Gabor frames and Wannier bases from groupoid Morita equivalences

Abstract: A key question in Gabor analysis is the reconstruction of elements in a Hilbert space via a Gabor frame. Gabor frames arise from a finite set of vectors acted upon by a canonically defined set of operators (typically translation and modulation). This data is often conveniently encoded in the algebraic structure of a groupoid. In this talk we will discuss how the natural notion of Morita equivalence of groupoids gives rise to Gabor frames for the Hilbert space localisation of the Morita equivalence bimodule of the reduced groupoid $C^*$-algebras. For finitely generated and projective submodules, we show these Gabor frames are orthonormal bases if and only if the module is free. If time allows, we will discuss an application of this result to spectral subspaces of Schroedinger operators with atomic potentials supported on (aperiodic) Delone sets.

This is joint work with Chris Bourne (Tohoku University)

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( paper | video )

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