Convexity, the spectrum of free Laplacians and geometric consequences: a free KLS bound
Abstract
We establish a compactness theorem for the resolvent of free Laplacians. More precisely, let X=(X1,…,Xn) be a self-adjoint tuple admitting Lipschitz conjugate variables in the sense of Dabrowski (2014). Under the non-commutative curvature--dimension condition CD(c,∞) with c>0, introduced in our previous work (2026), we prove that the free Laplacian Δ:=∂*∂̅ is proper, that is, has compact resolvent. Our argument is non-perturbative and uses a spectral variant of the free Bochner method based on Dabrowski's almost-commutation formulas. A further stochastic argument gives a normal unital trace-preserving embedding of M=W*(X1,…,Xn) into L(𝔽2). Together, the compactness and embedding theorems gives several operator-algebraic consequences, including the Haagerup approximation property, the CMAP, and strong solidity. We also obtain proper proximality and bi-exactness. We finally obtain several geometric consequences. We prove a free analogue of Klartag's improved log--Lichnerowicz inequality (2023), together with rigidity in the equality case. More surprisingly, under CD(0,∞), we prove a dimension-free Kannan--Lovász--Simonovits (KLS) bound in the free setting. As a consequence, we obtain a free thin-shell estimate for isotropic tuples. We conclude with further questions on spectral comparison with the canonical semicircular model.
Related
- Heat content and spectrum for subordinated sub-Laplacians
- Robin Laplacians with point interactions on unbounded domains: discrete spectrum and coupling asymptotics
- A Deterministic Game-Theoretic Approach to Convexity-Preserving Property and Finite-Time Extinction of Level Sets for Geometric Equations
- A geometric theory of visual adaptation and its perceptual consequences
- An exact computer-assisted lower bound for Fisher-information convexity along the two-dimensional heat flow
BibTeX
@misc{diez2026convexity,
title = {{Convexity, the spectrum of free Laplacians and geometric consequences: a free KLS bound}},
author = {Charles-Philippe Diez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22109},
eprint = {2610.22109},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22109},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Convexity, the spectrum of free Laplacians and geometric consequences: a free KLS bound
AU - Charles-Philippe Diez
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22109
UR - https://arxiv.org/abs/2610.22109
AB - We establish a compactness theorem for the resolvent of free Laplacians. More precisely, let X=(X_1,…,X_n) be a self-adjoint tuple admitting Lipschitz conjugate variables in the sense of Dabrowski (2014). Under the non-commutative curvature--dimension condition CD(c,∞) with c>0, introduced in our previous work (2026), we prove that the free Laplacian Δ:=∂^*∂̅ is proper, that is, has compact resolvent. Our argument is non-perturbative and uses a spectral variant of the free Bochner method based on Dabrowski's almost-commutation formulas. A further stochastic argument gives a normal unital trace-preserving embedding of M=W^*(X_1,…,X_n) into L(𝔽_2). Together, the compactness and embedding theorems gives several operator-algebraic consequences, including the Haagerup approximation property, the CMAP, and strong solidity. We also obtain proper proximality and bi-exactness. We finally obtain several geometric consequences. We prove a free analogue of Klartag's improved log--Lichnerowicz inequality (2023), together with rigidity in the equality case. More surprisingly, under CD(0,∞), we prove a dimension-free Kannan--Lovász--Simonovits (KLS) bound in the free setting. As a consequence, we obtain a free thin-shell estimate for isotropic tuples. We conclude with further questions on spectral comparison with the canonical semicircular model.
ER -