Duality of Gabor frames and Heisenberg modules

Abstract

Given a locally compact abelian group $G$ and a closed subgroup $\Lambda$ in $G\times\widehat{G}$, 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 ${\textbf{S}}_{0}(\mathbb{R}^m)$.