Conjugate operators for finite maximal subdiagonal algebras
Narcisse Randrianantoanina
Let ${\mathcal M}$ be a von Neumann algebra with a faithful normal trace $\tau$, and let $H^\infty$ be a finite, maximal, subdiagonal algebra of ${\mathcal M}$. 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 $L^p({\mathcal M}, \tau)$ into $L^p({\mathcal M}, \tau)$ for $1 \lt p \lt \infty$, and to be a continuous map from $L^1({\mathcal M},\tau)$ into $L^{1, \infty}({\mathcal M},\tau)$. We also obtain that if an operator $a$ is such that $|a|\log^+|a| \in L^1({\mathcal M},\tau)$ then its conjugate belongs to $L^1({\mathcal M},\tau)$. Finally, we present some partial extensions of the classical Szegö's theorem to the non-commutative setting.