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arXiv · 2103.02944

A metric characterization of freeness

Léonard Cadilhac·Benoit Collins
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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 $$\left\|\sum_{i=1}^N u_i \otimes (u_i^{\mathrm{op}})^* + u_i^*\otimes u_i^{\mathrm{op}}\right\|_{\mathcal{M}\overline{\otimes}\mathcal{M}^{\mathrm{op}}} = 2\sqrt{2N - 1}.$$

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Léonard Cadilhac, Benoit Collins. 2021-03-04. A metric characterization of freeness. https://doi.org/10.1016/j.jfa.2022.109562

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