Proposition 33.1.4.label Let $G$ be a locally compact Abelian group. If $G$ is discrete, then $\wh G$ is compact. If $G$ is compact, then $\wh G$ is discrete.
Proof, [Proposition 4.5, Fol16]. If $G$ is discrete, then $L^{1}(G; \complex)$ is a unital Banach algebra. By Definition 38.7.4, $\wh G \iso \Omega(L^{1}(G; \complex))$ is compact.
If $G$ is compact, then $\one \in L^{1}(G; \complex)$, and $\bracsn{f \in L^\infty(G; \complex)|\int f| > 1/2}$ is weak*-open. By Proposition 33.1.3, for any $\phi \in \wh G \setminus \bracsn{\one}$, $\int \phi = 0$. Therefore $\bracsn{1}$ is open in $\wh G$, and $\wh G$ is discrete.$\square$
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