Let $A = \mathbb C[G]$ be the group ring of all finitely supported functions $f\colon G \to \mathbb C$ of a discrete abelian group $G$ with the usual convolution product, and involution defined by $f^*(a) = \overline{f(a^{-1})}$, so that $A$ is a $*$-algebra. Consider the norm which is given by $\|f\| := \sup_{\rho} \|\rho(f)\|$, where $\rho$ runs through all unital representations $\rho\colon A \to \mathcal B(H)$ on some Hilbert space $H$. The group $C^*$-algebra of $G$, denoted by $C^*(G)$, is the completion of $A$ with respect to this norm.
I'd like to show that $C^*(G)$ can be computed in that $C^*(G) \cong \mathcal C(\hat G)$, where $\hat G$ denotes the character group of all group homomorphisms of $G$ into the circle group $\mathbb T = \{z \in \mathbb C \mid |z| = 1\}$ and $\hat G$ is equipped with the topology of pointwise convergence, which makes it a compact space. On the right, we have the $C^*$-algebra of all continuous functions $\hat G \to \mathbb C$.
Now, each $f \in A$ can be written as $f = \sum_{a\in G} f(a) \delta_a$, and a map $A \to \mathcal C(\hat G)$, $f \mapsto \hat f$, can be defined by setting $\hat f(\phi) = \sum_{a \in G} \phi(a)f(a)$. Is there a way to see (more or less directly) that this extends to a isometric $*$-isomorphism $C^*(G) \to \mathcal C(\hat G)$?