Free nilpotent groups are C*-superrigid

Abstract

The free nilpotent group $G_{m,n}$ of class $m$ and rank $n$ is the free object on $n$ generators in the category of nilpotent groups of class at most $m$. We show that $G_{m,n}$ can be recovered from its reduced group $C^{\ast }$-algebra, in the sense that if $H$ is any group such that $C_r^{\ast }(H)$ is isomorphic to $C_r^{\ast }(G_{m,n})$, then $H$ must be isomorphic to $G_{m,n}$.

5 pages; minor revision; to appear in Proc. Amer. Math. Soc