# Gabor Duality Theory for Morita Equivalent $C^*$-algebras

## TL;DR

This paper generalizes Gabor duality principles to Morita equivalent $C^*$-algebras using Gabor bimodules, introducing $(n,d)$-matrix frames and establishing density theorems for these structures.

## Contribution

It extends Gabor duality to Morita equivalent $C^*$-algebras with Gabor bimodules and introduces $(n,d)$-matrix frames, broadening the scope of Gabor analysis.

## Key findings

- Established duality principle for Gabor bimodules.
- Introduced $(n,d)$-matrix frames generalizing superframes.
- Proved density theorems for $(n,d)$-matrix frames.

## Abstract

The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalent $C^*$-algebras where the equivalence bimodule is a finitely generated projective Hilbert $C^*$-module. These Hilbert $C^*$-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 $C^*$-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 $C^*$-module with respect to a trace.