Weak Banach-Saks property and Komlos theorem for preduals of JBW$^*$-triples

TL;DR

This paper proves that preduals of JBW*-triples possess the weak Banach-Saks property and satisfy the Komlós property, extending classical results about $L^1$ spaces to a broader non-commutative setting.

Contribution

It establishes the weak Banach-Saks and Komlós properties for preduals of JBW*-triples, linking geometric and probabilistic properties in this non-commutative context.

Findings

Preduals of JBW*-triples have the weak Banach-Saks property.

Preduals of JBW*-triples satisfy the Komlós property.

Subspaces containing uniform copies of $ ext{l}_1^n$ embed $ ext{l}_1$.

Abstract

We show that the predual of a JBW$^*$-triple has the weak Banach-Saks property, that is, reflexive subspaces of a JBW$^*$-triple predual are super-reflexive. We also prove that JBW$^*$-triple preduals satisfy the Koml\'os property (which can be considered an abstract version of the weak law of large numbers). The results rely on two previous papers from which we infer the fact that, like in the classical case of $L^1$, a subspace of a JBW$^*$-triple predual contains $\ell_1$ as soon as it contains uniform copies of $\ell_1^n$.