Recall 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 $F\subset \Gamma$.
Let $\Gamma$ be a discrete group, $\Lambda\subset\Gamma$ be a subgroup and $\pi: C^{*}(\Lambda)\rightarrow C^{*}(\Gamma)$ be a canonical *-homomorphism. We assume $\Lambda$ is countable. Then $C^{*}(\Lambda)$ has a faithful state $\phi$ which we think of as a positive definite function on $\Lambda$. Now extend $\phi$ to all of $\Gamma$ be defining $\phi(s)=0$, for all $s\not\in \Lambda$.
If the extension is positive definite on $\Gamma$, then $\pi$ is injective? (Hints: Since the GNS representation of $C^{*}(\Lambda)$ with respect to $\phi$ will be a subrepresentation of $\pi$ composed with the GNS representation of $C^{*}(\Gamma)$ with respect to the extension of $\phi$ to $\Gamma$)
Even though there are some hints, I still do not know how to verify this exercise. Could someone help me ?
Recall that one definition of a faithful state is that the associated GNS representation is faithful. So if the extension is positive definite then you can do GNS to get a unitary rep of $\Gamma$ which gives a rep of $C^*(\Gamma)$. Then as the hint says the GNS rep of $C^*(\Lambda)$ will be a subrep $\pi$ composed with the rep of $C^*(\Gamma)$. But this rep is faithful since $\phi$ is faithful. Thus you must have that $\pi$ is faithful (injective).