Heisenberg Modules as Function Spaces

Abstract

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

Gabor duality theory for Morita equivalent C∗-algebras

“…Remark 4.1. The fact that we get the same twisted group C * -algebras by using S 0 (Λ, c) as we get when using the more traditional approach with L 1 (Λ, c) was noted in [3]. on S 0 (Λ, c) and S 0 (Λ • , c) given by evaluation in 0.…”

The Balian–Low theorem for locally compact abelian groups and vector bundles

“…Moreover, an embedding of the Heisenberg module into L 2 (G) was demonstrated in [3]. Building upon these works, it was shown in [2] that finite generating sets in the Heisenberg module E ∆ (G) correspond exactly to multiwindow Gabor frames over ∆ with generators in E ∆ (G). It follows from an argument of Rieffel (see Proposition 3.3) that there exists a Gabor frame over ∆ with generator in E ∆ (G) if and only if there exists a Gabor frame over ∆ with generator in S 0 (G).…”

“…which is homeomorphic to S 2 p by Proposition 5.10. Thus, the vector bundleẼ is a line bundle over S 2 p , and by Proposition 5.10, S 2 p can be realized as the inverse limit of the sequence · · · / / (R/Z) 2 (·p,·p) / / (R/Z) 2 (·p,·p) / / (R/Z) 2 .…”

Modulation spaces as a smooth structure in noncommutative geometry

“…Furthermore, the Heisenberg module E G,Λ can be continuously embedded in L 2 (G) . This statement can be proved by ways of localization as in [3]. However, since we are working exclusively with lattices in phase space, we use a different and simpler proof.…”

“…Througout this section, we fix a second-countable LCA group G, and we will let Λ be a lattice in G ×Ĝ , that is, Λ is a cocompact and discrete subgroup in G ×Ĝ . Here Ĝ is the dual group of G. The group G ×Ĝ is sometimes called the time-frequency plane of G or the phase space of G. We will have to restrict to lattices Λ as we wish to make use of the localization procedure developed in [3] in a particular case. Namely, we need to be able to localize both the C * -algebra C * (Λ, c) and a Heisenberg module, defined in (3.2) and (3.3).…”