On Markushevich bases in preduals of von Neumann algebras (Q312324)

The paper's main result states that the predual \(\mathcal{M}_*\) of a von Neumann algebra \(\mathcal{M}\) is a complex 1-Plichko space and that, moreover, the predual is WCG (= weakly compactly generated) iff \(\mathcal{M}\) is \(\sigma\)-finite. For a Banach space \(X\) to be 1-Plichko means that it admits a Markushevich basis \((x_\alpha, x_\alpha^*)_{\alpha\in\Gamma}\) in \(X\times X^*\) such that the set \(D=\{x^*\in X^*:\{\alpha\in\Gamma:x^*(x_\alpha)\neq0\}\text{ is countable}\}\) is 1-norming for \(X\); WCG spaces are known to be 1-Plichko. For the proof it is first shown that for a cyclic projection \(p\in\mathcal{M}\) the space \(L_p\mathcal{M}_*\) is weakly compactly generated where \(L_p\) is the operator \(\varphi\mapsto\varphi(p\,\cdot)\) defined on \(\mathcal{M}_*\); then \(\mathcal{M}\) is decomposed into what is called a 1-un\-con\-ditio\-nal sum of \(L_{p_\lambda}\mathcal{M}_*\)'s with pairwise orthogonal \(p_\lambda\); finally, a stability result concerning 1-un\-con\-ditio\-nal sums allows to conclude. By a nice description of \(D\) as a \(^*\)-subalgebra and two-sided ideal of \(\mathcal{M}\), the authors extend the main result from \(\mathcal{M}\) to its self-adjoint part \(\mathcal{M}_{sa}\). A result of Haagerup stating that preduals of von Neumann algebras have the 1-separable complementation property (i.e., every separable subspace is contained in a separable complemented subspace) is recovered as a corollary of the fact that being 1-Plichko is equivalent to admitting a commutative 1-projectional skeleton. It should be noted that meanwhile the authors have generalized their main result to the wider class of JBW\(^*\)-algebras, but with different methods, see [\textit{M. Bohata} et al., J. Math. Anal. Appl. 446, No. 1, 18--37 (2017; Zbl 1360.46012)].