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A note on the density theorem for projective unitary representations


Author: Deguang Han
Journal: Proc. Amer. Math. Soc. 145 (2017), 1739-1745
MSC (2010): Primary 46C05, 46L10, 47D03
DOI: https://doi.org/10.1090/proc/13358
Published electronically: October 26, 2016
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Abstract: It is well known that a Gabor representation on $ L^{2}(\mathbb{R}^{d})$ admits a frame generator $ h\in L^{2}(\mathbb{R}^{d})$ if and only if the associated lattice satisfies the Beurling density condition, which in turn can be characterized as the ``trace condition'' for the associated von Neumann algebra. It happens that this trace condition is also necessary for any projective unitary representation of a countable group to admit a frame vector. However, it is no longer sufficient for general representations, and in particular not sufficient for Gabor representations when they are restricted to proper time-frequency invariant subspaces. In this short note we show that the condition is also sufficient for a large class of projective unitary representations, which implies that the Gabor density theorem is valid for subspace representations in the case of irrational types of lattices.


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Additional Information

Deguang Han
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: deguang.han@ucf.edu

DOI: https://doi.org/10.1090/proc/13358
Keywords: Gabor frames, projective unitary representations, time-frequency lattice, von Neumann algebras.
Received by editor(s): April 22, 2016
Received by editor(s) in revised form: June 20, 2016, and June 21, 2016
Published electronically: October 26, 2016
Additional Notes: The author was partially supported by the NSF grants DMS-1106934 and DMS-1403400
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2016 American Mathematical Society