Timeline for Trace Class Functions on locally compact groups
Current License: CC BY-SA 3.0
21 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Mar 13, 2013 at 14:51 | vote | accept | Joël | ||
| Mar 13, 2013 at 14:51 | vote | accept | Joël | ||
| Mar 13, 2013 at 14:51 | |||||
| Mar 13, 2013 at 14:51 | comment | added | Joël | Dear Marc, that's perfectly clear now. Thank you. | |
| Mar 12, 2013 at 21:07 | comment | added | Marc Palm | Okay, I deleted this remark. Sorry. | |
| Mar 12, 2013 at 21:04 | history | edited | Marc Palm | CC BY-SA 3.0 |
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| Mar 12, 2013 at 20:35 | comment | added | Yemon Choi | @Marc: I did not point out any such thing, since it did not seem particularly relevant to the original question. Please do not put words in my mouth | |
| Mar 12, 2013 at 18:03 | history | edited | Marc Palm | CC BY-SA 3.0 |
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| Mar 12, 2013 at 17:39 | comment | added | Marc Palm | I edited the question. Is it clearer now?? | |
| Mar 12, 2013 at 17:37 | history | edited | Marc Palm | CC BY-SA 3.0 |
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| Mar 12, 2013 at 13:16 | comment | added | Joël | (4) Then you quote theorem 6.2.6 of your thesis. I don't understand the proof of that theorem. You refer to Theoreme 18.8.1 of Dixmier, but that theorem, nor that book, doesn't mention your notion of smooth function, nor even of a trace class operator. It is indeed written in a completely different language. How do you do the translation ? | |
| Mar 12, 2013 at 13:07 | comment | added | Joël | (3) Then you suggest that I change my definition of trace class to... what exactly ? I presume you mean that I should say that $f$ is trace class if $\pi(f * f*)$ is trace class for Plancherel-almost all $\pi$. But I am not ready to do that. That changes completely my question. I agree to say that $f$ is of "almost reduced trace class" if $\pi(f)$ is trace class for all $\pi$ in the spectrum except for a set of measure 0 for the Plancherel measure. Since by definition, the support of the Plancherel measure is the reduced spectrum, "almost reduced trace class" implies "reduced trace class". | |
| Mar 12, 2013 at 13:00 | comment | added | Joël | (2) Then what is connection with the question I ask? For $G$ a general locally compact group, is the space of compact support smooth function dense in L^1(G)? Fo such a function $f$ is it true to false that $\pi(f)$ is trace class for every unitary irreducible $\pi$ (it would be enough to say that $\pi(f)$ is Hilbert-Schmidt, by Dixmier-Malliavin, wouldn't it)? If the answers to this questions is yes then the answer to my question is yes in full generality, and I suppose you would have said it. So which one(s) have answer "no"? | |
| Mar 12, 2013 at 12:54 | comment | added | Joël | Dear Marc, thanks for your reference. There are plenty of interesting stuff in your answer, but I really struggle to follow your thoughts. (1) In the first two paragraphs, you say with reference that (a) one can define (Bruhat) a "smooth" function on any locally compact group, in a way which extends the usual notions on Lie groups and p-adic groups, (b) that on any locally compact group a compact support smooth function is a finite sum of convolution products of two such functions (Dixmier-Malliavin). Am I correct? | |
| Mar 12, 2013 at 12:20 | history | edited | Marc Palm | CC BY-SA 3.0 |
added paragraph for reductive groups, corrected minor spelling mistakes
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| Mar 12, 2013 at 10:45 | history | edited | Marc Palm | CC BY-SA 3.0 |
Added references, extension of Dixmier-Malliavin; added 216 characters in body
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| Mar 11, 2013 at 22:45 | comment | added | Marc Palm | My thesis is on my homepage. I give more precise reference with page tomorrow. Dixmier Malliavin holds for general locally compact groups in this manner, because it holds for general Lie groups, not necessarily reductive. | |
| Mar 11, 2013 at 22:36 | comment | added | Joël | For the reference to Duflo-Labesse, you're right, I didn't remember it well and had no access to their paper. So they attribute the lemma to Dixmier-Malliavin, this I remember. And the result is "for any reductive real Lie group and any irreducible representation $\pi$, $\pi(f)$ is trace Class for $f$ smooth with compact support" ? | |
| Mar 11, 2013 at 22:29 | comment | added | Joël | Can you explicit a little bit? You say $G$ can be written as a projective limit of Lie groups ? what kind of group $G$ ? And why ? and where can I find your thesis ? | |
| Mar 11, 2013 at 22:06 | history | edited | Marc Palm | CC BY-SA 3.0 |
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| Mar 11, 2013 at 21:56 | history | answered | Marc Palm | CC BY-SA 3.0 |