The Balian-Low theorem for locally compact abelian groups and vector bundles

Let $Λ$ be a lattice in a second countable, locally compact abelian group $G$ with annihilator $Λ^{\perp} \subseteq \widehat{G}$. We investigate the validity of the following statement: For every $η$ in the Feichtinger algebra $S_0(G)$, the Gabor system $\{ M_τ T_λ η\}_{λ\in Λ, τ\in Λ^{\perp}}$ is not a frame for $L^2(G)$. When $G = \mathbb{R}$ and $Λ= α\mathbb{Z}$, this statement is a variant of the Balian-Low theorem. Extending a result of R. Balan, we show that whether the statement generalizes to $(G,Λ)$ is equivalent to the nontriviality of a certain vector bundle over the compact space $(G/Λ) \times (\widehat{G}/Λ^{\perp})$. We prove this equivalence using a connection between Gabor frames and Heisenberg modules. More specifically, we show that the Zak transform can be viewed as an isomorphism of certain Hilbert $C^*$-modules. As an application, we prove a new Balian-Low theorem for the group $\mathbb{R} \times \mathbb{Q}_p$, where $\mathbb{Q}_p$ denotes the $p$-adic numbers.

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