Hilbert Transform Associated with Finite Maximal Subdiagonal Algebras

Let M be a von Neumann algebra with a faithful normal trace τ , and let H∞ be a finite, maximal, subdiagonal algebra of M. Fundamental theorems on conjugate functions for weak∗-Dirichlet algebras are shown to be valid for non-commutative H∞. In particular the Hilbert transform is shown to be a bounded linear map from Lp(M, τ) into Lp(M, τ) for 1 < p < ∞, and to be a continuous map from L1(M, τ) into L1,∞(M, τ). We also obtain that if a positive operator a is such that a log a ∈ L1(M, τ) then its conjugate belongs to L1(M, τ).