Proof, [Proposition 1.3.9, AP]. (2) $\Rightarrow$ (1): Let $T \in Z(L(G))$, then for each $x \in G$,

so $T\delta_{e}$ is invariant under conjugation, and hence constant on conjugacy classes of $G$.

Since $G$ is ICC and $T\delta_{e} \in l^{2}(G; \complex)$, there exists $\lambda \in \complex$ with $T\delta_{e} = \lambda \delta_{e}$. As such, $T = \lambda I$, and $Z(L(G)) = \complex I$.

$\neg$ (2) $\Rightarrow$ $\neg$ (1): Let $C \subset G$ be a non-trivial finite conjugacy class, then for each $x \in G$, $\delta_{x} * \one_{C} = \one_{xC}= \one_{Cx}= \one_{C} * \delta_{x}$, so $L_{\one_C}$ is a non-trivial element of $Z(L(G))$.$\square$