  Let $M$ be a von Neumann algebra with a faithful normal finite trace $t$, and
$H^\infty$ be a finite, maximal, subdiagonal of $M$. Fundamental theorems on
conjugate functions for weak* Dirichlet algebras are 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 linear map from $L^1(M,t)$ into $L^{1,\infty}(M,t)$. We also obtain
that if a positive operator $a$ is such that $a\log^{+}a \in L^1(M,t)$, then
its conjugate belongs to $L^1(M,t)$.
