A fundamental theorem of Menchoff and Rademacher tells that for any orthogonal system (xj) in L2[0, 1] and any scalar sequence 2 the series
converges almost everywhere. It was observed by Kantorovicz that this result even holds for every weakly 2-summable sequence (xj) and is moreover a consequence of the following maximal inequality: There is a constant c > 0 such that for every choice of finitely many functions x1, ..., xn L2[0, 1] and scalars 1, ..., n
A sort of border case of convergence theorems of this type is due independently to Bennett and Maurey-Nahoum saying that for each unconditionally convergent series j = 1xj in L1[0, 1] the series
converges almost everywhere – again this result comes from a more general maximal inequality.
The aim of this talk is to discuss analogues of these important theorems for series in non-commutative Lq-spaces built over von Neumann algebras of operators acting on a Hilbert space H, together with a trace .
Joint work with Marius Junge (Urbana)