Let $G$ be a locally compact group and let $ C_r^\ast(G) $ denote its reduced group $C^\ast$-algebra. Many features of a $G$ can be realized from $L^1(G)$ or $C_r^\ast(G)$. For example, $G$ is discrete iff $L^1(G)$ (resp. $C_r^\ast(G)$) is unital, or $G$ is abelian iff $L^1(G)$ (resp. $C_r^\ast(G)$) is commutative.

I was wondering what kind of information about the group $G$ is encoded in $C_r^\ast(G)$ which is not in $L^1(G)$?

Is it correct if we say $C_r^\ast(G)$ is just a $C^\ast$-completion of $L^1(G)$ and in fact we lose some information about $G$ by considering $C_r^\ast(G)$ instead of $L^1(G)$?

conceivablybe some sarcasm here.) – Yemon Choi Mar 24 '13 at 22:18