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Kubiś, Wiesław

Banach spaces with projectional skeletons. (English) Zbl 1166.46008

J. Math. Anal. Appl. 350, No. 2, 758-776 (2009).
A partially ordered set \(\Gamma\) is directed if, for every \(\gamma_0,\gamma_1\in \Gamma\), there is \(\delta\in \Gamma\) such that \(\gamma_0\leq\delta\) and \(\gamma_1\leq\delta\). Let \(X\) be a Banach space. A projectional skeleton in \(X\) is a family \(\{P_{\gamma}\}_{\gamma\in\Gamma}\) of bounded projections of \(X\) indexed by a directed partially ordered set \(\Gamma\), satisfying the following conditions: (1) \(X=\bigcup_{\gamma\in\Gamma}P_{\gamma}X\) and each \(P{\gamma}X\) is separable; (2) \(\gamma\leq\delta\Rightarrow P_{\gamma}=P_{\gamma}\circ P_{\delta}=P_{\delta}\circ P_{\gamma}\); (3) if \(\gamma_0\leq\gamma_1\leq\dots\), then \(\delta=\sup_n\gamma_n\) exists in \(\Gamma\) and \(P_{\delta}X=\mathrm{cl}(\bigcup_nP_{\gamma_n}X)\). For \(\mathcal{C}(K)\) spaces, this notion is dual to the retractional skeletons in compact spaces, introduced by W.Kubiś and H.Michalewski [Topology Appl.153, No.14, 2560–2573 (2006; Zbl 1138.54024)].
In the paper under review, a systematic investigation of the projectional skeletons is presented. In particular, every space with a projectional skeleton has a projectional resolution of the identity. The class of Banach spaces with projectional skeletons is strictly larger than the class of spaces with countably norming Markushevich bases (\(M\)-bases), and a Banach space has a commutative projectional skeleton if and only if it has a countably norming \(M\)-basis. An essential method of proof is using elementary substructures in logic. The paper contains open problems.
Reviewers remark. The answer to Question 3 (Does every closed subspace of a Banach space with a projectional skeleton have the separable complementation property (SCP)?) is negative. T.Figiel, W.Johnson and A.Pełczyński [“Some approximation properties of Banach spaces and Banach lattices”, Isr.J.Math.(to appear)] have constructed a Banach space \(\mathcal{C}(K)\) whose dual contains a subspace without the SCP. The dual \(\mathcal{C}(K)^*\) has a countably norming \(M\)-basis [O.Kalenda, J. Math.Anal.Appl.340, 81–101 (2008; Zbl 1138.46013)], hence a projectional skeleton.
Reviewer: Anatolij M. Plichko (Krakow)

Cited in 4 Reviews
Cited in 31 Documents

MSC:

46B26 Nonseparable Banach spaces
54C15 Retraction
03C98 Applications of model theory

Keywords:

projection; projectional skeleton; norming set; projective sequence; Plichko space

Citations:

Zbl 1138.54024; Zbl 1138.46013
PDFBibTeX XMLCite
\textit{W. Kubiś}, J. Math. Anal. Appl. 350, No. 2, 758--776 (2009; Zbl 1166.46008)
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References:

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