A generalization of Macaev's theorem to non-commutative \(L^ p\)-spaces (Q1822311)

For \(1<p<\infty\) the non-commutative \(L^ p\)-spaces associated with a von Neumann algebra are shown to belong to the class UMD (that is, to possess the unconditionality property for martingale differences). With the aid of a recent result of the authors, which permits the classical Hilbert transform to be transferred to UMD spaces, a generalization of Macaev's theorem to non-commutative \(L^ p\)-spaces is introduced. This generalization utilizes the Hilbert kernel in a central role, broadens the ''harmonic conjugation'' aspects of Macaev's theorem, and provides a universal bound depending only on p.