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

Abstract

Let $\Lambda$ be a lattice in a second countable, locally compact abelian group $G$ with annihilator $\Lambda^{\perp} \subseteq \widehat{G}$. We investigate the validity of the following statement: For every $\eta$ in the Feichtinger algebra $S_0(G)$, the Gabor system $\{ M_{\tau} T_{\lambda} \eta \}_{\lambda \in \Lambda, \tau \in \Lambda^{\perp}}$ is not a frame for $L^2(G)$. When $G = \mathbb{R}$ and $\Lambda = \alpha \mathbb{Z}$, this statement is a variant of the Balian-Low theorem. Extending a result of R. Balan, we show that whether the statement generalizes to $(G,\Lambda)$ is equivalent to the nontriviality of a certain vector bundle over the compact space $(G/\Lambda) \times (\widehat{G}/\Lambda^{\perp})$. 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.