In the following paper https://www.jstor.org/stable/2001458?seq=1#metadata_info_tab_contents Jolissaint introduces the property RD of a group G if the space $H_L^\infty(G)$ is contained in the reduced $C^*$-algebra of $G$ for some length function $L$ (where $H_L^\infty(G)$ are the functions on $\psi$ G such that the $L^2$ norm of $\psi(1+L)^s$ is bounded for every $s\in \mathbb{R}$) Then he claims that this happens if and only if for every $\phi$ in the group algebra of $G$ the following inequality holds for some $s,c,L$ $$||\phi||\leq c||\phi||_{2,s,L}$$ where the norm $||.||$ is the reduced norm and the norm $||.||_{2,s,L}$ is the $L^2$ norm of $\phi(1+L)^s$.
I don't see why this should hold the group algebra is the same as the compactly supported functions over $G$ which is dense in both $H_L^\infty(G)$ (Well this is not really a Banach space but I mean dense in each $H_L^s(G)$) and $C^*_r(G)$ and the equation between the norms just seems to say that there is a surjection from $H_L^s(G)\rightarrow C^*_r(G)$ for some $s$ and not an inclusion as asked for. Jolissaint appeals to the closed graph theorem but I don't understand where it comes from as you're applying it to the group algebra which is not a Banach space in those norms. Could anyone please explain what is meant here?
Basically I guess if you have the norm bound every cauchy sequence in $H_L^s$ is cauchy in $C_r^*$ and hence you can define a map from $H_L^s\rightarrow C_r^*$ but why is this map injective when seen from $H_L^\infty$
Moreover in the opposite direction I am not sure why an inclusion of $H^\infty_L(G)\rightarrow C_r^*$ will imply such a norm bound.