There is a quotation below (C*-Algebras and Finite-Dimensional Approximations):
$ \qquad$We denote the $group~ring$ of $\Gamma$ by $\mathbb{C}[\Gamma]$. By definition, it is the set of formal sums $$\sum_{s\in \Gamma}a_{s}s,$$ where only finitely many of the scalar coefficients $a_{s}\in \mathbb{C}$ are nonzero, and multiplication is defined by
$$\Big(\sum_{s\in \Gamma}a_{s}s\Big)\Big(\sum_{t\in \Gamma}a_{t}t\Big)=\sum_{s, t\in \Gamma}a_{s}a_{t}st.$$ The group ring $\mathbb{C}[\Gamma]$ acquires an involution by declaring $\displaystyle\Big(\sum_{s\in \Gamma}a_{s}s\Big)^*=\sum{s\in \Gamma}\bar{a_{s}}s^{-1}$.
$\qquad$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 $\pi: \mathbb{C}[\Gamma]\rightarrow B(H)$. Evidently we have the following property.
Proposition 2.5.2. Let $u: \Gamma \rightarrow B(H)$ be any unitary representation of $\Gamma$. Then, there is a unique *-homomorphism $\pi_{u}: C^{*}(\Gamma)\rightarrow B(H)$ such that $\pi_{u}(s)=u_{s}$ for all $s\in \Gamma.$
I can not see the "evidently" of the Proposition above. Could someone show me more details?
Suppose that $\pi_1,\pi_2:C^*(\Gamma)\to B(H)$ are $*$-homomorphisms that satisfy $\pi_1(s)=u_s=\pi_2(s)$ for all $s\in \Gamma$. Then $$ \pi_1(\sum_{s\in \Gamma}a_ss)=\sum_sa_2\pi_1(s)=\sum_sa_s\pi_2(s)=\pi_2(\sum_sa_ss). $$ Also, $$ \|\pi_j(x)\|\leq\sup\{\|\pi(x)\|:\ \pi\}=\|x\|_u, $$ so both $\pi_1,\pi_2$ are contractions (in particular, continuous). As they agree on the dense subalgebra $\mathbb C\Gamma$, the continuity guarantees that they agree on all of $C^*(\Gamma)$.