Suppose that $C^*_r(\Gamma)$ admits some character (homomorphism into $\mathbb{C}$)-here $\Gamma$ is discrete group and $C^*_r(\Gamma)$ is the closure of the image of the group ring $\mathbb{C}\Gamma$ under the left regular representation $\lambda$ acting on the Hilbert space $\ell^2(\Gamma)$. Assuming that $C^*_r(\Gamma)$ admits a character $\tau$ one takes this $\tau$ and regard it as a state: as the state it can be extended to the whole of $B(\ell^2(\Gamma))$ (in particular its domain contains $\ell^{\infty}(\Gamma)$). Note that if $s.f$ denotes the action $s.f(t)=f(s^{-1}t)$ for $s \in \Gamma$ and $ f \in \ell^{\infty}(\Gamma)$ then we have (regarded both side as multiplication operators) $\lambda_sf\lambda_s^{*}=s.f$-this can be checked by straightforward computation. Therefore 
$$\tau(s.f)=\tau(\lambda_sf\lambda_s^*)=\tau(\lambda_s)\tau(f)\tau(\lambda_s^*)=|\tau(\lambda_s)|^2\tau(f)=\tau(f).$$
So our group has to be amenable! Also the converse is true: when $\Gamma$ is amenable then $C^*_r(\Gamma)$ admits character. So these conditions are equivalent-nevertheless
I have a problem in understanding in which moment we really have used the assumption that $C^*_r(\Gamma)$ admits a character: more precise  (since obviusly we use this assumption at the beginning of the argument) I'm asking 
> why this argument won't work for $C^*(\Gamma)$-the universal $C^*$-algebra of the group?   

Well, yes, you can say that it is due to the presence of representation $\lambda$ or the fact that we can do the computations in $\ell^2(\Gamma)$, whence for $C^*(\Gamma)$ there is a lack of preferred Hilbert space on which $C^*(\Gamma)$ acts but still I have an impression that in the proof we haven't used our assumption heavily and that it can be somehow improved provided we assume some weaker (for the first sight-but in fact equivalent) condition. 
So in other words: 
>Is it possible to deduce from the above argument _and the knowledge that we cannot improve our proof in order to work with $C^*(\gamma)$_ something about the representation theory of $C^*(\Gamma)$?