There is a quotation below (C*-Algebras and Finite-Dimensional Approximations):
$ \qquad$For a discrete group $\Gamma$ we let $\lambda:\Gamma\rightarrow B(l^{2}(\Gamma))$ denote the left regular representation: $\lambda_{s}(\delta_{t})=\delta_{st}~$ for all $s, t\in \Gamma$, where $\{\delta_{t}: t\in \Gamma\}\subset l^{2}(\Gamma)$ is the canonical orthonormal basis.
$ \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 $\displaystyle\Big(\sum_{s\in \Gamma}a_{s}s\Big)^*=\sum{s\in \Gamma}\bar{a_{s}}s^{-1}$. Note that the left regular representation can be extended to an injective *-homomorphism $\mathbb{C}[\Gamma]\rightarrow \mathbb{B}(l^{2}(\Gamma))$, which we also denote by $\lambda$. Evidently, there is a one-to-one correspondence between unitary representations of $\Gamma$ and *-representations of $\mathbb{C}[\Gamma]$.
My question is : Could someone explain the "Evidently" above to me ? Thanks.
If $H$ is a Hilbert space, given a unitary representation $\gamma:\Gamma\rightarrow\mathcal{U}(H)$, how do you induce a (at first, simply linear) function $\widetilde{\gamma}:\mathbb{C}(\Gamma)\rightarrow\mathcal{B}(H)$? What about the converse: if $c:\mathbb{C}(\Gamma)\rightarrow\mathcal{B}(H)$ is a (unital) *-representation, how do you obtain a unitary representation $\overline{c}:\Gamma\rightarrow\mathcal{U}(H)$? (Hint: For $s\in \Gamma$, we have $s=1s\in\mathbb{C}(\Gamma)$)
Answer: