Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces

TL;DR

This paper connects noncommutative geometry with Gabor analysis, showing that projective modules over noncommutative tori correspond to multi-window Gabor frames for modulation spaces, and establishing a duality framework via Morita-Rieffel equivalence.

Contribution

It extends Connes and Rieffel's results to Banach algebras in Gabor analysis, linking projective modules to Gabor frames and duality theory.

Findings

Projective modules over noncommutative tori correspond to multi-window Gabor frames.

Existence of Gabor frames with atoms in Feichtinger's algebra and Schwartz space.

Morita-Rieffel equivalence underpins the duality theory of Gabor frames.

Abstract

In the present investigation we are linking noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. Therefore we are in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to exploit the deeper properties of Gabor frames. Furthermore we are able to extend results due to Connes and Rieffel on projective modules over noncommutative tori to Banach algebras, which arise in a natural manner in Gabor analysis. The main goal of this investigation is twofold: (i) an interpretation of projective modules over noncommutative tori in terms of Gabor analysis and (ii) that the Morita-Rieffel equivalence between noncommutative tori is the natural framework for the duality theory of Gabor frames. More concretely, we interpret generators of projective modules over noncommutative tori as the Gabor atoms of multi-window Gabor frames for modulation spaces. Moreover, we show that this implies the {\it existence} of good multi-window Gabor frames for modulation spaces with Gabor atoms in Feichtinger's algebra and in Schwartz space.