*Wissenschaftliches Programm* † *Liste der Vortragenden*

Sektion 4
Dienstag, 19.09.2000, 15.00–15.20 Uhr, POT 251

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

Andreas Defant, Carl-von-Ossietzky Universitšt Oldenburg

A fundamental theorem of Menchoff and Rademacher tells that for any orthogonal system (xj) in L2[0, 1] and any scalar sequence a  (- l2 the series

 oo  sum      aj
    ----------xj
j=1 log(j + 1)

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 a1, ..., an

||     | m                |||                   (  n         )
||     | sum    ----aj----   |||                      sum    '    2 1/2
|| smu<pn |     log(j + 1) xj |||2 < c ||(aj)||2||sxu'p||<1     |x (xj)|    .
        j=1                                      j=1

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  sum j = 1 oo xj in L1[0, 1] the series

 oo  sum       1
    ----------xj
j=1 log(j + 1)

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 t.

Joint work with Marius Junge (Urbana)