We show that the construction of Gabor frames in L 2 (R) with generators in S 0 (R) and with respect to time-frequency shifts from a rectangular lattice αZ × βZ is equivalent to the construction of certain Gabor frames for L 2 over the adeles over the rationals and the group R × Q p . Furthermore, we detail the connection between the construction of Gabor frames on the adeles and on R × Q p with the construction of certain Heisenberg modules.

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Let be a closed, cocompact subgroup of G × G, where G is a second countable, locally compact abelian group. Using localization of Hilbert C *-modules, we show that the Heisenberg module E (G) over the twisted group C *-algebra C * (, c) due to Rieffel can be continuously and densely embedded into the Hilbert space L 2 (G). This allows us to characterize a finite set of generators for E (G) as exactly the generators of multi-window (continuous) Gabor frames over , a result which was previously known only for a dense subspace of E (G). We show that E (G) as a function space satisfies two properties that make it eligible for time-frequency analysis: Its elements satisfy the fundamental identity of Gabor analysis if is a lattice, and their associated frame operators corresponding to are bounded.

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Let Λ be a lattice in a second countable, locally compact abelian group G with annihilator Λ ⊥ ⊆ G. We investigate the validity of the following statement: For every η in the Feichtinger algebra S 0 (G), the Gabor system {Mτ T λ η} λ∈Λ,τ ∈Λ ⊥ is not a frame for L 2 (G). When G = R and Λ = αZ, 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) × ( G/Λ ⊥ ). 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 R × Qp, where Qp denotes the p-adic numbers.