# Duality of Gabor frames and Heisenberg modules

### Mads S. Jakobsen

Norwegian University of Science and Technology (NTNU), Trondheim, Norway### Franz Luef

Norwegian University of Science and Technology (NTNU), Trondheim, Norway

## Abstract

Given a locally compact abelian group $G$ and a closed subgroup $Λ$ in $G×G^$, Rieffel associated to $Λ$ a Hilbert $C_{∗}$-module $E$, known as a Heisenberg module. He proved that $E$ is an equivalence bimodule between the twisted group $C_{∗}$-algebra $C_{∗}(Λ,c)$ and $C_{∗}(Λ_{∘},cˉ)$, where $Λ_{∘}$ denotes the adjoint subgroup of $Λ$. Our main goal is to study Heisenberg modules using tools from time-frequency analysis and pointing out that Heisenberg modules provide the natural setting of the duality theory of Gabor systems. More concretely, we show that the Feichtinger algebra $S_{O}(G)$ is an equivalence bimodule between the Banach subalgebras $S_{O}(Λ,c)$ and $S_{O}(Λ_{∘},cˉ)$ of $C_{∗}(Λ,c)$ and $C_{∗}(Λ_{∘},cˉ)$, respectively. Further, we prove that $S_{O}(G)$ is finitely generated and projective exactly for co-compact closed subgroups $Λ$. In this case the generators $g_{1},…,g_{n}$ of the left $S_{O}(Λ)$-module $S_{O}(G)$ are the Gabor atoms of a multi-window Gabor frame for $L_{2}(G)$. We prove that this is equivalent to $g_{1},…,g_{n}$ being a Gabor super frame for the closed subspace generated by the Gabor system for $Λ_{∘}$. This duality principle is of independent interest and is also studied for infinitely many Gabor atoms. We also show that for any non-rational lattice $Λ$ in $R_{2m}$ with volume $s(Λ)<1$ there exists a Gabor frame generated by a single atom in $Λ_{∘}(R_{m})$.

## Cite this article

Mads S. Jakobsen, Franz Luef, Duality of Gabor frames and Heisenberg modules. J. Noncommut. Geom. 14 (2020), no. 4, pp. 1445–1500

DOI 10.4171/JNCG/413