Projections in noncommutative tori and Gabor frames

Abstract

We describe a connection between two seemingly different problems: (a) the construction of projections in noncommutative tori and (b) the construction of tight Gabor frames for L 2 (ℝ). The present investigation relies on 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. © 2010 American Mathematical Society.