Timeline for Urysohn's lemma for Bochner functions?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| May 15, 2022 at 23:28 | comment | added | Yemon Choi | @Andromeda Yes, that's what I meant with the second approach. | |
| May 15, 2022 at 20:48 | vote | accept | Andromeda | ||
| May 15, 2022 at 20:47 | comment | added | Andromeda | @YemonChoi Alright thanks for the further comment! If I understand you well, here are the details to show that $\mathcal{B}^1(G)$ is closed under translation. Let $y \in G$ and $f\in \mathcal{P}^1(G)$. If $f$ corresponds to the measure $\mu \in M(\widehat{G})$, then a calculation shows that $(L_y f)(x) = f(y^{-1}x) = \int \xi(y^{-1}x)d\mu(\xi) = \int \overline{\xi(y)}\xi(x) d\mu(\xi)$ so $L_y f$ corresponds with the measure $\nu$ defined by $d\nu(\xi) = \overline{\xi(y)}d\mu(\xi)$ and thus $L_y f \in \mathcal{B}^1(G)$, as desired. | |
| May 15, 2022 at 20:17 | comment | added | Matthew Daws | Ah, yes, you are correct about the normalisation: so you can build suitable functions for Urysohn, but the natural norms have little control. | |
| May 15, 2022 at 19:15 | comment | added | Yemon Choi | @Andromeda It depends slightly which definition one is using of ${\mathcal B}(G)$. I tend to define it as the space of all possible coefficient functions of all possible unitary representations, in which case translating such a coefficient function just corresponds to shifting one of the defining vectors. In the part of Folland's book you're referring to, G is LCA, and so you can use Bochner's theorem (together with the fact that translation on $G$ corresponds by Fourier transform to a phase shift on the dual group $\widehat{G}$) | |
| May 15, 2022 at 19:12 | comment | added | Yemon Choi | @MatthewDaws That sounds plausible, but I'm not sure about the normalizations: there are known results for the Fourier algebras of ${\mathbb Z}$ or ${\mathbb R}$ which say that a function which is 1 on a large compact interval and zero outside some "tight" neighbourhood must have large Fourier-algebra norm, while in your suggestions we're getting something with Fourier-algebra norm 1, right? | |
| May 15, 2022 at 18:42 | comment | added | Andromeda | @Yemon Choi: Why is $\mathcal{B}^1(G)$ translation invariant? | |
| May 15, 2022 at 18:35 | comment | added | Matthew Daws | If instead of taking $\xi=\eta=|K|^{-1/2} 1_K$, we instead take $\eta = |L|^{-1/2} 1_L$ for a really "small" compact $L$, then the resulting coefficient functional will be identically $1$ on a "large" subset of $K$, and zero off $KL^{-1}$ (or similar). In this way, I think one can prove a result which is close to Urysohn's lemma (which of a locally compact space would normally ask for a function identically 1 on a compact subset, and $0$ on some specified disjoint closed subset). | |
| May 15, 2022 at 17:15 | comment | added | Andromeda | Thanks for your answer! | |
| May 15, 2022 at 17:11 | history | answered | Yemon Choi | CC BY-SA 4.0 |