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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.

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© Copyright 1998, Australian Mathematical Society
TeXAdel Scientific Publishing
1998-09-18