Banach spaces of continuous functions without norming Markushevich bases

Abstract

We investigate the question whether a scattered compact topological space K such that $C\left(K\right)$ has a norming Markushevich basis (M-basis, for short) must be Eberlein. This question originates from the recent solution, due to Hájek, Todorčević and the authors, to an open problem from the 1990s, due to Godefroy. Our prime tool consists in proving that $C\left([0,{\omega }_1]\right)$ 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 K for $C\left(K\right)$ not to embed in a Banach space with a norming M-basis. Examples of such conditions are that K is a zero-dimensional compact space with a P-point, or a compact tree of height at least ${\omega }_1+1$. In particular, this allows us to answer the said question in the case when K 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 ω₂ are Valdivia.