Maximal theorems of Menchoff-Rademacher type in non-commutative L_q-spaces

Estimates for maximal functions provide the fundamental tool for solving problems on pointwise convergence. This applies in particular for the Menchoff-Rademacher theorem on orthogonal series in L₂[0, 1] and for results due independently to Bennett and Maurey-Nahoum on unconditionally convergent series in L₁ [0, 1]. We prove corresponding maximal inequalities in non-commutative L_q-spaces over a semifinite von Neumann algebra. The appropriate formulation for non-commutative maximal functions originates in Pisier's recent work on non-commutative vector valued L_q-spaces.