- Is every hyperstonean space a Stone-Čech compactification of a discrete space?
- Is there a closed subset of Stone-Čech boundary that is extremally disconnected?
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$\begingroup$ $\beta\omega-\omega$ is not extremally disconnected. Indeed the Boolean algebra $2^\omega/2^{(\omega)}$ is not complete. $\endgroup$– YCorCommented May 12, 2022 at 8:16
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$\begingroup$ Is every proper closed subset of \beta \omega -\omega not extremally disconnected? $\endgroup$– SherlokCommented May 12, 2022 at 8:58
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$\begingroup$ As for 1, hyperstonean space need not have isolated points. $\endgroup$– Narutaka OZAWACommented May 12, 2022 at 9:02
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$\begingroup$ Thank you very much for your answer. As we know the abelian von Neumann algebra $L^{\infty}(\mathbb{T})=C(X)$. Is $X$ a Stone-Čech compactification of a discrete space? $\endgroup$– SherlokCommented May 12, 2022 at 10:00
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As for Q1, the answer is no. As remarked by Narutaka, the hyperstonean cover of $[0,1]$ does not have isolated points (Corollary 2.22) and all points of a discrete space $\Gamma$ are isolated in $\beta \Gamma$. Hyperstonean spaces can be way more wilder; all relevant information can be found in the book Banach spaces of continuous functions as dual spaces by Dales, Dashiell, Lau, and Strauss.
As for Q2, if you mean $\beta \mathbb N \setminus \mathbb N$, then yes, there are lots of copies of $\beta \mathbb N$ therein.
answered May 12, 2022 at 10:40
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$\begingroup$ Thank you very much for your answer. As we know the abelian von Neumann algebra $L^{\infty}(\mathbb{T})=C(X)$. Is X a Stone-Čech compactification of a discrete space? $\endgroup$– SherlokCommented May 12, 2022 at 10:51
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$\begingroup$ @Sherlok, no, it is not as it does not have isolated points. This is deducible from the linked result. $\endgroup$ Commented May 12, 2022 at 10:53
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$\begingroup$ Thank you very much for your answer. Silov boundary of $H^{\infty}(\mathbb{D})$ is a retraction of a Stone-Čech compactification of a discrete space. Does Silov boundary of $H^{\infty}(\mathbb{D})$ contains isolated points? $\endgroup$– SherlokCommented May 12, 2022 at 11:41