Talk:Gelfand representation

"Natural"

Doing a disambig run on Natural, and I'm not enough of a mathematician to know if the word "natural" used in this article should go to Nature or Natural transformation. Could someone please help? Thanks. Tevildo 01:25, 16 December 2006 (UTC) Natural transformation.--CSTAR 01:44, 16 December 2006 (UTC)

Several inaccuracies

The link to "natural transformation" is incorrect; in this particular example, it would be better to have a link to "functor". But since the word "natural" often simply means "canonical" and does not in any way refer to abstract nonsense, it would IMHO be better to remove the link completely. Secondly, the Gelfand map as described in the article is *not* functorial; the problem is that the zero character is not included in the spectrum. For example when composing a character B --> \C with the zero *-homomorphism A --> B, we always get the zero character A --> \C, which does not lie in Sp(A). A better way to do this is to include the zero character in the spectrum and then have it take values in the category of compact pointed spaces. Then the continuous functions functor C_0 has compact pointed spaces as its domain and maps a space to the algebra of all functions vanishing at the basepoint. This gives an equivalence of categories. - 80.143.109.238 13:46, 28 January 2007 (UTC)

Reply to 1st objection about use of natural.

For purposes of argument, let's stick to commutative C*-algebras with unit (your second objection doesn't apply here). Then the dual object is a (contravariant) functor

$\operatorname {dual} :A\mapsto {\hat {A}}$;

composing the dual object functor with the continuous function functor on the category of compact spaces, gives a covariant functor

$F:A\mapsto C({\hat {A}})$

but there is also a (canonically defined) map The Gelfand map for each commutative A""

$G(A):A\rightarrow C({\hat {A}})$

This map is a natural transformation in the sense that a certain square diagrams are commutative. So in my IMHO I think the use of natural as occurs in the article is correct. Am I missing something in your objection?

Let me think about ypur second objection and I will reply to it soon.--CSTAR

Reply to CSTAR: wow, that was a quick response ! I agree that the Gelfand map -- in the unital case -- is a natural transformation. However, I can't see where this is stated in the article... Also, the natural i was referring to is in the sentence

For any locally compact Hausdorff topological space X, the space C_0(X) of continuous complex-valued functions on X which vanish at infinity is in a natural way a commutative C*-algebra

sorry for being unclear. Where is the natural transformation supposed to be here? Doesn't natural mean something like canonical or functorial in this case? Actually the other occurrences around the Gelfand-Mazur theorem seem more in place to me. - 80.143.74.46 09:57, 29 January 2007 (UTC)

Reply. Perhaps the link to natural transformation should occur in the discussion of the Gelfand map. Nevertheless, I think it is sill possible to regard the dual object functor as being "natural" - in the sense of natural transformation--as opposed to just canonical. I have not done enough functoring around in my lifetime to explicitly write down how this can be done, but I have done enough to think to think it should be possible.--CSTAR 16:09, 29 January 2007 (UTC)

Reply to second point. It is clear that on the category of locally compact spaces and continuous, maps there is no "naturally defined" (ahem) functor into the category of C*-algebras and morphisms. We would have to consider proper continuous maps. In the converse direction, to get a functor from C*-algebras to spaces, we would have to at the very least consider morphisms on C*-algebras that don't map into a proper ideal. This of course is true for unit-preserving maps, since the unit is contained in no proper ideal.--CSTAR 19:51, 31 January 2007 (UTC)

Reply to both points: I see what you mean. In fact, now I know that there are two possibilities to extend the duality between compact spaces and unital algebras to the non-unital case. The first one is the one you described; the *-homomorphisms have to be proper in the sense that they map approximate units to approximate units; see http://planetmath.org/encyclopedia/GelfandNaimarkTheorem.html or http://www.science.uva.nl/~npl/CK.pdf . The second possibility is a duality between compact pointed spaces with continuous maps and C*-algebras with all *-homomorphisms. Without the literature at hand right now, I couldn't find a better reference than Theorem C.1.10 in http://wwwmath1.uni-muenster.de/sfb/about/publ/heft289.ps which just states what I said. I guess it's a matter of taste which one to include in the article. As for the naturality discussion, i couldn't figure out as yet what you mean by the "dual object functor" being "natural"...

Reply:ReAs for the naturality discussion, i couldn't figure out as yet what you mean by the "dual object functor" being "natural"... I am supposing it can be defined by using general categorical constructions such as adjoint functors and so on -- although don't ask me how, I don't have too much desire to figue it out.--CSTAR 15:57, 1 February 2007 (UTC)

First the general then the specifc?

I think it would be better the most general definition we want (we could define it for a general complex algebra without reference to a topology but I think commutative Banach algebra is plenty) and THEN restrict to the special case of commutative C*-algebras. As it stands the more general section is rather vague saying "we could have worked in more generality". Does anyone not approve of this? A Geek Tragedy 21:46, 26 May 2007 (UTC)

Reply to AGT: I agree, and since no one has voiced objections I've made a first attempt at giving the general CBA case first and only then moving on to the particular C*-case. I've put in some links/references to Wiener's lemma, and the Fourier and Laplace transforms, to try and give some context/motivation. More editing for style/typos is no doubt needed.

I think it might also be of interest/use to readers to say a little about functorial aspects of the Gelfand transform, does anyone else want to have a go at this? NowhereDense (talk) 12:44, 27 September 2008 (UTC)

Is Φ_A compact for any unital algebra A?

Let A = ℓ^∞. Then $\Phi _{A}$ is contains an infinite discrete space (of natural numbers), which is not compact. ℓ^∞ has a unit, but is not separable, though. Incnis Mrsi (talk) 15:45, 25 October 2009 (UTC)

The answer to your question is yes. In the example you give $\Phi _{A}$ is not the set N of natural numbers, but its Stone–Čech compactification (which, as the name suggest, is compact). In general one has to be careful to check if, aside from the 'obvious' characters on a Banach algebra, there might be more 'exotic' ones NowhereDense (talk) 06:10, 26 October 2009 (UTC)

As I realized, you claim existence of such characters which are zero on all finite sequences but are not zero on ℓ^∞. IMHO this topic should be described in the article. Incnis Mrsi (talk) 08:52, 26 October 2009 (UTC)

As the article on Stone–Čech points out, you need the axiom of choice to show the compactification exists. You need to assume AC or possibly some slightly weaker form to assert existence of those characters, I guess. If the relation with the compactification is of interest (I'd say it is) it can be included in the article as an example. I don't see much point in quibbling about whether one assumes AC in this branch of functional analysis, since it is entirely standard to use it. Charles Matthews (talk) 11:30, 26 October 2009 (UTC)

Inconsistent notation

The text uses Φ_A to refer to the set of characters. But in math mode, in the equation defining \widehat{a}, it uses the notation \Phi_A, which generates a different glyph. I found this very confusing, because I didn't realize it was refering to the same object. I could not figure out how to generate the Φ character within a math mode expression. If someone who knows how could make the notation consistent here, it would be an improvement. Andylatto (talk) 15:41, 23 September 2020 (UTC)