A note on the density theorem for projective unitary representations
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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.References
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Additional Information
- Deguang Han
- Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
- Email: deguang.han@ucf.edu
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1739-1745
- MSC (2010): Primary 46C05, 46L10, 47D03
- DOI: https://doi.org/10.1090/proc/13358
- MathSciNet review: 3601564