Obata's rigidity theorem in free probability
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911489437204480 |
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| author | Diez, Charles-Philippe |
| author_facet | Diez, Charles-Philippe |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_05466 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Obata's rigidity theorem in free probability Diez, Charles-Philippe Operator Algebras Differential Geometry Probability 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. |
| title | Obata's rigidity theorem in free probability |
| topic | Operator Algebras Differential Geometry Probability |
| url | https://arxiv.org/abs/2603.05466 |