Free groups and quasidiagonality

TL;DR

This paper investigates the quasidiagonality of operators associated with free groups, providing quantitative bounds and exploring the structure of related operator spaces, thereby clarifying obstructions to quasidiagonality.

Contribution

It introduces new bounds for the modulus of quasidiagonality for free group generators and connects free group representations to operator space structures, advancing understanding of quasidiagonality obstructions.

Findings

qd({λ_a, λ_b}) ∈ [1/2, √3/2] for free group generators

Proper isometries have modulus of quasidiagonality equal to 1

qd(Ω) can be arbitrarily close to zero for certain unitaries

Abstract

We use free groups to settle a couple questions about the values of the Pimsner-Popa-Voiculescu modulus of quasidiagonality for a set of operators $\Omega$, denoted by qd$(\Omega)$. Along the way we deduce information about the operator space structure of finite dimensional subspaces of $\mathbb{C}[\mathbb{F}_d]\subseteq C^*_{\ell^p}(\mathbb{F}_d)$ where $C^*_{\ell^p}(\mathbb{F}_d)$ is the so-called $\ell^p$-completion of $\mathbb{C}[\mathbb{F}_d].$ Roughly speaking, we use free groups and qd$(\Omega)$ to put a quantitative face on the two known qualitative obstructions to quasidiagonality; absence of an amenable trace or the presence of a proper isometry. The modulus of quasidiagonality for a proper isometry is equal to 1. We show that qd$(\{\lambda_a,\lambda_b\})\in [1/2,\sqrt{3}/2]$ where $a$ and $b$ are free group generators and $\lambda$ is the left regular representation. In another direction, we use certain $\ell^p$ representations of free groups constructed by Pytlik and Szwarc and a recent result of Ruan and Wiersma to show that qd$(\Omega)$ may be positive, yet arbitrarily close to zero when $\Omega$ is a set of unitaries.