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I am splitting off Gelfand duality from Gelfand spectrum. Want to state the actual equivalence theorem here. But just a moment…
…now with statement of the theorem…
…and now the statement of the non-unital version should even be correct ;-). I have also expanded a bit more.
Well is this desired? I mean the non-unitality is geometrically about the noncompact case, i.e. for locally compact Hausdorff. I see quoted some theorem on pointed slice category, this is a known trick (I learned it from a short discussion of Maszczyk with Janelidze in Warszawa), but it is really useful to have rather the standard Gel’fand duality for locally compact Hausdorff spaces with function spaces of functions vanishing at infinity. For that case, one has to be careful with morphisms.
Qiaochu Yuan points out here that Gelfand duality as stated in the nLab here is not correct. I checked out Gelfand duality and the mistake is repeated in the Idea section, but does not reappear in the more formal statements and proofs below.
Particularly, corollary 1 which states a contravariant equivalence between pointed compact Hausdorff spaces and non-unital commutative $C^\ast$-algebras is certainly (and clearly) correct. I’m guessing the mistake was thinking the category of pointed compact Hausdorff spaces is equivalent to the category of locally compact Hausdorff spaces and proper continuous maps. Yes, the one-point compactification does give a functor from $LCH$ (and proper maps) to $\ast/CH$, and this functor is faithful and essentially surjective, but it isn’t full since it doesn’t hit any map that sends more than one point to the basepoint at infinity. Maybe we could identify the category of pointed CH spaces with the category of LCH spaces and partial proper maps with open domain, but I haven’t thought that through to the end.
I don’t know how far this mistake has propagated through the nLab. I’m guessing not too far.
Thanks for the alert, that’s probably my bad. Sorry.
I have removed the wrong statement and added (for the moment) a remark that with due care on the morphisms there is the desired generalization, with a pointer to (for the moment) the note Brandenburg 07.
Should be expanded…
added pointer to Johnstone 82 and added remark that this discusses Gelfand duality with real-valued function algebras.
I am looking for a reference that makes explicit first the general adjunction between topological spaces and (star-)algebras, and then obtains Gelfand duality as the fixed point equivalence of this adjunction. I see lots of chat about this perspective, but what’s a citable reference that spells it out?
added pointer to Negropontis 71. I suppose, historically, that’s the origin of category-theoretic discussion of Gelfand duality?
added pointer to Porst-Tholen 91, section 4-c for Gelfand duality as the center of an adjunction between general (compactly generated Hausdorff) topological spaces and general (topological) complex algebras
8: clicking on doi for Negropontis did not work because the doi has brackets inside what does not work with our markup syntax. So I redone the link with writing html directly. Clicking on doi now works.
I’d also like to mention the old unpublished observation of Joel W. Robbin that the set theoretic version of Gel’fand-Neimark duality holds iff we assume set theory satisfying the axiom of existence of measurable cardinals. Nowdays there are wider generalizations of this result.
Thanks.
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