Title of dissertation : GABOR FRAMES FOR QUASICRYSTALS AND K - THEORY

Title of dissertation: GABOR FRAMES FOR QUASICRYSTALS AND K-THEORY Michael Kreisel, Doctor of Philosophy, 2015 Dissertation directed by: Professor Jonathan Rosenberg Department of Mathematics We study the connection between Gabor frames for quasicrystals, the topology of the hull ΩΛ of a quasicrystal Λ and the K-theory of an associated twisted groupoid algebra. In particular, we construct a finitely generated projective module over this algebra, and any multiwindow Gabor frame for Λ can be used to construct a projection representing this module in K-theory. For the case of lattices, modules of this kind were first constructed over noncommutative tori in [31]. Luef developed connections with Gabor analysis and showed that the operator algebraic framework tied together many unique aspects of lattice Gabor frames [25], [26]. Our work adapts their results to the setting of quasicrystals. Along the way, we prove a variety of compatibility conditions between the topology of ΩΛ and the associated Gabor frame operators. We give a version of Janssen’s representation for the Gabor frame operator when Λ is a model set. We also prove that certain quasicrystals can never be the support of a tight multiwindow Gabor frame when the window functions are in the modulation space M1(R). As an application to noncommutative topology, we are able to deduce results on the twisted version of Bellissard’s gap labeling conjecture. We show that the twisted gap labeling group of Λ always contains the image of the trace map of an associated noncommutative torus and we identify modules in K-theory corresponding to these gap labels. For lattice subsets in dimension two we prove that this constitutes the entire gap labeling group. As a byproduct of our analysis, we also show that when ΩΛ is viewed as a fiber bundle over a torus with projection p, the pullback map p∗ is injective on K. GABOR FRAMES FOR QUASICRYSTALS AND K-THEORY