Conjugate operators for finite maximal subdiagonal algebras

arXivmath/9605227MaRDI QIDQ6503630

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Abstract: Let $M$ be a von Neumann algebra with a faithful normal trace $T$, and let $H^{i}nfty$ be a finite, maximal, subdiagonal algebra of $M$. Fundamental theorems on conjugate functions for weak${}^{\ast }$!-Dirichlet algebras are shown to be valid for non-commutative $H^{i}nfty$. 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 $|a|lo{g}^{+}|a|in{L}^1(M,T)$ 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.

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