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Free monotone transport

  • Author(s): Guionnet, A
  • Shlyakhtenko, D
  • et al.
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 Z1,..., Zn satisfies a regularity condition (its conjugate variables ξ1,..., ξn should be analytic in Z1,..., Zn and ξj should be close to Zj in a certain analytic norm), then there exist invertible non-commutative functions Fj of an n-tuple of semicircular variables S1,..., Sn, so that Zj=Fj(S1,..., Sn). Moreover, Fj can be chosen to be monotone, in the sense that [InlineEquation not available: see fulltext.] and g is a non-commutative function with a positive definite Hessian. In particular, we can deduce that C*(Z1,..., Zn)≅C*(S1,..., Sn) and {Mathematical expression}. 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 {Mathematical expression} 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. © 2013 Springer-Verlag Berlin Heidelberg.

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