Hilbert transform associated with finite maximal subdiagonal algebras

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.