Preduals of JBW${}^{\ast }$-triples are 1-Plichko spaces

Abstract

We prove that the predual, $M_{\ast }$, of a JBW${}^{\ast }$-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${}^{\ast }$-algebras. Consequently, dual spaces of JB${}^{\ast }$-triples also are 1-Plichko spaces. We also show that $M_{\ast }$ is weakly Lindelöf determined if and only if $M$ is $\sigma$-finite if and only if $M_{\ast }$ is weakly compactly generated. Moreover, contrary to the proof for JBW${}^{\ast }$-algebras, our proof dispenses with the use of elementary submodels theory.

References in corpus (2)

• Decompositions of preduals of JBW and JBW${}^{\ast }$ algebras
• On Markushevich bases in preduals of von Neumann algebras

Cited by in corpus (6)

• Measures of weak non-compactness in preduals of von Neumann algebras and JBW${}^{\ast }$-triples
• Finite tripotents and finite JBW${}^{\ast }$-triples
• Projectional skeletons and Markushevich bases
• Smooth and polyhedral norms via fundamental biorthogonal systems
• Characterizations of weakly $\mathcal{K}$-analytic and Vašák spaces using projectional skeletons and separable PRI
• Characterization of (semi-)Eberlein compacta using retractional skeletons