# Heisenberg modules as function spaces

## TL;DR

This paper demonstrates that Heisenberg modules over twisted group C*-algebras can be embedded into L^2 spaces, enabling their use as function spaces in time-frequency analysis with Gabor frames.

## Contribution

It establishes a continuous embedding of Heisenberg modules into L^2(G) and characterizes their generators as multi-window Gabor frames, extending previous results.

## Key findings

- Heisenberg modules can be embedded into L^2(G) as function spaces.
- Generators of these modules correspond to multi-window Gabor frames.
- Modules satisfy properties suitable for time-frequency analysis, including the fundamental identity and bounded frame operators.

## Abstract

Let $\Delta$ 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 $\mathcal{E}_{\Delta}(G)$ over the twisted group $C^*$-algebra $C^*(\Delta,c)$ 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 $\mathcal{E}_{\Delta}(G)$ as exactly the generators of multi-window (continuous) Gabor frames over $\Delta$, a result which was previously known only for a dense subspace of $\mathcal{E}_{\Delta}(G)$. We show that $\mathcal{E}_{\Delta}(G)$ 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 $\Delta$ is a lattice, and their associated frame operators corresponding to $\Delta$ are bounded.