# 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.)

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Kaplansky showed that every non-commutative C*-algebra contains a non-zero nilpotent element. I don't have a reference I'm afraid.

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Ah, so I should have looked harder before asking, I guess. Thanks! –  Yemon Choi Mar 29 '13 at 23:18
For a proof for the above statement see, Page 110 (Proposition II.6.4.14) of Blackadar's book "Operator algebras". –  Vahid Shirbisheh Mar 30 '13 at 13:12
Thanks, Vahid - I will look that up. –  Yemon Choi Mar 30 '13 at 19:16
"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."