Non-isomorphism of reduced free group $C^{\ast }$-algebras

Using a new approach involving embedding spaces in II${}_1$ factors with plenty of freely independent Haar unitaries, we prove that $C_r^{\ast }({\mathbb{F}}_n)≆C_r^{\ast }({\mathbb{F}}_m)$ for $n\mathrm{\ne }m$. This recovers the seminal result of Pimsner and Voiculescu with a short new proof.