Free monotone transport

Abstract

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 [subscript 1],…,Z [subscript n] satisfies a regularity condition (its conjugate variables ξ [subscript 1],…,ξ [subscript n] should be analytic in Z [subscript 1],…,Z [subscript n] and ξ[subscript j] should be close to Z [subscript j] in a certain analytic norm), then there exist invertible non-commutative functions F [subscript j] of an n-tuple of semicircular variables S [subscript 1],…,S [subscript n], so that Z [subscript j] =F [subscript j] (S [subscript 1],…,S [subscript n] ). Moreover, F [subscript 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[superscript ∗](Z[subscript 1],…,Z [subscript n] )≅C[superscript ∗](S [subscript 1],…,S [subscript n] ) and W[superscript ∗](Z[subscript 1],…,Z[subscript 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[subscript q](R[superscript 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.