Quasinilpotent elements of group C-star algebras

If $G$ is a discrete torsion-free group, can its (reduced or full) group C-star algebra contain non-zero quasinilpotent elements? I've seen various examples in the group von Neumann algebra setting (usually in the context of finding quasinilpotent generators of II-1 factors) but if I remember correctly these usually don't belong to the reduced group C-star algebra.

(My reason for assuming torsion-free is that if $G$ contains a finite non-abelian subgroup $H$, then the C-star algebra generated by H will contain a non-trivial matrix algebra and hence will contain nilpotent elements.)

• A question on nilpotency: Is there a $C^{*}$ algebra such that each nilpotentce element has all its roots: that is for every nilpotent $a$ and $n\in \mathbb{N}$ there is $b$ with $b^{n}=a$? – Ali Taghavi Nov 11 '14 at 22:06
• @AliTaghavi Surely this should be asked as a separate question? – Yemon Choi Nov 11 '14 at 22:14

"The aim of this note is to prove the following theorem: The group algebra $L^1(G)$ of a locally compact group $G$ has nilpotent elements if and only if $G$ is non abelian."