Timeline for Is there a relation between Gelfand duality and the spectrum of a ring (with its Zariski topology)?
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23 events
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| Jan 30, 2022 at 11:12 | vote | accept | Gabriel | ||
| Jan 14, 2022 at 13:41 | comment | added | NameNo | For functional analysts: say you are interested in commutative radical Banach algebras deepdyve.com/lp/de-gruyter/… You need more than the maximal spectrum, and the algebraic results apply, but you dislike the non-closed ideals you need. mathoverflow.net/questions/20268/… | |
| Jan 14, 2022 at 9:03 | answer | added | Gabriel | timeline score: 1 | |
| Jan 14, 2022 at 8:41 | history | edited | Gabriel | CC BY-SA 4.0 |
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| Jan 14, 2022 at 0:34 | comment | added | Yemon Choi | @LSpice I have definitely seen variants on this question before, but I am less enthusiastic about this genre than you are, perhaps because I was trained in settings where the Gelfand transform is just an adjunction not a duality, and because the answers to these questions seem to get increasingly abstract without showing analysts like me whether Gelfand theory for CBAs is being superseded or just taken for granted. (No offence intended to the OP; it's a natural question, I just find that the answers lauded by everyone else are not my cup of tea) | |
| Jan 14, 2022 at 0:20 | comment | added | LSpice | This is a good question—so good that, I could swear, it's been asked before. Is there someone who has a sufficiently broader memory of MO arcana than mine to dig it up? | |
| Jan 13, 2022 at 23:51 | answer | added | Dmitri Pavlov | timeline score: 16 | |
| Jan 13, 2022 at 17:09 | comment | added | NameNo | Purely real commutative (identity as involution) C$^*$-algebra are rings (order, real algebra, norm are uniquely defined from the ring). Their structure is also equivalent to (partially ordered abelian gropus with strong unit, which are then vector lattices used in Stone - Weierstrass) their "complexification" $A\oplus iA$ ("polarized" real $C^*$-algebras, where a central antihermitian squre root of $-1$ is fixed; they are a sligthtly more precise structure than complex $C^*$-algebras, See mathoverflow.net/questions/160872/…) | |
| Jan 13, 2022 at 16:12 | comment | added | Yemon Choi | @FernandoMuro I am not sure either, but the definition of Zariski topology always reminds me of the definition in Banach algebra world of the hull-kernel topology. It is known that the hull-kernel topology restricted to the max ideals of a commutative Cstar algebra coincides with the Gelfand topology (there are many "natural" examples of commutative Banach algebras where the restriction of the HK topology is much coarser) | |
| Jan 13, 2022 at 16:09 | comment | added | Yemon Choi | There is an obvious obstacle which needs to be addressed: Thm A involves objects with an involution and Thm B does not. (I insert my usual tired comment that commutative Banach algebras can look nothing like commutative Cstar algebras, and even the semisimple ones can behave differently.) So if you want to get Thm A from Thm B one has to work out what special properties are enjoyed by the underlying ring of a commutative Cstar algebra, and by that stage I feel like one may just as well develop classical Gelfand theory for Banach algebras anyway | |
| Jan 13, 2022 at 14:51 | comment | added | NameNo | If you instead are only interested in commutative rings and in a spectrum between the prime and the maximal ones, look at this mahani.uk.ac.ir/documents/1000262/1220532/… (unfortunately it gives no proofs, but I hope that a google scholar search might help) | |
| Jan 13, 2022 at 13:31 | comment | added | NameNo | If you are searching a common categorical generalization, you might start here: andrew.cmu.edu/user/awodey/preprints/lambek.pdf | |
| Jan 13, 2022 at 13:13 | comment | added | afh | All Zarsiki open subsets of the sepctrum of a C-star algebra are open (complememt of vanishing locus of a continuous function is open). So the Zarsiki topology is coarser. If the spectrum of a unital algebra is compact and Hausdorff, then it is normal. (T4). In that case it seems that you could use the extension theorem to prove the that the Zarsiki topology is also finer. Namely, for any point and neighborhood of it, there is a continuous function that does not banish on the point, but vanishes on the complement of the neighborhood. | |
| Jan 13, 2022 at 11:56 | comment | added | Fernando Muro | The Zariski topology is quasi-compact, so the Gelfand spectrum of a non-unital C-star algebra won't carry the Zariski topology in general. I currently don't have an argument for the unital case, I'm also curious to know the answer, I would be surprised if both topologies agreed. | |
| Jan 13, 2022 at 11:46 | comment | added | Gabriel | @BenjaminSteinberg I knew that but I don't know if its topology coincides with the Zariski topology. | |
| Jan 13, 2022 at 11:16 | comment | added | Paul Taylor | Chapters IV and V of Stone Spaces by Peter Johnstone have material about these things. | |
| Jan 13, 2022 at 10:55 | comment | added | Benjamin Steinberg | The spectrum of a commutative C*-algebra is the space of maximal ideals (max spec). This is only the prime spectrum when the space X os totally disconnected. | |
| Jan 13, 2022 at 10:29 | comment | added | Gabriel | This answer seems to suggest otherwise? mathoverflow.net/a/90810/131975 | |
| Jan 13, 2022 at 10:22 | comment | added | Fernando Muro | The Gelfand spectrum doesn’t carry the Zariski topology, I believe. | |
| Jan 13, 2022 at 10:16 | comment | added | Gabriel | @FernandoMuro I'm sorry but I don't quite understand what you mean. I surely don't think that Thm B can be deduced from Thm A. I'm talking about the other direction. | |
| Jan 13, 2022 at 10:11 | history | edited | YCor |
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| Jan 13, 2022 at 10:11 | comment | added | Fernando Muro | Affine varieties are not locally compact and Hausdorff, nor all rings are commutative C-star algebras, so no, but both results have the same flavour. | |
| Jan 13, 2022 at 10:01 | history | asked | Gabriel | CC BY-SA 4.0 |