Timeline for Ring of continuous functions, reference request.
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 1, 2011 at 4:31 | vote | accept | Marty | ||
| Apr 29, 2011 at 14:53 | comment | added | Marty | Thank you for getting this started. Perhaps I'll use an upsilon for the first time since Greek class. | |
| Apr 29, 2011 at 7:15 | comment | added | Igor Khavkine | Great to hear that you've found the answer. A tiny comment though is that Gillman and Jerison insist that the realcompactification notation $\upsilon$ is an upsilon and not a v. :-) | |
| Apr 29, 2011 at 6:40 | comment | added | Marty | The proof finishes: A unital continuous ring homom $\phi$ from $C(Y)$ to $C(X)$ determines a unique continuous map $f$ from $X$ to $vY$ such that for all $g \in C(Y)$, and all $x \in X$, $g^v(f(x)) = (\phi(g))(x)$ by Thm 10.8 of Gillman-Jerison (here $g^v$ is the extension of $g$ to $vY$). The map $ev_x \circ \phi$, sending $g \mapsto (\phi(g))(x)$ is continuous since $\phi$ is continuous and evaluation at $x$ is continuous. All such continuous functionals on $C(Y)$ arise from evaluation at some (unique) $y \in Y$, when $Y$ is loc.cpt. (R.E.Edwards,1957,Mathematika). So $f(x) = y \in Y$. | |
| Apr 29, 2011 at 3:14 | comment | added | Marty | Theorem 10.8 (with your comments) goes most of the way towards what I want. Thank you again! I think I'll check in some earlier papers of Hewitt, and see if I can finish off the proof. | |
| Apr 29, 2011 at 0:07 | comment | added | Marty | When I say "rings", I mean unital rings (and unital homomorphisms). I definitely don't want any connectedness hypothesis. But thank you for the Theorem 10.8 reference. I'll check it out this evening. | |
| Apr 29, 2011 at 0:03 | history | answered | Igor Khavkine | CC BY-SA 3.0 |