Decompositions of preduals of JBW and JBW$^*$ algebras
Martin Bohata, Jan Hamhalter, Ond\v{r}ej F.K. Kalenda

TL;DR
This paper proves that preduals of JBW and JBW$^*$ algebras are complex and real 1-Plichko spaces, respectively, extending previous results on von Neumann algebras using set-theoretical methods.
Contribution
It establishes the 1-Plichko property for preduals of JBW and JBW$^*$ algebras, generalizing earlier work on von Neumann algebras.
Findings
Preduals of JBW$^*$-algebras are complex 1-Plichko spaces.
Preduals of JBW-algebras are real 1-Plichko spaces.
A set-theoretical method of elementary submodels is used in the proof.
Abstract
We prove that the predual of any JBW-algebra is a complex -Plichko space and the predual of any JBW-algebra is a real -Plichko space. I.e., any such space has a countably -norming Markushevich basis, or, equivalently, a commutative -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.
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