  We construct first examples of non-trivial groups without non-cyclic free
subgroups whose reduced $C^\ast$-algebra is simple and has unique trace. This
answers a question of de la Harpe. Both torsion and torsion free examples are
provided. In particular, we show that the reduced $C^\ast$-algebra of the free
Burnside group $B(m,n)$ of rank $m\ge 2$ and any sufficiently large odd
exponent $n$ is simple and has unique trace.
