Heisenberg modules as function spaces

Abstract

Let $\Delta$ be a closed, cocompact subgroup of $G\times \hat{G}$, where $G$ is a second countable, locally compact abelian group. Using localization of Hilbert $C^{\ast }$-modules, we show that the Heisenberg module ${\mathcal{E}}_{\Delta }(G)$ over the twisted group $C^{\ast }$-algebra $C^{\ast }(\Delta ,c)$ 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.

24 pages; several changes have been made to the presentation, while the content remains essentially unchanged; to appear in Journal of Fourier Analysis and Applications

References in corpus (1)

• Time-frequency analysis on the adeles over the rationals

Cited by in corpus (3)

• Smooth lattice orbits of nilpotent groups and strict comparison of projections
• Localised module frames and Wannier bases from groupoid Morita equivalences
• The Balian-Low theorem for locally compact abelian groups and vector bundles