132 research outputs found
Banach spaces of continuous functions without norming Markushevich bases
We investigate the question whether a scattered compact topological space
such that has a norming Markushevich basis (M-basis, for short) must be
Eberlein. This question originates from the recent solution, due to H\'ajek,
Todor\v{c}evi\'c, and the authors, to an open problem from the Nineties, due to
Godefroy. Our prime tool consists in proving that does not
embed in a Banach space with a norming M-basis, thereby generalising a result
due to Alexandrov and Plichko. Subsequently, we give sufficient conditions on a
compact for not to embed in a Banach space with a norming M-basis.
Examples of such conditions are that is a -dimensional compact space
with a P-point, or a compact tree of height at least . In
particular, this allows us to answer the said question in the case when is
a tree and to obtain a rather general result for Valdivia compacta. Finally, we
give some structural results for scattered compact trees; in particular, we
prove that scattered trees of height less than are Valdivia
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