Gabor Duality Theory for Morita Equivalent $C^{\ast }$-algebras

Abstract

The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalent $C^{\ast }$-algebras where the equivalence bimodule is a finitely generated projective Hilbert $C^{\ast }$-module. These Hilbert $C^{\ast }$-modules are equipped with some extra structure and are called Gabor bimodules. We formulate a duality principle for standard module frames for Gabor bimodules which reduces to the well-known Gabor duality principle for twisted group $C^{\ast }$-algebras of a lattice in phase space. We lift all these results to the matrix algebra level and in the description of the module frames associated to a matrix Gabor bimodule we introduce $(n,d)$-matrix frames, which generalize superframes and multi-window frames. Density theorems for $(n,d)$-matrix frames are established, which extend the ones for multi-window and super Gabor frames. Our approach is based on the localization of a Hilbert $C^{\ast }$-module with respect to a trace.

36 pages