Let G be a locally compact group with Haar measure $\mu$. The left regular representation of G on $L^2(G, \mu)$ is given by $L(g)(f): x \mapsto f(g^{-1}x)$.
If G is abelian, the Plancherel theorem says that the Fourier transform can be extended to an unitary equivalence $L^2(G) \cong L^2(\widehat{G})$, so this implies that L can be associated to a unitary representation $\widehat{L}: G \to U(L^2(\widehat{G}))$ that sends $g \in G$ to $\mathcal{F} \circ L(g) \circ \mathcal{F}^*$, where $\mathcal{F}$ is the unitary extension of the Fourier transform.
How does $\widehat{L}(g)$ act in $L^2(\widehat{G})$ for a given $g \in G$? Besides that, is there any nice decomposition of $L$ or $\widehat{L}$, like the one given by the third part of the Peter-Weyl theorem in the compact case?
Edit: As an extra question, if I'm not mistaken, the theory of commutative Banach algebras implies that $L^1(G)$ is Jacobson semi-simple (because the Fourier transform is injective and it's also the Gelfand transform of $L^1(G)$). Also taking into account that there is a correspondence between unitary representations of G and non-degenerate *-representations of $L^1(G)$, does it follow that unitary representations of G are semi-simple?