Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
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 $\mathbb {R}^n$.

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Peter A. Linnell
Affiliation: Department of Mathematics, Virginia Polytech Institute and State University, Blacksburg, Virginia 24061–0123

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