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 Z_1,..., Z_n satisfies a regularity condition (its conjugate variables ξ_1,..., ξ_n should be analytic in Z_1,..., Z_n and ξ_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 = D_jg and g is a non-commutative function with a positive definite Hessian. In particular, we can deduce that C~?(Z_1,..., Z_n)～= C~?(S_1,..., S_n) and W~?(Z_1,..., Z_n)= L(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 Γ_q (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.
展开▼
