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 noncommutative $H^\infty$. 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 Szegö's theorem to the noncommutative setting.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 1996
 DOI:
 10.48550/arXiv.math/9605227
 arXiv:
 arXiv:math/9605227
 Bibcode:
 1996math......5227R
 Keywords:

 Mathematics  Operator Algebras