Timeline for Is the left-regular representation of a locally compact group a homeomorphism onto its image?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Nov 3, 2022 at 23:27 | comment | added | Yemon Choi | Yes, this is what I had in mind, I was writing in a hurry between meetings. So Nick's answer supplies the details. | |
| Nov 3, 2022 at 12:42 | comment | added | Lau | @YemonChoi Your idea is basically what Nick suggested in his answer, right? | |
| Nov 1, 2022 at 0:02 | answer | added | Nick | timeline score: 3 | |
| Oct 31, 2022 at 13:06 | vote | accept | Lau | ||
| Oct 31, 2022 at 0:36 | history | edited | Lau | CC BY-SA 4.0 |
added 10 characters in body
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| Oct 31, 2022 at 0:35 | comment | added | Lau | Thanks! Yes, you are right. I'll edit it. | |
| Oct 30, 2022 at 23:13 | answer | added | YCor | timeline score: 8 | |
| Oct 30, 2022 at 17:12 | comment | added | YCor | I understand the reduction to such sequences only assuming that $G$ is second countable. In this case, using local compactness, we reduce to proving that $g_n\to\infty$ implies that $\lambda_{g_n}$ does not converge to the identity. But indeed for all $f$ it holds that $\langle\lambda_{g_n}f,f\rangle$ tends to zero (the regular representation is $C^0$. So $\lambda_{g_n}f$ can tend to $f$. The same argument works for arbitrary $G$, using nets instead of sequences, and works for every faithful $C^0$ representation. | |
| Oct 30, 2022 at 15:59 | comment | added | Yemon Choi | I suspect the answer is yes, by taking the contrapositive. (Minor nitpick: in general you should use nets rather than sequences.) That is, suppose $g_\alpha$ is a net which does not converge to $e$; then take a basic WOT-open-nhd ${\mathcal V}\ni I$, use it to define a suitable open neighbourhood $U\ni e$; we know that $g_\alpha\notin U$ infinitely often, and then by building bump functions supported in $U$ using vectors in $L^2(G)$, we should get $\lambda_{g_n} \notin {\mathcal V}$ infinitely often. (We can use WOT, since WOT and SOT agree on the group of unitaries of a Hilbert space.) | |
| Oct 30, 2022 at 14:49 | history | edited | Lau |
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| Oct 30, 2022 at 14:31 | history | asked | Lau | CC BY-SA 4.0 |