There is something that is nagging me, I thought that the operator norm of the group ring inside the $C^*_r(G)$ was the same as it's $l^2$ norm but this seems to not be true, clearly the $l^2$ norm is less than the operator norm by evaluating by the function which is one at the identity and 0 elsewhere. I just realised that this could not be the case for the norms to be the same as the definition of RD group given in Josillaint's paper below would not be interesting.
https://www.jstor.org/stable/2001458?seq=1#metadata_info_tab_contents
could anyone give me an example of why this is not true? I feel like I am missing something.
Suppose $g\in G$ is a non-trivial torsion element, and let $n>1$ be the smallest number such that $g^n=1$. Then the element $p=\frac1n\sum_{k=0}^{n-1}g^k$ of $\mathbb CG$ is a projection (i.e. $p^2=p=p^*$), so $\|p\|_{op}=1$. But we can compute directly that $\|p\|_2=\frac{1}{\sqrt n}$, so the two norms are not equal (in general).
If instead $G$ is torsion-free, and $g\neq1$, then we have $$\|1+g\|_2^2=2<\sqrt 6=\|(1+g^{-1})(1+g)\|_2\leq\|(1+g^{-1})(1+g)\|_{op}=\|1+g\|_{op}^2.$$