2012/04/10 by Alice Guionnet, A. Guionnet, Guionnet, A. +3 · 3 citations
Mathematics · #46L54 #Advanced Operator Algebra Research #FOS: Mathematics #Geometry and complex manifolds #Operator Algebras (math.OA) #Probability (math.PR) #Random Matrices and Applications #math.OA #math.PR #msc:46L54
paper · pdf · doi:10.48550/arxiv.1204.2182
More corrections of typos as suggested by referees and a simplified proof of Lemma 3.4
openalex publication_date 2012/04/10 · arxiv created 2013/10/08 · arxiv updated 2013/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
By solving a free analog of the Monge-Ampère equation, we prove a non-commutative analog of Brenier's monotone transport theorem: if an n-tuple of self-adjoint non-commutative random variables Z1,...,Zn satisfies a regularity condition (its conjugate variables ξ1,...,ξn should be analytic in Z1,...,Zn and ξj should be close to Zj in a certain analytic norm), then there exist invertible non-commutative functions Fj of an n-tuple of semicircular variables S1,...,Sn, so that Zj=Fj(S1,...,Sn). Moreover, Fj can be chosen to be monotone, in the sense that Fj=\mathscrDjg and g is a non-commutative function with a positive definite Hessian. In particular, we can deduce that C*(Z1,...,Zn)≅ C*(S1,...,Sn) and W*(Z1,...,Zn)≅ L(\mathbbF(n)). Thus our condition is a useful way to recognize when an n-tuple of operators generate a free group factor. We obtain as a consequence that the q-deformed free group factors Γq(ℝn) are isomorphic (for sufficiently small q, with bound depending on n) to free group factors. We also partially prove a conjecture of Voiculescu by showing that free Gibbs states which are small perturbations of a semicircle law generate free group factors. Lastly, we show that entrywise monotone transport maps for certain Gibbs measure on matrices are well-approximated by the matricial transport maps given by free monotone transport.