Preduals of JBW$^*$-triples are 1-Plichko spaces

TL;DR

This paper proves that the predual of a JBW*-triple is a 1-Plichko space, extending known results and providing a natural description of its $\

Contribution

It establishes that preduals of JBW*-triples are 1-Plichko spaces and describes their $\

Findings

Preduals of JBW*-triples are 1-Plichko spaces.

Dual spaces of JB*-triples are also 1-Plichko.

Characterization of when $M_*$ is weakly Lindelöf determined.

Abstract

We prove that the predual, $M_*$, of a JBW$^*$-triple $M$ is a 1-Plichko space (i.e. it admits a countably 1-norming Markushevich basis or, equivalently, it has a commutative 1-projectional skeleton), and obtain a natural description of the $\Sigma$-subspace of $M$. This generalizes and improves similar results for von Neumann algebras and JBW$^*$-algebras. Consequently, dual spaces of JB$^*$-triples also are 1-Plichko spaces. We also show that $M_*$ is weakly Lindel\"{o}f determined if and only if $M$ is $\sigma$-finite if and only if $M_*$ is weakly compactly generated. Moreover, contrary to the proof for JBW$^*$-algebras, our proof dispenses with the use of elementary submodels theory.