Timeline for Two Definitions of "Character" of topological groups
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jan 19, 2012 at 22:53 | comment | added | Georges Elencwajg | Dear Hiro: if I can help you, I'll certainly do it with pleasure. | |
| Jan 19, 2012 at 22:16 | comment | added | Hiro | Dear Georges, thank you for your immediate response. I greatly appreciate your kindness. Please lend your hand next time, too! | |
| Jan 19, 2012 at 22:05 | comment | added | Georges Elencwajg | Dear Hiro, everything you write is correct. The sad truth is that, much as I would like to help you, I don't know how to prove the statement you ask about: that's why I wrote "Xandi explains..." I hope he will answer the question you ask in your comment to him. | |
| Jan 19, 2012 at 21:47 | comment | added | Hiro | Dear @ Georges Elencwajg: Thank you for your answer. Since any proper closed subgroup of the torus T is a finite discrete cyclic subgroup, what I have to show is the following statement: "for any profinite group G and any continuous homomorphism f:G -> T, the image of f is a proper subgroup of T". This statement cannot be true without the hypotheses that G is profinite, since we have a counter example G=T. However, I do not know how to use the profinite assumption. Will you please tell me how to show the statement above? | |
| Jan 19, 2012 at 21:12 | comment | added | Georges Elencwajg | Dear @ Xandi: 1) Neither Hiro nor I ever mention local compactness. So it is true that my $G$ in the second paragraph is not locally compact, but this does not affect my answer . 2) It is of course true that a character on a compact group does not have discrete image. I meant a character with values in $\mathbb Q/\mathbb Z$, so that the image has two induced topologies which can be compared. I modified my badly written last paragraph thanks to your comment: thanks a lot. | |
| Jan 19, 2012 at 20:59 | history | edited | Georges Elencwajg | CC BY-SA 3.0 |
added reference to Xandi
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| Jan 19, 2012 at 20:38 | history | edited | Georges Elencwajg | CC BY-SA 3.0 |
added 41 characters in body
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| Jan 19, 2012 at 18:49 | comment | added | Xandi Tuni | N.b. The group $G$ in the 2nd paragraph is not locally compact. @georges: If $G$ is compact, its image under a character isn't necessarily discrete in the circle... | |
| Jan 19, 2012 at 16:48 | history | edited | Georges Elencwajg | CC BY-SA 3.0 |
deleted 22 characters in body
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| Jan 19, 2012 at 16:39 | history | answered | Georges Elencwajg | CC BY-SA 3.0 |