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Hilbert transform associated with finite maximal subdiagonal algebras

Part of: Abstract harmonic analysis Linear function spaces and their duals

Published online by Cambridge University Press:  09 April 2009

Narcisse Randrianantoanina
Narcisse Randrianantoanina
Affiliation:
Department of Mathematics and Statistics, Miami University, Oxford, OH 45056 e-mail: [email protected]
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Abstract

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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.

Keywords

von-Neumann algebrasconjugate functionsHardy spaces

MSC classification

Secondary: 46E15: Banach spaces of continuous, differentiable or analytic functions 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 3 , December 1998 , pp. 388 - 404
Copyright
Copyright © Australian Mathematical Society 1998

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