A metric characterization of freeness

TL;DR

This paper provides a precise metric criterion involving tensor norms to characterize when a set of unitaries in a finite von Neumann algebra generate a free group factor, linking algebraic freeness to a specific norm condition.

Contribution

It introduces a new metric-based characterization of freeness in finite von Neumann algebras using tensor norms, offering a concrete criterion for freeness.

Findings

Freeness of unitaries characterized by a specific tensor norm equality.

Provides a necessary and sufficient condition for generating free group factors.

Connects algebraic freeness with a computable norm condition.

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