On Markushevich bases in preduals of von Neumann algebras

TL;DR

This paper demonstrates that the predual of any von Neumann algebra possesses a countably 1-norming Markushevich basis, establishing important structural properties and simplifying existing proofs related to their separable complementation.

Contribution

It proves that all preduals of von Neumann algebras are 1-Plichko spaces, extending previous results to the general case and providing new insights into their structure.

Findings

Preduals of all von Neumann algebras are 1-Plichko.

Preduals have a countably 1-norming Markushevich basis.

The self-adjoint part of the predual is also 1-Plichko.

Abstract

We prove that the predual of any von Neumann algebra is $1$-Plichko, i.e., it has a countably $1$-norming Markushevich basis. This answers a question of the third author who proved the same for preduals of semifinite von Neumann algebras. As a corollary we obtain an easier proof of a result of U.~Haagerup that the predual of any von Neumann algebra enjoys the separable complementation property. We further prove that the self-adjoint part of the predual is $1$-Plichko as well.