  Let $\M$ be a von Neumann algebra with a faithful normal trace $\T$, and let
$H^\infty$ be a finite, maximal, subdiagonal algebra of $\M$. Fundamental
theorems on conjugate functions for weak$^*$\!-Dirichlet algebras are shown to
be valid for non-commutative $H^\infty$. In particular the conjugation operator
is shown to be a bounded linear map from $L^p(\M, \T)$ into $L^p(\M, \T)$ for
$1 < p < \infty$, and to be a continuous map from $L^1(\M,\T)$ into $L^{1,
\infty}(\M,\T)$. We also obtain that if an operator $a$ is such that
$|a|\log^+|a| \in L^1(\M,\T)$ then its conjugate belongs to $L^1(\M,\T)$.
Finally, we present some partial extensions of the classical Szeg\"o's theorem
to the non-commutative setting.
