Metrized Quantum Vector Bundles over Quantum Tori Built from Riemannian Metrics and Rosenberg's Levi-Civita Connections

Huang, Leonard

doi:10.3842/sigma.2018.079

Cited by 1 publication

(1 citation statement)

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“…In this paper we revisit Rieffel's original construction of Heisenberg modules for locally compact abelian groups [51]. Heisenberg modules have been the subject of many investigations in noncommutative geometry [5,8,12,30,36,37,41,42,35,38,39,40], but the interplay between the left and the right module structure has not been addressed at all. A link between applied harmonic analysis and noncommutative geometry was described in [44,45] by relating the Heisenberg modules over noncommutative tori, [51], with (multi-window) Gabor frames for L 2 (R m ).…”

Section: Introductionmentioning

confidence: 99%

Duality of Gabor frames and Heisenberg modules

Jakobsen¹,

Luef²

2018

Preprint

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Given a locally compact abelian group G and a closed subgroup Λ in G× G, 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, c) and C * (Λ • , c), where Λ • denotes the adjoint subgroup of Λ. Our main goal is to study Heisenberg modules using tools from timefrequency 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 S 0 (G) is an equivalence bimodule between the Banach subalgebras S 0 (Λ, c) and S 0 (Λ • , c) of C * (Λ, c) and C * (Λ • , c), respectively. Further, we prove that S 0 (G) is finitely generated and projective exactly for co-compact closed subgroups Λ. In this case the generators g 1 , . . . , g n of the left S 0 (Λ)-module S 0 (G) are the Gabor atoms of a multi-window Gabor frame for L 2 (G). We prove that this is equivalent to g 1 , . . . , g n 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 R 2m with volume s(Λ) < 1 there exists a Gabor frame generated by a single atom in S 0 (R m ).

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