Von Neumann algebras and linear independence of translates

Proc. Amer. Math. Soc., to appear

Peter A. Linnell
http://www.math.vt.edu/people/linnell/

For x,y in R (where R denotes the real numbers) and f in L^2(R),
define (x,y)f(t) = e^{2 pi i yt}f(t+x) and if L is a subset of R^2,
define S(f,L) = {(x,y)f | (x,y) in L}.  It has bee conjectured that if
f is not 0, then S(f,L) is linearly independent over C; one motivation
for this problem comes from Gabor analysis.  We shall prove that
S(f,L) is linearly independent if f is nonzero and L is contained in a
discrete subgroup of R^2, and as a byproduct we shall obtain some
results on the gruop von Neumann algebra generated by the operators
{(x,y) | (x,y) in L}.  Also we shall prove these results for the
obvious generalization to R^n.  This can be considered a tiny
contribution to the REPRESENTATION theory of the Heisenberg group.