On Markushevich bases in preduals of von Neumann algebras

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.

13 pages

Cited by in corpus (9)

• Measures of weak non-compactness in preduals of von Neumann algebras and JBW${}^{\ast }$-triples
• Decompositions of preduals of JBW and JBW${}^{\ast }$ algebras
• Preduals of JBW${}^{\ast }$-triples are 1-Plichko spaces
• Rich families and projectional skeletons in Asplund WCG spaces
• Finite tripotents and finite JBW${}^{\ast }$-triples
• Projectional skeletons and Markushevich bases
• Smooth and polyhedral norms via fundamental biorthogonal systems
• Characterizations of weakly $\mathcal{K}$-analytic and Vašák spaces using projectional skeletons and separable PRI
• Characterization of (semi-)Eberlein compacta using retractional skeletons