Timeline for Function spaces over pseudocompact spaces
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 4, 2012 at 13:14 | answer | added | KP Hart | timeline score: 2 | |
| Jul 31, 2012 at 3:24 | comment | added | MTS | RadekM: In my first comment I was referring to the morphism of algebras that you have denoted $\beta : C(K) \to C(\beta K)$. I think that map should go in the other direction. Of course, if it is an isomorphism then there is an inverse, so you may equally well talk about the inverse map (as you have done). But the functorial way of defining it goes the other way. The relationship between spaces and algebras is contravariant. Also: yes, $\beta$ is a $\ast$-homomorphism, assuming that by $\ast$ you mean complex conjugation of functions. | |
| Jul 31, 2012 at 3:10 | comment | added | MTS | @Ralph: That's what I meant. I was just regarding $K$ as a subset of $\beta K$, but you're right: it is more properly regarded as a subobject with an inclusion map $i : K \to \beta K$. Then the morphism of algebras $f \mapsto f \circ i$ morally the map given by restriction of functions on $\beta K$ to functions on $K$. | |
| Jul 31, 2012 at 0:20 | comment | added | Joseph Van Name | The quotient algebra $A$ may be infinite dimensional. Consider the space $\omega_{1}\times[0,1]$. The space $\omega_{1}\times[0,1]$ is pseudocompact since every function on this space is eventually constant. In this case, we have $f\in C_{0}(X)$ iff $supp(f)\subseteq\alpha\times[0,1]$ for some ordinal $\alpha$ since every continuous function is eventually constant. Therefore, we have $f+C_{0}(X)=g+C_{0}(X)$ iff $f-g$ is eventually zero. Therefore, the quotient $f+C_{0}(X)$ only considers the tail of the function $f$. Therefore, the algebra $A$ is isomorphic to $C([0,1])$. | |
| Jul 30, 2012 at 21:46 | comment | added | Ralph | @MTS: The other direction $C(\beta K) \to C(K),\; f \mapsto f \circ i: K \to \beta K \to \mathbb{C}$ works as well. | |
| Jul 30, 2012 at 21:26 | comment | added | RadekM | Sure. Corrected. | |
| Jul 30, 2012 at 21:26 | history | edited | RadekM | CC BY-SA 3.0 |
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| Jul 30, 2012 at 21:08 | comment | added | MTS | The space $K$ includes into its Stone-Cech compactification, so your map between algebras should go the other direction (a function on $\beta K$ restricts to a function on $K$). | |
| Jul 30, 2012 at 20:44 | history | edited | RadekM | CC BY-SA 3.0 |
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| Jul 30, 2012 at 20:38 | history | asked | RadekM | CC BY-SA 3.0 |