M-bases for $C(K)$-spaces, $K$ -scattered

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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.

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1 Answer

I suggest that you read Zizler's article "Nonseparable Banach spaces" in volume 2 of the Handbook of the Geometry of Banach Spaces. Therein he describes the space he calls $JL_0$, constructed by Lindenstrauss and me in "Some remarks on weakly compactly generated spaces," Israel J. Math. 17 (1974), 219-230. $JL_0$ as well as its nonseparable subspaces do not have $M$ bases but $JL_0$ is a $C(K)$ space for some scattered $K$; in fact, $JL_0$ is a twisted sum of $c_0(\Gamma)$ with $c_0$ with $\Gamma$ uncountable.

If I understand Question 2, $c(\aleph)$ (that is, the closed span in $\ell_\infty(\aleph)$ of $c_0(\aleph)$ and the constant functions) seems to be a counterexample as long as $\aleph$ is large enough so that every linear order of $\aleph$ has a well ordered (or reverse well ordered) subset of order type $\omega_2$.

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