There is a quotation below:
$\qquad$The $full~group$ C*-algebra of $\Gamma$, denoted $C^{*}(\Gamma)$, is the completion of $\mathbb{C}(\Gamma)$ with respect to the norm $$\|x\|_{u}=\sup\limits_{\large{\pi}}\|\pi(x)\|,$$ where the supremum is taken over all (cyclic) *-representations $\pi: \mathbb{C}[\Gamma]\rightarrow B(H)$. (Note that, since unitaries have norm one, the supremum is finite.)
Well, my question is: in the note, how can we conclude the supremum is finite from unitaries having norm one?
Each $x$ is a $\textit{finite}$ linear combination of elements from $\Gamma$. So since elements of $\Gamma$ get mapped to unitaries you have that
$x=\sum a_gu_g\Rightarrow\|x\|\leq \sum |a_g|\cdot\|u_g\|=\sum|a_g|<\infty $ since the sum if finite.