\mathcal{A}}_{\sigma })$ representing ${\mathcal{H}}_{\text{L}}$ in $K_0({\mathcal{A}}_{\sigma })\mathrm{.}$ 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.