Maximal theorems of Menchoff-Rademacher type in non-commutative Lq-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 L-2 [0,1] and for results due independently to Bennett and Maurey-Nahourn on unconditionally convergent series in L-1 [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. (C) 2003 Elsevier Inc. All rights reserved.