Almost everywhere convergence of series in non-commutative L_p-spaces
by
Andreas Defant
University of Oldenburg
Coauthors: Marius Junge (Urbana University)

A fundamental theorem of Menchoff and Rademacher tells that for every orthogonal system (x_n) in L₂[0, 1] and any scalar sequence \alpha in l₂ the series

\infty å k=1     \alphak log(k+1)   xk

converges almost everywhere. It was observed by Kantorovic that this result even holds for every weakly 2-summable sequence (x_n) 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 in L₂[0, 1] and scalars \alpha₁, ... , \alpha_n

|| sup m <= n    | m å k=1      \alphak log(k+1)    xk|||2 <= c   ||(\alphak)||2 sup ||x'|| <= 1  æ è n å k=1  |x'(xk)|2 ö ø 1/2    .

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^\infty_k=1 x_k in L₁[0, 1] the series

\infty å k=1     1 log(k+1)   xk

converges almost everywhere - again this result comes from a more general maximal theorem.

The aim of this talk is to discuss analogues of these important theorems for series in non-commutative L_p-spaces built over von Neumann algebras M of operators (acting on a Hilbert space H) together with a trace \tau.

According to results going back to Segal, Nelson, Terp and others such a non-commutative space L_p(M, \tau), 1 <= p <= \infty, can be realized as the Banach space of all unbounded operators x acting on H which are affiliated to M (commute with each operator in the commutant of M) and satisfy ||x||_p : = \tau(|x|^p)^1/p < \infty (here M=L_\infty(M, \tau) and M_*=L₁(M, \tau)). In this object almost everywhere convergence of a sequence (x_n) to 0 means that there is a sequence (p_m) of orthogonal projections in M such that lim_m \tau(1-p_m) = 0 and lim_n ||x_n p_m||_\infty = 0 for all m (Egoroff's theorem!).

We obtain the complete analogue of the Menchoff-Rademacher theorem in L₂(M, \tau), counterexamples to the Bennett-Maurey-Nahoum theorem in non-commutative L₁-spaces, and positive results for this case after a suitable modification of the notion of almost everywhere convergence given here.

The proofs are heavily based on a discrete formulation of the commutative case in terms of absolutely summing operators (originally anticipated by Kwapién's and Pelczynski's work on the ``main triangle projection'') as well as a non-commutative version of the Maurey-Rosenthal factorization theorem (a consequence of Pisier's non-commutative little Grothendieck theorem).

(T)

Andreas Defant

Date received: April 10, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caei-89.