Free monotone transport

By solving a free analog of the Monge-AmpSre 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),aEuro broken vertical bar,Z (n) satisfies a regularity condition (its conjugate variables xi (1),aEuro broken vertical bar,xi (n) should be analytic in Z (1),aEuro broken vertical bar,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),aEuro broken vertical bar,S (n) , so that Z (j) =F (j) (S (1),aEuro broken vertical bar,S (n) ). Moreover, F (j) can be chosen to be monotone, in the sense that and g is a non-commutative function with a positive definite Hessian. In particular, we can deduce that C (au)(Z (1),aEuro broken vertical bar,Z (n) )a parts per thousand...C (au)(S (1),aEuro broken vertical bar,S (n) ) and . 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 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.