Timeline for What is your favorite isomorphism?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Sep 3, 2010 at 6:45 | comment | added | Jonas Meyer | @Paul: Thank you for the nice addendum to this answer. | |
| Sep 3, 2010 at 6:19 | comment | added | Paul Siegel | That being said, the Gelfand-Naimark theorem really is the key idea that gets Connes' program started and inspires many of the relevant constructions. | |
| Sep 3, 2010 at 6:17 | comment | added | Paul Siegel | The correct statement is indeed that the Gelfand transform, regarded as a functor from the category of commutative C*-algebras to the category of locally compact Hausdorff spaces, is an equivalence of categories. And strictly speaking Connes' noncommutative geometry program is not about "extending the isomorphism" - so far as I know there is no sensible way to extend the Gelfand transform to a functor from the category of all C*-algebras to some other category. Rather, it is about using C*-algebras as proxies for spaces and extending geometric tools to the noncommutative case. | |
| Sep 3, 2010 at 0:04 | comment | added | Tracy Hall | I'm no expert either, of the subject or of its history, and I'm sure the slogan is an overstatement depending on how it is interpreted. I just meant that this deep equivalence at the level of axioms, or isomorphism of categories, is the reason that it even makes sense to speak of "non-commutative" topological spaces, since the algebraic axiomatization of the category of locally compact Hausdorff topological spaces includes an axiom of commutativity that can be relaxed. | |
| Sep 2, 2010 at 23:18 | comment | added | Jonas Meyer | I think this is overstated. First of all, which noncommutative geometry? Even if we stick to what is covered in Connes's book, there's much more to it than what might be called "noncommutative (locally compact) topology". But maybe I miss the point, and I'm no NCG expert. | |
| Sep 2, 2010 at 22:55 | comment | added | Yemon Choi | Hmm. I thought (in my limited understanding) that the real point is that the isomorphism is a natural isomorphism at the level of functors... And I think that the answer given by Tracy makes a nice slogan but is perhaps eliding over some historical details (IMHO) | |
| Sep 2, 2010 at 22:03 | comment | added | Paul Siegel | This might be my favorite example with actual mathematical content. | |
| Sep 2, 2010 at 21:58 | history | answered | Tracy Hall | CC BY-SA 2.5 |