Duality of Gabor frames and Heisenberg modules

Given a locally compact abelian group $G$ and a closed subgroup $Lambda$ in $Gtimeswidehat{G}$, Rieffel associated to $Lambda$ a Hilbert $C^*$-module $mathcal{E}$, known as a Heisenberg module. He proved that $mathcal{E}$ is an equivalence bimodule between the twisted group $C^*$-algebra $C^*(Lambda,textsf{c})$ and $C^*(Lambda^circ,bar{textsf{c}})$, where $Lambda^{circ}$ denotes the adjoint subgroup of $Lambda$. 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 ${textbf{S}}_{0}(G)$ is an equivalence bimodule between the Banach subalgebras ${textbf{S}}_{0}(Lambda,textsf{c})$ and ${textbf{S}}_{0}(Lambda^{circ},bar{textsf{c}})$ of $C^*(Lambda,textsf{c})$ and $C^*(Lambda^circ,bar{textsf{c}})$, respectively. Further, we prove that ${textbf{S}}_{0}(G)$ is finitely generated and projective exactly for co-compact closed subgroups $Lambda$. In this case the generators $g_1,ldots,g_n$ of the left ${textbf{S}}_{0}(Lambda)$-module ${textbf{S}}_{0}(G)$ are the Gabor atoms of a multi-window Gabor frame for $L^2(G)$. We prove that this is equivalent to $g_1,ldots,g_n$ being a Gabor super frame for the closed subspace generated by the Gabor system for $Lambda^{circ}$. 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 $Lambda$ in $mathbb{R}^{2m}$ with volume ${s}(Lambda)<1$ there exists a Gabor frame generated by a single atom in ${textbf{S}}_{0}(mathbb{R}^m)$.