Theorem 33.1.2.label Let $G$ be a locally compact Abelian group and $\phi \in \wh G$. Under the identification that $\mathbb{T}= \partial B_{\complex}(0, 1)$ and $B(\complex) = \complex$, the representation of $L^{1}(G; \complex)$ determined by $\phi$ takes the form
and the mapping $\wh G \to \Omega(L^{1}(G; \complex))$ defined by $\phi \mapsto \pi_{\phi}$ is a homeomorphism. Moreover, for any $f \in L^{1}(G; \complex)$ with $\pi_{\phi}(f) \ne 0$, $\dpn{x, \phi}{G}= \pi_{\phi}(L_{x}f)/\pi_{\phi}(f)$ for all $x \in G$.
Proof, [Theorem 4.3, Fol16]. (Continuous): Let $\seqf{f_j}\subset L^{1}(G; \complex)$ and $\eps > 0$. By inner regularity of the Haar measure, there exists $K \subset G$ compact such that $\int_{G\setminus K}|f_{j}| < \eps$ for all $1 \le j \le n$. In which case, for any $\phi, \psi \in \wh G$ with $\sup_{x \in K}|\phi(x) - \psi(x)| < \eps$,
(Injective): By (3) of Definition 32.2.1, $\phi$ is uniquely determined by $\pi_{\phi}$.
(Surjective): Let $\pi \in \Omega(L^{1}(G; \complex))$. Since $G$ is Abelian, $L^{1}(G; \complex)$ is also Abelian by Proposition 31.4.2. As such, for any $x \in G$ and $f, g \in L^{1}(G; \complex)$, $f * (L_{x}g) = L_{x}(f * g) = (L_{x}f) * g$ and $\pi(L_{x}f)\pi(g) = \pi(f)\pi(L_{x}g)$. In particular, if $\pi(f), \pi(g) \ne 0$, then $\pi(L_{x}f)/\pi(f) = \pi(L_{x}g)/\pi(g)$. Let[1]
then $\phi$ is defined independently of the choice of $f$. By continuity of translation, $\phi$ is a continuous homomorphism from $G$ to $\complex \setminus \bracsn{0}$. Moreover, by Proposition 38.7.3, $\norm{\pi}_{L^1(G; \complex)^*}\le 1$, and $\phi$ is bounded. Given that $\phi$ is a homomorphism, $\phi(G) \subset \partial B_{\complex}(0, 1)$, and $\phi \in \wh G$.
Since $\pi \ne 0$, there exists $\psi \in L^{1}(G; \complex)$ with $\pi(\psi) = 1$. Using Riemann sums,
(Embedding): By Proposition 38.7.3, every multiplicative linear functional on $L^{1}(G; \complex)$ is contractive. Hence $\Omega(L^{1}(G; \complex))$ is equicontinuous. By the Arzelà-Ascoli Theorem, the topology of compact convergence on $\Omega(L^{1}(G; \complex))$ coincides with the weak* topology.
Let $\phi, \psi \in \wh G$. Since $\phi \ne 0$, there exists $f \in L^{1}(G; \complex)$ such that $\pi_{\phi}(f) \ne 0$. If $|\pi_{\phi}(f) - \pi_{\psi}(f)|$ is sufficiently small, then $\pi_{\psi}(f) \ne 0$ as well. Let $K \subset G$ be compact, then $\bracsn{L_xf|x \in K}$ is compact in $L^{1}(G; \complex)$ by continuity of translation. In which case,
Thus for any $\eps > 0$, $\bracsn{\pi_\psi|\psi \in \wh G, \sup_{x \in K}|\phi(x) - \psi(x)| \le \eps}$ is a neighbourhood of $\pi_{\phi}$ in the weak* topology, and the mapping $\psi \mapsto \pi_{\psi}$ is an embedding.$\square$
- It may be tempting to construct $\phi$ directly as an element of $L^{\infty}$. However, this requires either reducing to the $\sigma$-compact case, or expanding the measurable sets of the Haar measure.keyboard_return
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