Heisenberg Modules as Function Spaces

Abstract

AbstractLet $$\Delta $$Δ be a closed, cocompact subgroup of $$G \times \widehat{G}$$G×G^, where G is a second countable, locally compact abelian group. Using localization of Hilbert $$C^*$$C∗-modules, we show that the Heisenberg module $$\mathcal {E}_{\Delta }(G)$$EΔ(G) over the twisted group $$C^*$$C∗-algebra $$C^*(\Delta ,c)$$C∗(Δ,c) due to Rieffel can be continuously and densely embedded into the Hilbert space $$L^2(G)$$L2(G). This allows us to characterize a finite set of generators for $$\mathcal {E}_{\Delta }(G)$$EΔ(G) as exactly the generators of multi-window (continuous) Gabor frames over $$\Delta $$Δ, a result which was previously known only for a dense subspace of $$\mathcal {E}_{\Delta }(G)$$EΔ(G). We show that $$\mathcal {E}_{\Delta }(G)$$EΔ(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 $$\Delta $$Δ is a lattice, and their associated frame operators corresponding to $$\Delta $$Δ are bounded.