Localised module frames and Wannier bases from groupoid Morita equivalences

Abstract

Following the operator algebraic approach to Gabor analysis, we construct frames of translates for the Hilbert space localisation of the Morita equivalence bimodule arising from a groupoid equivalence between Hausdorff groupoids, where one of the groupoids is étale and with a compact unit space. For finitely generated and projective submodules, we show these frames are orthonormal bases if and only if the module is free. We then apply this result to the study of localised Wannier bases of spectral subspaces of Schrödinger operators with atomic potentials supported on (aperiodic) Delone sets. The noncommutative Chern numbers provide a topological obstruction to fast-decaying Wannier bases and we show this result is stable under deformations of the underlying Delone set.

v2: name changed and other minor changes. To appear in J. Fourier Anal. Appl. 28 pages

References in corpus (4)

• Renault's Equivalence Theorem for Reduced Groupoid C*-algebras
• Tight frames of exponentially decaying Wannier functions
• The Iterated Projected Position Algorithm for Constructing Exponentially Localized Generalized Wannier Functions for Periodic and Non-Periodic Insulators in Two Dimensions and Higher
• Algebraic localization implies exponential localization in non-periodic insulators

Cited by in corpus (3)

• Localization of generalized Wannier bases implies Chern triviality in non-periodic insulators
• Ultra-generalized Wannier bases: Are they relevant to topological transport?
• On topological obstructions to the existence of non-periodic Wannier bases