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 $C^*(\Gamma)$?

EDIT: According to the comment below: I know that $C^*(\Gamma)$ *always* admits a character therefore it is clear that our proof cannot work in general since it would give that every group is amenable. My question was about what is essential in the above *argument* which prevents us to argue like this for $C^*(\Gamma)$.