Timeline for Realize a homomorphism $\mathcal{C}(X) \to \mathbb{R}$ as an evaluation
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Mar 6, 2012 at 13:56 | vote | accept | Martin Brandenburg | ||
| Mar 3, 2012 at 17:14 | history | edited | Ralph |
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| Mar 3, 2012 at 1:17 | answer | added | Ralph | timeline score: 5 | |
| Jan 23, 2012 at 19:59 | comment | added | Yemon Choi | @Martin: yes, but I tend to view A) as the restriction to an appropriate subcategory of the unit of the adjunction from topological spaces to compact Hausdorff ones, which is just the Stone-Cech compactification. I agree that my previous comment does not really give an answer to (A) as stated. | |
| Jan 23, 2012 at 17:31 | comment | added | Martin Brandenburg | @Yemon: Isn't this rather B)? | |
| Jan 23, 2012 at 3:28 | comment | added | Yemon Choi | @Martin, this is not quite what you were asking, but one use (for me at least) of the point evaluation mindset is when one considers $C^\ast$-subalgebras of $C_b(X)$ where $X$ is locally compact. In the case where $X$ is a locally compact group, such algebras can arise in harmonic analysis, and then it can be useful to think of them as $C(Y)$ where $Y$ is some kind of compactification of the original group -- roughly like thinking of "evaluate at a point of the Stone-Cech compactification" instead of "take a limit along an ultrafulter". | |
| Jan 22, 2012 at 1:00 | answer | added | Dmitri Pavlov | timeline score: 1 | |
| Jan 21, 2012 at 22:46 | comment | added | Martin Brandenburg | What is unclear about 2) and 3)? I agree with Qfwfq, please post an elaborate version of this as an answer, it sounds interesting. I don't know what it means to integrate with respect to a character. In any case it would be interesting to give a specific G and χ instead of another general Theorem. | |
| Jan 21, 2012 at 13:52 | comment | added | Dmitri Pavlov | @Qfwfq: I have doubts whether my comment can be seen as an answer. In particular, conditions (2) and (3) are rather vague and I am not sure how to interpret them. Let's see what Martin has to say. | |
| Jan 21, 2012 at 13:35 | comment | added | Qfwfq | (@DP, Why not to post as an answer something wich is an answer?) | |
| Jan 21, 2012 at 12:20 | comment | added | Dmitri Pavlov | However, perhaps this example will do. Consider a locally compact abelian group G and some character χ. Integrating with respect to χ gives you a multiplicative functional (with respect to the convolution of functions, not the usual multiplication). On the first glance it is unclear why this functional is evaluation at some point. This is only revealed by the Fourier transform. | |
| Jan 21, 2012 at 12:14 | comment | added | Dmitri Pavlov | The second part asserts that compact regular locales are spatial, in particular the category of compact regular locales is equivalent to the category of compact Hausdorff topological spaces. This statement is equivalent to a weak form of the axiom of choice. Your question belongs to the second part, but the connection to Gelfand-Neumark duality seems to be rather weak. | |
| Jan 21, 2012 at 12:05 | comment | added | Dmitri Pavlov | The classical statement of Gelfand-Neumark duality can be split into two parts. The first part is Gelfand-Neumark duality proper and asserts a contravariant equivalence between the categories of commutative C*-algebras and compact regular locales. It can be proved in any elementary topos, in particular the proof is constructive: ncatlab.org/nlab/show/constructive+Gelfand+duality+theorem. | |
| Jan 21, 2012 at 11:00 | history | asked | Martin Brandenburg | CC BY-SA 3.0 |