Let

be a von Neumann algebra with a faithful normal trace

,
and let

be a finite, maximal, subdiagonal algebra of

.
We prove that
the Hilbert transform associated with

is a linear
continuous map from

into

.
This provides a non-commutative version of a classical
theorem of Kolmogorov on weak type boundedness of the Hilbert transform.
We also show
that if a
positive measurable operator

*b* is such that

then its conjugate

,
relative to

belongs to

.
These
results generalize classical facts from function algebra theory
to a non-commutative setting.