CONJUGATE OPERATORS FOR FINITE MAXIMAL SUBDIAGONAL ALGEBRAS

NARCISSE RANDRIANANTOANINA

Abstract. 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 noncommutative H^{∞}. In particular the conjugation operator is shown to be a bounded linear map from L^{p}(M, τ) into L^{p}(M, τ) for 1 < p < ∞, and to be a continuous map from L^{1}(M, τ) into L^{1,∞}(M, τ). We also obtain that if an operator a is such that |a| log^{+} |a| ∈ L^{1}(M, τ) then its conjugate belongs to L^{1}(M, τ). Finally, we present some partial extensions of the classical Szegö’s theorem to the non-commutative setting.