Talk:Spectrum of a C*-algebra
 | This article is rated B-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
| |||||||||||
"Natural"?
[edit]Doing a disambig run on Natural, and I'm not enough of a mathematician to know if the word "natural" used in the "Examples" section of this article should go to Nature or Natural transformation - or, indeed, somewhere else. Could someone please help? Thanks. Tevildo 04:06, 16 December 2006 (UTC)
- Neither of the articles mentioned, including the one currently linked to, is really relevant. I can't see any non-trivial way that natural transformations (in the category theoretical sense) apply, i.e, other than letting Fbe the identity functor and G be I, and the homeomorphisms are not natural in the way that forests, volcanos and rain are. The homeomorphisms are natural in the sense that they are relatively simple and intuitive, but this is not the sense discussed in the articles mentioned. In light of this, I've unlinked it. James pic 16:34, 10 July 2007 (UTC)
- The association F: X -> X (the identity functor, as you point out) and and G: X -> Prim(C(X)) are covariant functors the category of compact topological spaces into itself. It is a significant and not entirely trivial observation that there is a natural transformation (in fact, a natural isomorphism) from the functor F to the functor G, where natural has exactly the meaning you object to. Note that G is not the identity functor, even though it is naturally isomorphic to it. If this sounds like mumbo jumbo, well maybe that's why it's called abstract nonsense. At least it's better than mambo jimbo--CSTAR (talk) 18:41, 3 February 2008 (UTC)
- I must confess, I'm no expert on category theory. I guess the fact that I is a natural transformation is not as trivial as I'd supposed. Also, just to clarify, when I said G was I, I of course meant I as defined in the article, rather than the identity functor. I am overusing the letter I. All the best. James pic (talk) 18:04, 4 February 2008 (UTC)
- Would you object if I restored the link to natural transformation?--CSTAR (talk) 21:23, 4 February 2008 (UTC)
- Not at all, feel free to restore it, you made your case well, and it sounds like the right thing to do. Also, don't forget WP:OWN and WP:BOLD; you don't need anyone's permission to improve an article, least of all mine! James pic (talk) 11:38, 5 February 2008 (UTC)
- Would you object if I restored the link to natural transformation?--CSTAR (talk) 21:23, 4 February 2008 (UTC)
- I must confess, I'm no expert on category theory. I guess the fact that I is a natural transformation is not as trivial as I'd supposed. Also, just to clarify, when I said G was I, I of course meant I as defined in the article, rather than the identity functor. I am overusing the letter I. All the best. James pic (talk) 18:04, 4 February 2008 (UTC)
- The association F: X -> X (the identity functor, as you point out) and and G: X -> Prim(C(X)) are covariant functors the category of compact topological spaces into itself. It is a significant and not entirely trivial observation that there is a natural transformation (in fact, a natural isomorphism) from the functor F to the functor G, where natural has exactly the meaning you object to. Note that G is not the identity functor, even though it is naturally isomorphic to it. If this sounds like mumbo jumbo, well maybe that's why it's called abstract nonsense. At least it's better than mambo jimbo--CSTAR (talk) 18:41, 3 February 2008 (UTC)
Error in article
[edit]There is an obvious error in this page where it claims that an open continuous map is a homeomorphism. Since the map in question (from the pure state space to the equivalence classes of irreducible representatons) is not injective this is not true. —Preceding unsigned comment added by 91.125.123.56 (talk) 18:19, 3 February 2008 (UTC)
- Yes, obviously it was a very crass error. The error was inserted by an anonymous user in this edit.
- Thanks for noticing and fixing it. This shows either (a) the need for constant vigilance to detect and stamp out these errors by editors or (b) That readers that believe anything written here do so at there own peril. Or perhaps it shows (a) and (b). --CSTAR (talk) 18:26, 3 February 2008 (UTC)
B*-algebras
[edit]There was a sentence warning that a certain result, true for C*-algebras, was not true in general for B*-algebras. However B*-algebras are in fact the same thing as C*-algebras according to Wikipedia and Wolfram mathworld.
Thus I assume the sentence was in error.MathHisSci (talk) 11:43, 9 August 2010 (UTC)