Let M be a von Neumann algebra with a faithful normal trace tau, and let H-
infinity be a finite, maximal, subdiagonal algebra of M. We prove that the
Hilbert transform associated with H-infinity is a linear continuous map fro
m L-1 (M, tau) into L-1,L-infinity(M, tau). This provides a non-commutative
version of a classical theorem of Kolmogorov on weak type boundedness of t
he Hilbert transform. We also show that if a positive measurable operator b
is such that b log(+) b is an element of L-1(M, tau) then its conjugate (h
) over bar, relative to H-infinity belongs to L-1(M, tau). These results ge
neralize classical facts from function algebra theory to a non-commutative
setting.