Decompositions of preduals of JBW and JBW⁎ algebras

Abstract

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.

Preduals of JBW*-triples are 1-Plichko spaces

“…Our main result reads as follows. The approach in this paper resembles more the one of [4] than the one of [5]. One reason for this has already been mentioned, in the present paper the proofs and arguments do not make use of the set theoretic tool of submodels.…”

“…A generalization to JBW * -algebras appeared to be non-trivial. In [5] the same authors showed that the predual of any JBW * -algebra is 1-Plichko. The proof was quite different from that given in the setting of von Neumann algebras.…”

“…The proof in the Jordan case was based on constructing a special projection skeleton with the help of the set theoretical tool of elementary submodels. Obviously, the question whether, as in the case of von Neumann algebra preduals [4], the result can be obtained without any use of submodels theory is a gap which is not fulfilled by the results in [5].…”

Projectional skeletons and Markushevich bases

“…Indeed, for example is 1‐Plichko for any compact space (see, for example, [, Example 4.10(a)] or [, Theorem 5.5]), but not every space is Asplund. More generally, dual to any ‐algebra is 1‐Plichko by [, Corollary 1.3] (for further generalizations see ). (2) Let be an Asplund space.…”

“…Several equivalences from this theorem are already known. The equivalence (1), (5), and (7) is contained in [16,Theorem 8.3.3 and the following remark]. The equivalence (1)⇔(3) is proved in [13,Theorem 15].…”