Decompositions of preduals of JBW and JBW${}^{\ast }$ algebras

Abstract

We prove that the predual of any JBW${}^{\ast }$-algebra is a complex $1$-Plichko space and the predual of any JBW-algebra is a real $1$-Plichko space. I.e., any such space has a countably $1$-norming Markushevich basis, or, equivalently, a commutative $1$-projectional skeleton. This extends recent results of the authors who proved the same for preduals of von Neumann algebras and their self-adjoint parts. However, the more general setting of Jordan algebras turned to be much more complicated. We use in the proof a set-theoretical method of elementary submodels. As a byproduct we obtain a result on amalgamation of projectional skeletons.

Some formulations were corrected or simplified, some proofs were slighlty simplified and some references were added

References in corpus (1)

• On Markushevich bases in preduals of von Neumann algebras

Cited by in corpus (7)

• Measures of weak non-compactness in preduals of von Neumann algebras and JBW${}^{\ast }$-triples
• Preduals of JBW${}^{\ast }$-triples are 1-Plichko spaces
• Finite tripotents and finite JBW${}^{\ast }$-triples
• Projectional skeletons and Markushevich bases
• Characterizations of weakly $\mathcal{K}$-analytic and Vašák spaces using projectional skeletons and separable PRI
• Boundedness of completely additive measures with application to 2-local triple derivations
• On projectional skeletons in Vašák spaces