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Von Neumann algebras and
linear independence of translates


Author: Peter A. Linnell
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
Published electronically: May 4, 1999
MathSciNet review: 1637388
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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 [Enhancements On Off] (What's this?)

  • 1. W. Arveson, An invitation to C*-algebra, Graduate Texts in Mathematics, vol. 39, Springer-Verlag, Berlin-New York, 1976. MR 58:23621
  • 2. S. K. Berberian, Baer $*$-rings, Grundlehren, vol. 195, Springer-Verlag, Berlin-New York, 1972. MR 55:2983
  • 3. -, The maximal ring of quotients of a finite von Neumann algebra, Rocky Mountain J. Math. 12 (1982), 149-164. MR 83i:16005
  • 4. C. Heil, J. Ramanathan, and P. Topiwala, Linear independence of time-frequency translates, Proc. Amer. Math. Soc. 124 (1996), 2787-2795. MR 96k:42039
  • 5. R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras, volume 1, elementary theory, Pure and Applied Mathematics Series, vol. 100, Academic Press, London-New York, 1983. MR 85j:46099
  • 6. P. A. Linnell, Zero divisors and group von Neumann algebras, Pacific J. Math. 149 (1991), 349-363. MR 92e:22013
  • 7. Marc A. Rieffel, Von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann. 257 (1981), 403-418. MR 84f:22010
  • 8. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, N.J., 1993. MR 95c:42002

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Additional Information

Peter A. Linnell
Affiliation: Department of Mathematics, Virginia Polytech Institute and State University, Blacksburg, Virginia 24061–0123
Email: linnell@math.vt.edu

DOI: https://doi.org/10.1090/S0002-9939-99-05102-3
Keywords: Group von Neumann algebra, Gabor analysis, Heisenberg group
Received by editor(s): January 30, 1998
Published electronically: May 4, 1999
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society

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