Preduals of JBW*-triples are 1-Plichko spaces

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 Σ-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 Lindelöf determined if and only if M is σ-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.2010 Mathematics Subject Classification. 17C65, 46L70, 46B26.

Projectional skeletons and Markushevich bases

“…Since α<κ (P α+1 − P α )X is linearly dense, we deduce that P * s x * = x * . (6)⇒ (7) Let (x * α , x α ) α∈Λ be the Markushevich basis provided by (6). It follows immediately that D is a Σ-subspace, thus it is weak * -countably closed by the already proved implication (1)⇒ (2).…”

“…(c) Assertion (6) can be strengthened by requiring that the Markushevich bases in question is moreover strong. The proof can be done by transfinite induction exactly in the same way as the proof of (5)⇒ (6). Indeed, separable spaces admit a strong Markushevich basis by [46] (see also [22,Theorem 1.36]) and this property is preserved in the induction step as remarked in the proof of [22, Theorem 5.1].…”

“…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.…”

“…By the choice of Γ , we have P s P A = P A P s , hence P * A P * s = P * s P A * as well. Therefore, (6) Note that under the assumptions of (4) the system (P s ) s∈Γ is a 1-projectional skeleton on X with induced subspace D. Lemma 2.1(a) applied to the skeleton (P s ) s∈Γ yields that (P s | PAX ) s∈Γ is a 1-projectional skeleton on P A X with induced subspace…”

“…(e) Let (x α , x * α ) α∈Λ be a Markushevich basis with the properties from assertion (6). Then D is a Σ-subspace, as the set M = {x α ; α ∈ Λ} witnesses it.…”