2018/06/14 by Jakobsen, Mads S., Luef, Franz · 1 citation
#42C15 #58B34 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1806.05616
Given a locally compact abelian group G and a closed subgroup Λ in G×\widehatG, Rieffel associated to Λ a Hilbert C^*-module E, known as a Heisenberg module. He proved that E is an equivalence bimodule between the twisted group C^*-algebra C^*(Λ,\textsfc) and C^*(Λ^∘,\textsfc), where Λ∘ denotes the adjoint subgroup of Λ. Our main goal is to study Heisenberg modules using tools from time-frequency analysis and pointing out that Heisenberg modules provide the natural setting of the duality theory of Gabor systems. More concretely, we show that the Feichtinger algebra S0(G) is an equivalence bimodule between the Banach subalgebras S0(Λ,\textsfc) and S0(Λ∘,\textsfc) of C^*(Λ,\textsfc) and C^*(Λ^∘,\textsfc), respectively. Further, we prove that S0(G) is finitely generated and projective exactly for co-compact closed subgroups Λ. In this case the generators g1,…,gn of the left S0(Λ)-module S0(G) are the Gabor atoms of a multi-window Gabor frame for L2(G). We prove that this is equivalent to g1,…,gn being a Gabor super frame for the closed subspace generated by the Gabor system for Λ∘. This duality principle is of independent interest and is also studied for infinitely many Gabor atoms. We also show that for any non-rational lattice Λ in ℝ2m with volume s(Λ)<1 there exists a Gabor frame generated by a single atom in S0(ℝm).