Timeline for Banach algebra of smooth functions
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 12, 2020 at 18:42 | comment | added | Yemon Choi | @AymanMoussa In fact, Thomas's result tells us that we don't even need to assume continuity of the derivation, but I prefer to avoid appealing to that result since it is very hard and since continuity of the derivation is usually "clear from inspection" in most cases of interest. | |
| Nov 12, 2020 at 18:41 | comment | added | Yemon Choi | @AymanMoussa Yes, as long as you want the derivation to be continuous on A then the Singer-Wermer theorem (Fact 1) forces $A$ to be one-dimensional or the zero algebra. The only reason I stated Corollary 2 in the form I did was because I wasn't sure if you needed the derivation to be continuous, and the easiest way to force this to hold was to require the inclusion $A\hookrightarrow C^\infty$ to be continuous. | |
| Nov 12, 2020 at 10:45 | vote | accept | Ayman Moussa | ||
| Nov 12, 2020 at 10:45 | comment | added | Ayman Moussa | Maxime Breden (who's not able to post for some reason) suggests to apply directly your FACT 1 to the hypothetic algebra A of my post. If instead of $\Delta_x(A)\subset A$ we have $\partial_j(A)\subset A$ then it works (which seems fairly reasonnable), so I accept your answer : thanks ! | |
| Nov 10, 2020 at 21:50 | history | answered | Yemon Choi | CC BY-SA 4.0 |