Von Neumann algebras and linear independence of translates
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- by Peter A. Linnell PDF
- Proc. Amer. Math. Soc. 127 (1999), 3269-3277 Request permission
Abstract:
For $x,y \in \mathbb {R}$ and $f \in L^2(\mathbb {R})$, define $(x,y) f(t) = e^{2\pi iyt} f(t+x)$ and if $\Lambda \subseteq \mathbb {R}^2$, define $S(f, \Lambda ) = \{(x,y)f \mid (x,y) \in \Lambda \}$. It has been conjectured that if $f\ne 0$, then $S(f,\Lambda )$ is linearly independent over $\mathbb {C}$; one motivation for this problem comes from Gabor analysis. We shall prove that $S(f, \Lambda )$ is linearly independent if $f \ne 0$ and $\Lambda$ is contained in a discrete subgroup of $\mathbb {R}^2$, and as a byproduct we shall obtain some results on the group von Neumann algebra generated by the operators $\{(x,y) \mid (x,y) \in \Lambda \}$. Also, we shall prove these results for the obvious generalization to $\mathbb {R}^n$.References
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Additional Information
- Peter A. Linnell
- Affiliation: Department of Mathematics, Virginia Polytech Institute and State University, Blacksburg, Virginia 24061–0123
- MR Author ID: 114455
- Email: linnell@math.vt.edu
- Received by editor(s): January 30, 1998
- Published electronically: May 4, 1999
- Communicated by: David R. Larson
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 3269-3277
- MSC (1991): Primary 46L10; Secondary 42C99
- DOI: https://doi.org/10.1090/S0002-9939-99-05102-3
- MathSciNet review: 1637388