Timeline for When the reduced $C^*$-algebra of $\Gamma$ admits character then $\Gamma$ is amenable [closed]
Current License: CC BY-SA 3.0
10 events
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| Feb 13, 2016 at 16:41 | history | closed |
Yemon Choi YCor Wolfgang Andreas Thom Myshkin |
Needs details or clarity | |
| Feb 13, 2016 at 16:14 | comment | added | Yemon Choi | Look: how are you proposing to define a functional on $\ell^\infty(\Gamma)$ if you are given a character on the full Cstar algebra and not on the reduced one? I think your previous comment answers your own question, inasmuch as I can find an actual well-defined question | |
| Feb 13, 2016 at 2:57 | answer | added | user75274 | timeline score: 2 | |
| Feb 13, 2016 at 2:29 | review | Close votes | |||
| Feb 13, 2016 at 16:41 | |||||
| Feb 13, 2016 at 2:27 | comment | added | truebaran | For example something like this: I have an action of $\Gamma$ on $\mathbb{C}\Gamma$ which comes from the group multiplication: this defines a homomorphism from $\mathbb{C}\Gamma$ into itself. We can extend it to the whole $C^*(\Gamma)$: let us call this $\alpha$-so why not to use the same trick with $\alpha(s)f \alpha(s)^*$? But this expression doesn't make sense unless $\alpha(s)$ lives in $B(\ell^2(\Gamma))$. | |
| Feb 13, 2016 at 2:24 | comment | added | Yemon Choi | The argument for the reduced Cstar algebra uses the fact that this is the left regular representation of the group. That is a specific assumption with specific consequences. Having a character on the full group Cstar algebra does not single out special features of the left regular representation, so I personally don't find it at all surprising that the argument for the reduced Cstar case does not extend to an argument for the full Cstar case | |
| Feb 13, 2016 at 2:21 | comment | added | Yemon Choi | Your question, as stated above, asks: "Is it possible to deduce from the above argument ... something about the representation theory of $C^*(\Gamma)$?" The representation theory of $C^*(\Gamma)$ is the same as the unitary representation theory of $\Gamma$; what are you hoping to deduce about this representation theory? | |
| Feb 13, 2016 at 2:18 | history | edited | truebaran | CC BY-SA 3.0 |
added 330 characters in body
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| Feb 13, 2016 at 2:11 | comment | added | Yemon Choi | The augmentation character is always a character on $C^*(\Gamma)$ for any discrete group $\Gamma$, so I do not see what one could deduce merely from knowing the existence of a character on the full $C^*$-algebra | |
| Feb 13, 2016 at 1:31 | history | asked | truebaran | CC BY-SA 3.0 |