Projectional skeletons and Markushevich bases

Abstract

We prove that Banach spaces with a $1$-projectional skeleton form a $\mathcal{P}$-class and deduce that any such space admits a strong Markushevich basis. We provide several equivalent characterizations of spaces with a projectional skeleton and of spaces having a commutative one. We further analyze known examples of spaces with a non-commutative projectional skeleton and compare their behavior with the commutative case. Finally, we collect several open problems.

67 pages, references were updated, remark on solution of one questions was added, few minor changes were made

References in corpus (7)

• Decompositions of preduals of JBW and JBW${}^{\ast }$ algebras
• Monotone retractability and retractional skeletons
• On Markushevich bases in preduals of von Neumann algebras
• Separable reduction of Frechet subdifferentiability in Asplund spaces
• Preduals of JBW${}^{\ast }$-triples are 1-Plichko spaces
• Rich families and projectional skeletons in Asplund WCG spaces
• Separable determination in Banach spaces

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• Rotund Gateaux smooth norms which are not locally uniformly rotund
• Banach spaces of continuous functions without norming Markushevich bases
• Weakly Corson compact trees
• On the weak${}^{\ast }$ separability of the space of Lipschitz functions
• Note on almost isometric ideals and local retracts in Banach and metric spaces