Abstract of a paper by Wieslaw Kubis
This is an announcement for the paper "Banach spaces with projectional skeletons" by Wieslaw Kubis. Abstract: 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. Archive classification: math.FA math.GN Mathematics Subject Classification: 46B26; 46B03; 46E15; 54C35 Remarks: 30 pages (including index and toc), submitted The source file(s), projs_survey-ver2e.bbl: 7090 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0802.1109 or http://arXiv.org/abs/0802.1109 or by email in unzipped form by transmitting an empty message with subject line uget 0802.1109 or in gzipped form by using subject line get 0802.1109 to: math@arXiv.org.
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alspach@fourier.math.okstate.edu