A metric characterization of freeness

Abstract

Let $\mathcal{M}$ be a finite von Neumann algebra and $u_1,\dots ,u_N$ be unitaries in $\mathcal{M}$. We show that $u_1,\dots ,u_N$ freely generate $L({\mathbb{F}}_N)$ if and only if $${\Vert \sum _{i=1}^N{u}_i\otimes (u_i^{\mathrm{op}}{)}^{\ast }+u_i^{\ast }\otimes u_i^{\mathrm{op}}\Vert }_{\mathcal{M}\overline{\otimes }{\mathcal{M}}^{\mathrm{op}}}=2\sqrt{2N-1}.$$

arXiv comment11 pages