Banach spaces of continuous functions without norming Markushevich bases

Abstract

We investigate the question whether a scattered compact topological space $K$ such that $C(K)$ 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 $C([0,\omega_1])$ 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 $C(K)$ not to embed in a Banach space with a norming M-basis. Examples of such conditions are that $K$ is a $0$-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 $\omega_2$ are Valdivia.