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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 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 $\kappa$.

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 $\kappa$.

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.

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 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 $\kappa$.

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 $K_m$.

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.