Timeline for Which is the correct ring of functions for a topological space?
Current License: CC BY-SA 2.5
11 events
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| Nov 29, 2009 at 7:16 | comment | added | Jonas Meyer | Well, that shows that there isn't even a Banach algebra norm on C(X) in general, but I just thought of something else. For commutative C*-algebras, the norm of an element is equal to its spectral radius, so the only possible candidate for a C*-norm on C(X) is the sup norm. | |
| Nov 29, 2009 at 7:02 | comment | added | Jonas Meyer | There isn't one. An unbounded element of C(X) has unbounded spectrum, but the spectrum of an element x of a Banach algebra is always contained in the closed disk of radius ||x|| centered at 0. | |
| Nov 29, 2009 at 6:51 | comment | added | Theo Johnson-Freyd | Well, yes, for compact spaces, they are all the same. When $X$ is not compact, what is the C* structure on $C(X)$? Certainly it is not the sup norm. | |
| Nov 29, 2009 at 6:45 | comment | added | Dave Penneys | @Jonas - you have every reason to harp away. somehow my brain is not working clearly right now. i edited again. i think i need to get off of MO right now and get some sleep. hopefully i haven't made any more ridiculous errors. please point them out if you see them. | |
| Nov 29, 2009 at 6:41 | history | edited | Dave Penneys | CC BY-SA 2.5 |
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| Nov 29, 2009 at 6:13 | comment | added | Dave Penneys | @Jonas - thank you. don't know what i was thinking... | |
| Nov 29, 2009 at 6:12 | history | edited | Dave Penneys | CC BY-SA 2.5 |
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| Nov 29, 2009 at 5:40 | comment | added | Greg Kuperberg | This answer is standard and has the ring of truth to it. It suggests that lots of bad things inevitably happen if X is not locally compact. But are there good arguments for that? | |
| Nov 29, 2009 at 4:42 | history | edited | Dave Penneys | CC BY-SA 2.5 |
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| Nov 29, 2009 at 4:35 | history | edited | Dave Penneys | CC BY-SA 2.5 |
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| Nov 29, 2009 at 4:17 | history | answered | Dave Penneys | CC BY-SA 2.5 |