Almost everywhere convergence of series in non-commutative L_q-spaces

Andreas Defant, Carl-von-Ossietzky Universität Oldenburg

A fundamental theorem of Menchoff and Rademacher tells that for any orthogonal system (x_j) 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 (x_j) 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, ..., x_n 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 = 1}x_j 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 L_q-spaces built over von Neumann algebras of operators acting on a Hilbert space H, together with a trace .

Joint work with Marius Junge (Urbana)