  By solving a free analog of the Monge-Amp\`ere equation, we prove a
non-commutative analog of Brenier's monotone transport theorem: if an $n$-tuple
of self-adjoint non-commutative 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 non-commutative 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 non-commutative
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 q-deformed 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 well-approximated by the matricial transport maps given by free monotone
transport.
