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Free monotone transport

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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 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 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 \(W^{*}(Z_{1},\dots,Z_{n})\cong L(\mathbb{F}(n))\). 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 \(\varGamma_{q}(\mathbb{R}^{n})\) 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.

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Acknowledgements

The authors are grateful to Yoann Dabrowski for many useful comments and discussions. We also wish to mention that the idea of looking for a free analog of optimal transport (although via some duality arguments) was considered some 10 years ago by Cédric Villani and the authors; although the precise connection between optimal and monotone transport is still missing for the moment, such considerations have been an inspiration for the present work. We also thank the anonymous referees for their numerous comments which helped us to greatly improve our article.

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Authors and Affiliations

  1. Department of Mathematics, MIT, Cambridge, MA, 02139-4307, USA

    A. Guionnet

  2. CNRS & École Normale Supérieure de Lyon, 69342, Lyon Cedex 07, France

    A. Guionnet

  3. Department of Mathematics, UCLA, Los Angeles, CA, 90095-1555, USA

    D. Shlyakhtenko

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  1. A. Guionnet
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  2. D. Shlyakhtenko
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Correspondence to D. Shlyakhtenko.

Additional information

Research of A. Guionnet was supported by ANR-08-BLAN-0311-01 and Simons foundation.

Research of D. Shlyakhtenko was supported by NSF grants DMS-0900776 and DMS-1161411 and DARPA HR0011-12-1-0009.

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Guionnet, A., Shlyakhtenko, D. Free monotone transport. Invent. math. 197, 613–661 (2014). https://doi.org/10.1007/s00222-013-0493-9

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