Let $\Gamma$ be a discrete group, and consider the reduced group $C_\lambda^* $-algebra $C^* (\Gamma)$. Let $1 \in \ell^\infty(\Gamma)$ be the function which is identically 1 on $\Gamma$.
My question is when does $1 \in C_\lambda^* (G)$, in the sense that there exist $f_n \in C_c(\Gamma)$ such that $f_n \rightarrow 1$ in the reduced norm, that is $$ \|1-f_n\|_\lambda \rightarrow 0. $$
I thought about approximation properties, such as amenability, weak amenability, rapid decay property but none of them speak of convergence in the reduced norm.
Let $f\in C_c(\Gamma).$ Then $$\|f\|_2=\|f*\delta_e\|_2\le \|f\|_\lambda\|\delta_e\|_2=\|f\|_\lambda$$ Hence if $f_n\in C_c(\Gamma)$ is convergent in $C_\lambda(\Gamma)$ then it is convergent in $\ell^2(\Gamma).$ Therefore $f_n\to {\bf 1}$ implies ${\bf 1}\in \ell^2(\Gamma),$ i.e. $\Gamma$ must be finite.