Von Neumann algebras and linear independence of translates

Abstract

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 been 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 group 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.