Definition 2.5.6. A function $\phi:\Gamma \rightarrow \mathbb{C}$ is said to be positive definite if the matrix$$[\phi(s^{-1}t)]_{s,t\in F}\in M_{F}(\mathbb{C})$$ is positive for every finite set $F\subset \Gamma$. ($\Gamma$ is a group here.)
$\qquad$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 $\pi: \mathbb{C}[\Gamma]\rightarrow B(H)$.
My question is: If we assume $\Gamma$ is countable. Then $C^{*}(\Gamma)$ has a faithful state $\phi$? Why?
Fix $F=\{s_1,\ldots,s_n\}$. Let $\bar s=(s_1,\ldots,s_n)$. Then, as a positive functional is completely positive, $$ 0\leq\phi^{(n)}(\bar s^*\bar s)=[\phi(s^{-1}t)]_{s,t\in F} $$