A projectional skeleton in a Banach space is a sigma-directed family of
projections onto separable subspaces, covering the entire space. The class of
Banach spaces with projectional skeletons is strictly larger than the class of
Plichko spaces (i.e. Banach spaces with a countably norming Markushevich
basis). We show that every space with a projectional skeleton has a
projectional resolution of the identity and has a norming space with similar
properties to Sigma-spaces. We characterize the existence of a projectional
skeleton in terms of elementary substructures, providing simple proofs of known
results concerning weakly compactly generated spaces and Plichko spaces.
We prove a preservation result for Plichko Banach spaces, involving
transfinite sequences of projections. As a corollary, we show that a Banach
space is Plichko if and only if it has a commutative projectional skeleton.Comment: 32 pages (including index and toc); revised (added example, comments,
references
We develop a category-theoretic framework for universal homogeneous objects, with some applications in the theory of Banach spaces, linear orderings, and in the topology of compact Hausdorff spaces.MSC (2010) Primary: 18A22, 18A35. Secondary: 03C50, 46B04, 46B26, 54C15.
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