2010/06/22 by Yoann Dabrowski, Dabrowski, Yoann · 5 citations
Economics, Econometrics and Finance · Mathematics · #46L54 (Primary) 46L10 46L57 (Secondary) #Advanced Operator Algebra Research #Algebra over a field #Applied mathematics #Commutative property #Computer science #Differential equation #Discrete mathematics #FOS: Mathematics #Free probability #Free space #Geometric Analysis and Curvature Flows #Markov chain #Mathematical analysis #Mathematical and Theoretical Analysis #Mathematics #Operator Algebras (math.OA) #Ordinary differential equation #Path (computing) #Physics #Pure mathematics #Random Matrices and Applications #Rigidity (electromagnetism) #Semigroup #Space (punctuation) #Stochastic differential equation #Stochastic processes and financial applications #Subalgebra #Uniqueness #Von Neumann architecture #advanced mathematical theories #math.OA #msc:46L10 #msc:46L54 #msc:46L57
paper · pdf · doi:10.48550/arxiv.1006.4351
published in arXiv (Cornell University) (Cornell University) · 75 pages; new results : more resolutions of SDEs from our dilations, free Talagrand inequality generalized to relative case; (slightly) improved exposition in section 2, typos corrected
openalex publication_date 2010/06/22 · arxiv created 2010/09/25 · arxiv updated 2010/09/28 · openalex created_date 2022/09/04 · openalex updated_date 2026/08/06
By defining tracial states on a non-commutative analogue of a path space, we\nconstruct Markov dilations for a class of conservative completely Markov\nsemigroups on finite von Neumann algebras. This class includes all symmetric\nsemigroups. For well chosen semigroups (for instance with generator any\ndivergence form operator associated to a derivation valued in the coarse\ncorrespondence) those dilations give rise to stationary solutions of certain\nfree SDEs previously considered by D. Shlyakhtenko. Among applications, we\nprove a non-commutative Talagrand inequality for non-microstates free entropy\n(relative to a subalgebra B and a completely positive map \η:B\→ B). We also\nuse those new deformations in conjunction with Popa's deformation/rigidity\ntechniques. For instance, combining our results with techniques of Popa-Ozawa\nand Peterson, we prove that the von Neumann algebra of a countable discrete\ngroup with CMAP and positive first L2 Betti number has no Cartan subalgebras.\n