TITLE: Quasinilpotent elements of group C-star algebras
QUESTION [11 upvotes]: 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.
Can we get such examples for free groups, for instance?
(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.)

REPLY [10 votes]: Prompted by Douglas Somerset's answer to look harder, I've found a paper of Behncke that also gives what I need, according to MathSciNet, by proving a stronger result:
MR0283582 (44 #813)
Behncke, H.
Nilpotent elements in group algebras. (Russian summary)
Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971) 197-198.

"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."

(I haven't yet got to the library to check the proof; in particular, I don't know if it bootstraps from Kaplansky's result as mentioned in the other answer, or mimics the proof in a different setting, or uses something more closely related to the group structure.)