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 be a von Neumann algebra with a faithful normal trace , and let be a finite, maximal, subdiagonal algebra of . We prove that the Hilbert transform associated with is a linear continuous map from into . 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 then its conjugate , relative to belongs to . These results generalize classical facts from function algebra theory to a non-commutative setting.

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