This is a proposition $2.5.8$ in Brown&Ozawa:
Let $H$ be a subgroup of $G$. There is a canonical inclusion $C^*(H)\subseteq C^*(G)$.
By universality we have a canonical $*$-homomorphism $\pi: C^*(H)\to C^*(G)$. We need to show it is injective.
We may assume $H$ is countable (why is that?). So let $\varphi$ be a faithful state on $H$. By the bijective correspondence between positive linear functionals and positive definite functions on $H$ we may think of $\varphi$ as a positive definite function on $H$. Extend $\varphi$ to a function on $G$ by sending each element of $G-H$ to zero. Denote the extension by $\phi$. Now, they claim it suffices to show that $\phi$ is positive definite. I followed their hints: If $\phi$ is positive definite then it defines a state on $C^*(G)$ (by the bijective correspondence). The GNS representation of $C^*(H)$ with respect to $\varphi$ is $\rho_{\varphi}: C^*(H)\to B(l^2_{\varphi}(H))$ is injective. And let $\rho_{\phi}: C^*(G)\to B(l^2_{\phi}(G))$ be the GNS representation with respect to $\phi$. I have shown we have a natural inclusion $l^2_{\varphi}(H)\subseteq l^2_{\phi}(G)$, and it is left to show that $\rho_{\varphi}$ is a subrepresentation of $\rho_{\phi}\circ \pi$. But, I don't know why it is true...
Thanks