Decompositions of preduals of JBW and JBW$^*$ algebras

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 $1$-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.