Theorem 44.2.6.label Let $G$ be a group, then $L(G) = LC(G) = R(G)'$.
Proof, [Theorem 1.3.6, AP]. ($\subset$): For any $x, y, z \in G$ and $f \in l^{2}(G; \complex)$,
\[(\lambda(x)\rho(y)f)(z) = (\rho(y)f)(x^{-1}z) = f(x^{-1}zy) = (\lambda(x)f)(zy) = (\rho(y)\lambda(x)f)(z)\]
so $\lambda(G) \subset R(G)'$. By Definition 44.2.5, $L(G) \subset LC(G) \subset R(G)'$.
($\supset$): Let $T \in R(G)'$ and $f = T\delta_{e} \in l^{2}(G; \complex)$, then for each $x \in G$,
\[T\delta_{x} = T\rho(x^{-1})\delta_{e} = \rho(x^{-1})T\delta_{e} = f * \delta_{x}\]
As such, $Tg = f * g$ for all $g \in l^{2}(G; \complex)$, and $T = L_{f}$.$\square$
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