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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Projections in noncommutative tori and Gabor frames


Author: Franz Luef
Journal: Proc. Amer. Math. Soc. 139 (2011), 571-582
MSC (2010): Primary 42C15, 46L08; Secondary 22D25, 43A20
Published electronically: July 16, 2010
MathSciNet review: 2736339
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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(\mathbb{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.


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Additional Information

Franz Luef
Affiliation: Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, 1090 Wien, Austria
Address at time of publication: Department of Mathematics, University of California, Berkeley, California 94720-3840
Email: franz.luef@univie.ac.at, luef@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10489-6
PII: S 0002-9939(2010)10489-6
Keywords: Gabor frames, noncommutative tori, projections in $C^{*}$-algebras
Received by editor(s): November 13, 2009
Received by editor(s) in revised form: March 10, 2010
Published electronically: July 16, 2010
Additional Notes: The author was supported by the Marie Curie Excellence grant MEXT-CT-2004-517154 and the Marie Curie Outgoing Fellowship PIOF-220464.
Communicated by: Marius Junge
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.