Free Monotone Transport
Abstract
By solving a free analog of the MongeAmpère equation, we prove a noncommutative analog of Brenier's monotone transport theorem: if an $n$tuple of selfadjoint noncommutative random variables $Z_{1},...,Z_{n}$ satisfies a regularity condition (its conjugate variables $\xi_{1},...,\xi_{n}$ should be analytic in $Z_{1},...,Z_{n}$ and $\xi_{j}$ should be close to $Z_{j}$ in a certain analytic norm), then there exist invertible noncommutative functions $F_{j}$ of an $n$tuple of semicircular variables $S_{1},...,S_{n}$, so that $Z_{j}=F_{j}(S_{1},...,S_{n})$. Moreover, $F_{j}$ can be chosen to be monotone, in the sense that $F_{j}=\mathscr{D}_{j}g$ and $g$ is a noncommutative function with a positive definite Hessian. In particular, we can deduce that $C^{*}(Z_{1},...,Z_{n})\cong C^{*}(S_{1},...,S_{n})$ and $W^{*}(Z_{1},...,Z_{n})\cong L(\mathbb{F}(n))$. 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 qdeformed free group factors $\Gamma_{q}(\mathbb{R}^{n})$ 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 wellapproximated by the matricial transport maps given by free monotone transport.
 Publication:

arXiv eprints
 Pub Date:
 April 2012
 DOI:
 10.48550/arXiv.1204.2182
 arXiv:
 arXiv:1204.2182
 Bibcode:
 2012arXiv1204.2182G
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  Probability;
 46L54
 EPrint:
 More corrections of typos as suggested by referees and a simplified proof of Lemma 3.4