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Obata's rigidity theorem in free probability

Charles-Philippe Diez

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 (Poincaré 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 Poincaré 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 $W^*(Y_1)\simeq L^{\infty}([-2,2],μ_{\rm sc})$, which is also maximal amenable in $M$. More generally, whenever the first eigenspace of the free Laplacian $Δ=\partial^*\bar\partial$ 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.

Obata's rigidity theorem in free probability

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 (Poincaré 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 -tuple 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 Poincaré inequality must be an affine function of the generators. Consequently, we deduce that the von Neumann algebra necessarily splits off a freely complemented semicircular component , which is also maximal amenable in . More generally, whenever the first eigenspace of the free Laplacian is finite-dimensional of rank , our rigidity argument shows that these extremal directions form a free semicircular family, yielding a free product decomposition with an factor. This provides a free-probability analogue of the classical Gaussian splitting phenomenon and reveals a rigidity mechanism under non-commutative curvature.
Paper Structure (15 sections, 27 theorems, 195 equations)

This paper contains 15 sections, 27 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Classical Rigidity and Stability in Spectral Geometry
  3. Free Probability Framework
  4. Free Poincaré Inequalities
  5. Outline of our Results
  6. Preliminaries
  7. Tracial von Neumann algebras
  8. Free difference quotients and conjugate variables
  9. Free Poincaré Inequalities
  10. The Gibbs Case and Lipschitz Conjugate Variables
  11. A free Obata's rigidity principle
  12. Open questions
  13. Compactness of the resolvent and discrete spectrum
  14. Free Ricci curvature and almost co-associativity
  15. Acknowledgments

Key Result

Theorem 1.1

Let $(M,g,\mu)$ be a smooth weighted Riemannian manifold satisfying $CD(1,\infty)$. If there exists a centered non-zero $f\in W^{1,2}(M,\mu)$ such that then $(M,g,\mu)$ splits isometrically and measure-theoretically as where $\gamma$ is the standard Gaussian measure on $\mathbb{R}$.

Theorems & Definitions (62)

  • Theorem 1.1: Cheng--Zhou ChengZhou; Gigli--Ketterer--Kuwada--Ohta GKKO
  • Definition 2.1: Partial free difference quotients
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Proposition 2.8: Cipriani-Sauvageaot SauvageCiprianiSauvageot2003, PetersonPeterson, Dabrowski Dab10
  • Proposition 2.9
  • ...and 52 more