Duality of Gabor frames and Heisenberg modules

Given a locally compact abelian group $G$ and closed subgroup $\Lambda$ in $G\times\widehat{G}$, Rieffel associated to Hilbert $C^*$-module $\mathcal{E}$, known as Heisenberg module. He proved that $\mathcal{E}$ is an equivalence bimodule between the twisted $C^*$-algebra $C^*(\Lambda,\textsf{c})$ $C^*(\Lambda^\circ,\bar{\textsf{c}})$, where $\Lambda^{\circ}$ denotes adjoint of $\Lambda$. Our main goal study modules using tools from time-frequency analysis pointing out provide natural setting duality theory Gabor systems. More concretely, we show Feichtinger algebra ${\textbf{S}}_{0}(G)$ Banach subalgebras ${\textbf{S}}_{0}(\Lambda,\textsf{c})$ ${\textbf{S}}_{0}(\Lambda^{\circ},\bar{\textsf{c}})$ respectively. Further, prove finitely generated projective exactly for co-compact subgroups In this case generators $g_1,\ldots,g_n$ left ${\textbf{S}}_{0}(\Lambda)$-module are atoms multi-window frame $L^2(G)$. We equivalent being super subspace by system $\Lambda^{\circ}$. This principle independent interest also studied infinitely many atoms. any non-rational lattice $\mathbb{R}^{2m}$ with volume ${s}(\Lambda)<1$ there exists single atom ${\textbf{S}}_{0}(\mathbb{R}^m)$.