Obata's rigidity theorem in free probability

TL;DR

This paper proves a free probability analogue of Obata's rigidity theorem, showing that under certain conditions, non-zero extremal functions are affine and the associated von Neumann algebra splits off a semicircular component, revealing a rigidity phenomenon in free probability.

Contribution

It extends Obata's rigidity theorem to free probability, demonstrating that extremal elements lead to algebraic splittings involving semicircular components.

Findings

Any non-zero saturator of Voiculescu's free Poincaré inequality is affine in generators.

The von Neumann algebra splits off a maximal amenable semicircular component.

Finite-dimensional eigenspaces of the free Laplacian correspond to free semicircular families.

Abstract

We establish a free analogue of Obata's rigidity theorem. More precisely, Cheng and Zhou (2017) proved that on a weighted Riemannian manifold, the sharp spectral gap (Poincar\'e constant) is achieved only when the space splits isometrically off a one-dimensional Gaussian factor, providing an infinite-dimensional counterpart of Obata's rigidity theorem. We obtain the corresponding phenomenon in free probability, extending it beyond the setting of analytic self-adjoint potentials: Assume a self-adjoint $n$-tuple $X=(X_1,\dots,X_n)$ admits Lipschitz conjugate variables in the sense of Dabrowski (2014). Under a suitable non-commutative curvature-dimension condition, we show that any non-zero saturator of Voiculescu's free Poincar\'e inequality must be an affine function of the generators. Consequently, we deduce that the von Neumann algebra $M=W^*(X_1,\dots,X_n)$ necessarily splits off a freely complemented semicircular component $W^*(Y_1)\simeq L^{\infty}([-2,2],\mu_{\rm sc})$, which is also maximal amenable in $M$. More generally, whenever the first eigenspace of the free Laplacian $\Delta=\partial^*\bar\partial$ is finite-dimensional of rank $r\ge 1$, our rigidity argument shows that these $r$ extremal directions form a free semicircular family, yielding a free product decomposition with an $L(\mathbb{F}_r)$ factor. This provides a free-probability analogue of the classical Gaussian splitting phenomenon and reveals a rigidity mechanism under non-commutative curvature.