My problem is the following: Firstly, let $G$ be a locally compact abelian group and denote by $L^{1}(G)$ the Banach $*$-algebra with respect to the convolution of functions as the multiplication and the involution given by $f^{*}(x)=\overline{f(x^{-1})}$. Now, consider as well the group C* algebra of $G$. I don't understand, how to prrove that $\widehat{C^{*}(G)}$ is homeomorphic to $\widehat{L^{1}(G)}$ (where I mean that the structure spaces of $C^{*}(G)$ and $L^{1}(G)$ are homeomorphic). This is the last peace missing for me in the proof that $C^{*}(G)\cong C_{0}(\hat{G})$ ( for $G$ as described in the beginning), hence I would highly appreciate, if anyone could help me.
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