Conjugate operators for finite maximal subdiagonal algebras

Abstract

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.