Timeline for Is the Čech–Stone compactification of the integers always a retract of an extremally disconnected space?

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Jan 17, 2022 at 17:08	comment	added	Hua Wang		$\beta \mathbb{N}$ is the spectrum of the abelian $W^*$-algebra $\ell^\infty(\mathbb{N})$, so it is a hyperstonean space, in particular, itself is already stonean, i.e. extremally disconnected and compact. See, e.g. section III.1 of Takesaki's book on operator algebras.
Jan 17, 2022 at 16:43	vote	accept	Tomasz Kania
Jan 17, 2022 at 16:43	comment	added	YCor		Could you define "absolute retract"? I guess it means "is a retract of any space in which it is embedded homeomorphically" but what kind of spaces are allowed? In any case, if $A$ is a free Boolean algebra with a surjective homomorphism onto $2^\omega$, this induces an embedding $\beta\omega\to X$ which is not split (i.e. $2^\omega$ is not a retract in $X$). So $2^\omega$ is not an absolute retract within Stone spaces.
Jan 17, 2022 at 16:42	answer	added	KP Hart		timeline score: 3
Jan 17, 2022 at 16:09	history	asked	Tomasz Kania	CC BY-SA 4.0