Heisenberg modules as function spaces

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.