Free monotone transport
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]...
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| Format: | Article |
| Language: | en_US |
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Springer-Verlag Berlin Heidelberg
2015
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| Online Access: | http://hdl.handle.net/1721.1/92887 https://orcid.org/0000-0003-4524-8627 |
| _version_ | 1826210108639543296 |
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| author | Guionnet, Alice Shlyakhtenko, D. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Guionnet, Alice Shlyakhtenko, D. |
| author_sort | Guionnet, Alice |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T14:43:24Z |
| format | Article |
| id | mit-1721.1/92887 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T14:43:24Z |
| publishDate | 2015 |
| publisher | Springer-Verlag Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/928872022-09-29T10:12:30Z Free monotone transport Guionnet, Alice Shlyakhtenko, D. Massachusetts Institute of Technology. Department of Mathematics Guionnet, Alice 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. France. Agence nationale de la recherche (ANR-08-BLAN-0311-01) Simons Foundation National Science Foundation (U.S.) (NSF grant DMS-0900776) National Science Foundation (U.S.) (Grant DMS-1161411) United States. Defense Advanced Research Projects Agency (DARPA HR0011-12-1-0009) 2015-01-15T17:23:51Z 2015-01-15T17:23:51Z 2013-11 2013-04 Article http://purl.org/eprint/type/JournalArticle 0020-9910 1432-1297 http://hdl.handle.net/1721.1/92887 Guionnet, A., and D. Shlyakhtenko. “Free Monotone Transport.” Invent. Math. 197, no. 3 (November 13, 2013): 613–661. https://orcid.org/0000-0003-4524-8627 en_US http://dx.doi.org/10.1007/s00222-013-0493-9 Inventiones mathematicae Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Springer-Verlag Berlin Heidelberg arXiv |
| spellingShingle | Guionnet, Alice Shlyakhtenko, D. Free monotone transport |
| title | Free monotone transport |
| title_full | Free monotone transport |
| title_fullStr | Free monotone transport |
| title_full_unstemmed | Free monotone transport |
| title_short | Free monotone transport |
| title_sort | free monotone transport |
| url | http://hdl.handle.net/1721.1/92887 https://orcid.org/0000-0003-4524-8627 |
| work_keys_str_mv | AT guionnetalice freemonotonetransport AT shlyakhtenkod freemonotonetransport |