Complete CCC Boolean algebras (or Stonean spaces)

Question

I am interested in what is known about complete Boolean algebras $B$ with the countable chain condition (ccc), i.e., every disjoint set is countable. Let $K$ be the Stone space of $B$; the completeness of $B$ is equivalent to the fact that $K$ is Stonean (extremally disconnected compact Hausdorff).

Let $K_d$ be the closure of the isolated points of $K$ and denote the complement by $K_c$, so $K = K_d \sqcup K_c$; ccc implies that there are countably many isolated points. On the other hand, [Banach spaces of continuous functions, Theorem 5.1.3] (originally by Dixmier) shows that $K = K_h \sqcup K_a$ where $K_h$ is hyperstonean and $K_a$ has no nontrivial normal measures, and $K_a = K_m \sqcup K_n$ with $K_m$ containing a dense meagre subset and in $K_n$ every meagre subset is nowhere dense and the support of every measure is nowhere dense. (All these notations were made up by me.) Denote the measure algebra of the unit interval $\require{enclose}\enclose{horizontalstrike}{\mathbb{I}}$ (Lebesgue measurable sets modulo null sets) by $\enclose{horizontalstrike}{\Sigma}$. By Maharan's Theorem $\enclose{horizontalstrike}{K_h = K_d \sqcup K_\Sigma}$ where $\enclose{horizontalstrike}{K_\Sigma}$ is either the Stone space of $\enclose{horizontalstrike}{\Sigma}$, or empty. The Boolean algebra corresponding to $K_h$ is classified by Maharan's Theorem (in particular, $K_d \subseteq K_h$). It follows that $$ K = K_h \sqcup K_m \sqcup K_n $$ The unique atomless separable complete Boolean algebra (see [Banach spaces of continuous functions, Theorem 1.7.11] or Description of atomless complete Boolean algebras with a countable $\pi$-base) has a Stone space homemorphic to $\mathbb{G}$, the Gleason cover of the unit interval $\mathbb{I}$, which has no normal measures, so if it exists, it is contained in $K_a = K_m \sqcup K_n$. In which part is it contained? ([Banach spaces of continuous functions, Example 5.1.4] does mention this fact about $\mathbb{G}$ but fails to answer this question.) What else is know about $K_a = K_m \sqcup K_n$ or equivalently, its Boolean algebra counterparts? Is some kind of characterization known?

EDIT: To elaborate on the Gleason part: let $B_c$ be the purely nonatomic part of $B$ and let $A := \{b \in B_c \colon [0,b] \text{ is separable}\}$. By ccc, $\sup A$ is a countable supremum and so $[0, \sup A]$ is also separable and anything disjoint with it is atomic or nonseparable. If $\sup A \not= 0$ then it corresponds to a clopen set in $K$ homeomorphic to $\mathbb{G}$.

(A Boolean algebra $B$ is separable if there exists a countable $D \subseteq B$ with $\forall b \in B \; \exists d \in D \; 0 < d \leq b$.)

Answer 1

A partial answer to the question about the Gleason cover of the unit interval: it can be found everywhere in every compact extremally disconnected space. The point is: in a compact extremally disconnected space the closure of every countably infinite relatively discrete subset is (homeomorphic to) $\beta\mathbb{N}$ and $\beta\mathbb{N}$ contains homeomorphic copies of all compact extremally disconnected spaces of weight $\mathfrak{c}$ (or less). Sketch of proof of the embedding result: the Cantor cube $2^\mathfrak{c}$ is separable, hence there is a continuous surjection $f$ of $\beta\mathbb{N}$ onto the cube. Take a copy in $2^\mathfrak{c}$ of the e.d. space $X$ that you want to embed and apply Zorn's Lemma to get a closed subset $K$ of $\beta\mathbb{N}$ such that the restriction $f\mathbin\upharpoonright K$ maps $K$ irreducibly onto $X$. Because $X$ is e.d. this irredicible map is a homeomorphism. Even nicer embeddings are constructed in this paper

Addendum: the Gleason cover of a separable space is separable, so if it has no isolated points then it has a meager dense subset. So the absolute of the unit interval, if clopen, is must be part of $K_m$.

Second Addendum It seems to me that the description of $K_h$ is a bit too simple. Maharam's theorem gives you not just the algebra $\Sigma$, but a direct sum of homogeneous measure algebras and that would mean that besides $K_\Sigma$ you get Stone spaces of the measure algebras of arbitrary powers of the unit interval as well.

More about this Every topological product of separable topological spaces is CCC (Marczewski Separabilité et multiplication cartesienne des espaces topologiques), so the CCC does not impose any bound on the number of factors $[0,1]$ that you can use: every topological power of $[0,1]$ is CCC. Furthermore, the homogeneous measure algebras in Maharam's theorem are the ones you obtain from the product measure (of Lebesgue measure) on powers of $[0,1]$, one algebra $M_\kappa$ for every cardinal number $\kappa$.