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 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 Lq-spaces over a semifinite von Neumann algebra. The appropriate formulation for non-commutative maximal functions originates in Pisierʹs recent work on non-commutative vector valued Lq-spaces