Timeline for Finite dimensional subrepresentations in a Fréchet space
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Feb 18, 2016 at 8:41 | history | edited | Romain Gicquaud | CC BY-SA 3.0 |
added 1281 characters in body
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| Feb 18, 2016 at 7:55 | comment | added | Romain Gicquaud | Argh... nope... Let me state the original problem. | |
| Feb 17, 2016 at 17:53 | comment | added | Romain Gicquaud | Thanks a lot guys! I finally found a way out of my problem without using the question I was mentioning :). | |
| Feb 17, 2016 at 15:39 | comment | added | paul garrett | Not without some further conditions. E.g., certain of the smooth principal series (not necessarily unitary or unitarizable) of $G=SL_2(\mathbb R)$ have the finite-dimensional irreducibles of $G$ as subrepns, but these are not complemented. There are intertwinings that have these finite-dimensional subs as kernels, yes, and the images are (anti-/) holomorphic discrete series, but there's no direct sum decomposition. (Yes, these repns are Frechet.) | |
| Feb 17, 2016 at 13:58 | comment | added | Yemon Choi | I am leaving this as a comment for now since I don't have time to check details, but I think that if G is any locally compact group acting on $L^\infty(G)$, then the one-dimensional copy of the trivial representation is complemented as a submodule in the sense you describe if and only if $L^\infty(G)$ has a (left or right) invariant mean, i.e. G would have to be amenable. (I'll come back to this later when I have more time, and either delete the comment if I've made a silly mistake, or turn it into a proper answer.) | |
| Feb 17, 2016 at 9:57 | history | asked | Romain Gicquaud | CC BY-SA 3.0 |