Von Neumann algebras and linear independence of translates

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 $\mathbb {R}^n$.