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.)



Quasinilpotent elements of group C-star algebras
Kaplansky showed that every non-commutative C\*-algebra contains a non-zero nilpotent element. I don't have a reference I'm afraid.


