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1 results

  • Hilbert transform associated with finite maximal subdiagonal algebras

  • Part of
  • Narcisse Randrianantoanina
  • Journal:
    Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics / Volume 65 / Issue 3 / December 1998
    Published online by Cambridge University Press:
    09 April 2009, pp. 388-404
    Print publication:
    December 1998
    • Article
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  • Let ℳ be a von Neumann algebra with a faithful normal trace τ, and let H∞ be a finite, maximal. subdiagonal algebra of ℳ. We prove that the Hilbert transform associated with H∞ is a linear continuous map from L1 (ℳ, τ) into L1.∞ (ℳ, τ). 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 ∈ L1 (ℳ, τ) then its conjugate b, relative to H∞ belongs to L1 (ℳ, τ). These results generalize classical facts from function algebra theory to a non-commutative setting.