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 Szegö's theorem to the non-commutative setting.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- May 1996
- DOI:
- 10.48550/arXiv.math/9605227
- arXiv:
- arXiv:math/9605227
- Bibcode:
- 1996math......5227R
- Keywords:
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- Mathematics - Operator Algebras