Journal of Operator Theory

Volume 48, Issue 2, Fall 2002 pp. 255-272.

Sequences in non-commutative

$L^p$ -spaces

Authors: N. Randrianantoanina
Author institution: Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056, USA

Summary: Let

$\cal M$ be a semi-finite von Neumann algebra equipped with a faithful, normal, semi-finite trace

$\tau$ . We introduce the notion of equi-integrability in non-commutative spaces and show that if a rearrangement invariant quasi-Banach function space

$E$ on the positive semi-axis is

$\alpha$ -convex with constant

$1$ and satisfies a non-trivial lower

$q$ -estimate with constant

$1$ , then the corresponding non-commutative space of measurable operators

$E({\cal M}, \tau)$ has the following property: every bounded sequence in

$E({\cal M}, \tau)$ has a subsequence that splits into an

$E$ -equi-integrable sequence and a sequence with pairwise disjoint projection supports. This result extends the well known Kadec-Pe\l czy\'nski subsequence splitting lemma for Banach lattices to non-commutative spaces. As applications, we prove that for

$1\leq p <\infty$ , every subspace of

$L^p(\cal M, \tau)$ either contains almost isometric copies of

$\ell^p$ or is strongly embedded in

$L^p(\cal M, \tau)$ .

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