Hilbert transform associated with finite maximal subdiagonal algebras

Journal of the Australian Mathematical Society

Narcisse Randrianantoanina¹ 

关键词: 46L50;    46E15;    secondary 43A15;    47D15;   

DOI  :  10.1017/S1446788700035953

学科分类：数学（综合）

来源: Cambridge University Press

【 摘 要 】

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.

【 授权许可】

Unknown