Journal of the Australian Mathematical Society - Series A Vol. 65 Part 3 (1998) Hilbert transform associated with finite maximal subdiagonal algebras Narcisse Randrianantoanina Department of Mathematics and Statistics Miami University Oxford OH 45056 e-mail: randrin@muohio.edu Abstract: 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 ${\mathcal M}$. We prove that the Hilbert transform associated with $H^\infty$ is a linear continuous map from $L^1({\mathcal M},\tau)$ into $L^{1, \infty}({\mathcal M},\tau)$. This provides a non-commutative version of a classical theorem of Kolmogorov on weak type boundedness of the Hilbert transform. We also show that if a positive measurable operator b is such that $b\log^+b \in L^1({\mathcal M},\tau)$then its conjugate $\tilde b$, relative to $H^\infty$ belongs to $L^1({\mathcal M},\tau)$. These results generalize classical facts from function algebra theory to a non-commutative setting. PDF file size: 112K © Copyright 1998, Australian Mathematical Society