Recall that a biorthogonal system $\{(x_i, x^*_i)\colon i\in I\}$ in a Banach space $E$ is a M-basis if $\{x_i\}_{i\in I}$ is linearly dense in $E$ and $\{x_i^*\}_{i\in I}$ separates points. Let me restrict my attention to $C(K)$-spaces only, where $K$ is a compact, scattered space.
(Naive) Question 1: Does every such a space admits a M-basis? (Note that W.B. Johnson observed that $\ell^\infty=C(\beta\omega)$ has no M-basis but $\beta\omega$ is not scattered).
I believe it's not true but I think I cannot produce any counter-example. Of course, when $\alpha$ is an ordinal, then $C([0,\alpha])$ admits a transfinite Schauder basis which one may use to extract a M-basis.
In fact, I am interested in special kinds of M-bases:
Question 2. Suppose that the space $C(K)$ admits a M-basis. Can we extract a M-basis which looks like the canonical Schauder basis in $C([0,\sigma])$ (that is $\{\mathbf{1}_{[0,\alpha]}\colon \alpha\leq \sigma\}$)?
More precisely, can we construct a new M-basis $\{(f_i, \mu_i)\colon i\in I\}$ indexed by some linearly ordered set $(I, <)$ with the property that if $i\leq j $ then $f_j(x) = f_i(x)$ for $x\in \mbox{supp}(f_i )$?
Any references to the papers studying the condition introduced above will be appreciated as well.
