Let $G$ be a group and $l^2(G)$ the Hilbert space on $G$. The complex group algebra $CG$ can be imbedded in $B(l^2(G))$, the set of all bounded linear operators, by left translation. The reduced group $C^*$ algebra $C_r^*(G)$ is the operator norm completion of $CG$. It's clear that $l^1(G)$ lies in $C_r^*(G)$.

Q1. Are there some typical elements in $C_r^*(G)\setminus l^1(G)$?

Q2. For a normed algebra $A$, let $C$ be the completion of the subalgebra $\{ab-ba|a,b\in A\}$ and $T(A)=A/C$. It's not hard to see $T(l^1(G))$ is the $l^1$ completion of $C[conj(G)]$, the set of all finitely supported functions on the conjugacy classes $conj(G)$. How to describle $T(C_r^*(G))$? In particular, if $g$ is an element of infinite order, can $g$ have the same image in the quotient algebra as $0$ or a finite order element in $G$? Thank you very much.