Projections in noncommutative tori and Gabor frames

TL;DR

This paper establishes a deep connection between projections in noncommutative tori and tight Gabor frames, showing their equivalence through Wexler-Raz relations and applying Gabor analysis results to construct and characterize projections.

Contribution

It demonstrates the equivalence between Rieffel's projections in noncommutative tori and Gabor frame biorthogonality, providing a new approach to constructing and understanding projections using Gabor analysis.

Findings

Projections in noncommutative tori are equivalent to tight Gabor frames via Wexler-Raz relations.

The projection generated by a Gaussian Gabor frame is Boca's projection.

The approach characterizes the existence range of Boca's projection.

Abstract

We describe a connection between two seemingly different problems: (a) the construction of projections in noncommutative tori, (b) the construction of tight Gabor frames. The present investigation relies an interpretation of projective modules over noncommutative tori in terms of Gabor analysis. The main result demonstrates that Rieffel's condition on the existence of projections in noncommutative tori is equivalent to the Wexler-Raz biorthogonality relations for tight Gabor frames. Therefore we are able to invoke results on the existence of Gabor frames in the construction of projections in noncommutative tori. In particular, the projection associated with a Gabor frame generated by a Gaussian turns out to be Boca's projection. Our approach to Boca's projection allows us to characterize the range of existence of Boca's projection. The presentation of our main result provides a natural approach to the Wexler-Raz biorthogonality relations in terms of Hilbert C*-modules over noncommutative tori.