$C^*$-Simplicity of $n$-Periodic Products

The $C^*$-simplicity of $n$-periodic products is proved for a large class of groups. In particular, the $n$-periodic products of any finite or cyclic groups (including the free Burnside groups) are $C^*$-simple. Continuum-many nonisomorphic 3-generated nonsimple $C^*$-simple groups are constructed in each of which the identity $x^n=1$ holds, where $n\ge 1003$ is any odd number. The problem of the existence of $C^*$-simple groups without free subgroups of rank 2 was posed by de la Harpe in 2007.