Heisenberg modules as function spaces

Let $Δ$ be a closed, cocompact subgroup of $G \times \widehat{G}$, where $G$ is a second countable, locally compact abelian group. Using localization of Hilbert $C^*$-modules, we show that the Heisenberg module $\mathcal{E}_Δ(G)$ over the twisted group $C^*$-algebra $C^*(Δ,c)$ 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 $\mathcal{E}_Δ(G)$ as exactly the generators of multi-window (continuous) Gabor frames over $Δ$, a result which was previously known only for a dense subspace of $\mathcal{E}_Δ(G)$. We show that $\mathcal{E}_Δ(G)$ 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 $Δ$ is a lattice, and their associated frame operators corresponding to $Δ$ are bounded.

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