Timeline for Can the full and reduced group $C^*$-algebras be "noncanonically" isomorphic?
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11 events
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| Jan 7, 2015 at 20:30 | comment | added | Yemon Choi | For the interest of those reading: in general $C_r^*(G)$ does not even "remember" if $G$ is unimodular or not: see, for instance, the remarks on page 190 of J. Rosenberg's paper The $C^*$-algebras of some real and $p$-adic solvable gropups, Pac. J. M. 65 (1976) | |
| Jan 7, 2015 at 19:50 | comment | added | Ali Taghavi | @YemonChoi thank you very much for your revision of the title of the question. | |
| Jan 7, 2015 at 19:47 | vote | accept | Ali Taghavi | ||
| Jan 7, 2015 at 18:58 | comment | added | Yemon Choi | @Phoenix87 I appreciate your point, and I have read your comments, but I think "This should be no surprise I think, as $C^*$-algebras are quite rigid" is not very solid reasoning. I agree that the statement sounds plausible, but many things sound plausible in mathematics without being true. | |
| Jan 7, 2015 at 18:51 | comment | added | Phoenix87 | @YemonChoi please reread my comment carefully. I've never said they are intuitive, but that it shouldn't be surprising. As for a reference there must be something in Brown-Ozawa (although just for the discrete case), but I can't check as I don't have a copy of it with me right now. There is a mention to this fact at en.wikipedia.org/wiki/…, although there is no reference cited there | |
| Jan 7, 2015 at 18:31 | comment | added | Yemon Choi | @Phoneix87 Moreover, I would argue that not all isomorphism questions about Cstar algebras are "intuitively" clear - I do not know of an "easy" proof that the reduced Cstar algebras of $F_2$ and $F_3$ are non-isomorphic, for instance. Also $C^*(G)\cong C^*(H)$ whenever $G$ and $H$ are finite abelian groups of the same cardinality... | |
| Jan 7, 2015 at 18:29 | comment | added | Yemon Choi | @Phoenix87 Can you provide a reference? (Alain Valette has already given a justification.) | |
| Jan 7, 2015 at 16:53 | comment | added | Phoenix87 | It is known that if $C^*(G)$ and $C^*_r(G)$ are isomorphic (through any -isomorphism) then $G$ is amenable (and vice-versa of course). This should be no surprise I think, as C-algebras are quite rigid because of the C*-identity and hence the existence of a unique C*-norm on a C*-algebra. | |
| Jan 7, 2015 at 16:10 | answer | added | Alain Valette | timeline score: 21 | |
| Jan 7, 2015 at 15:46 | history | edited | Yemon Choi | CC BY-SA 3.0 |
retagged and made title more descriptive
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| Jan 7, 2015 at 15:11 | history | asked | Ali Taghavi | CC BY-SA 3.0 |