I meet with some problems when I read a book about C*-algebra.
Definition 2.5.10. Let $\phi:\Gamma \rightarrow \mathbb{C}$ be a function ($\Gamma$ is a discrete group here). We define a corresponding linear functional $w_{\phi}: \mathbb{C}(\Gamma)\rightarrow\mathbb{C}$ by $$w_{\phi}(\sum\limits_{t\in\Gamma}\alpha_{t}t)=\sum\limits_{t\in\Gamma}\phi(t)\alpha_{t}.$$
My question is: If the functional $w_{\phi}$ can extend to a state on $C^{*}(\Gamma)$, can we identify $w_{\phi}$ with a vector state in the universal representation $C^{*}(\Gamma)\subset B(H)$(i.e., the direct sum of all GNS representations). Why?
Remark: Recall the $\mathbb{C}(\Gamma)$ is the group ring of $\Gamma$, it is the set of formal sums$$\sum\limits_{s\in \Gamma}a_{s}s$$ where only finitely many of the scalar cofficients $a_{s}\in \mathbb{C}$ are nonzzero.
The full group C*-algebra of $\Gamma$, denoted $C^{*}(\Gamma)$, is the completion of $\mathbb{C}(\Gamma)$ with respect to the norm $$||x||_{u}=sup||\pi(x)||,$$ where the supremum is taken over all *-representations $\pi:\mathbb{C}(\Gamma) \rightarrow B(H)$.
Yes, because when you do GNS your state becomes a vector state. So, if in the direct sum you put all zeroes and the cyclic vector for your $w_\phi$ in its corresponding component, you get a vector that implements your state.