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arXiv · 1609.00274

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

Martin Bohata·Jan Hamhalter·Ondrej F. K. Kalenda·Antonio M. Peralta·Hermann Pfitzner
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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.

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Martin Bohata, Jan Hamhalter, Ondrej F. K. Kalenda, Antonio M. Peralta, Hermann Pfitzner. 2016-09-01. Preduals of JBW$^*$-triples are 1-Plichko spaces. https://doi.org/10.1093/qmath/hax057

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