Vol. 2, No. 2, 2009

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Lower estimates on microstates free entropy dimension

Dimitri Shlyakhtenko

Vol. 2 (2009), No. 2, 119–146

By proving that certain free stochastic differential equations with analytic coefficients have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain n-tuples X1,,Xn. In particular, we show that δ0(X1,,Xn) dimM¯MoV , where M = W(X1,,Xn) and V ={((X1),,(Xn)) : C} is the set of values of derivations A = [X1,Xn] A A with the property that (A) A. We show that for q sufficiently small (depending on n) and X1,,Xn a q-semicircular family, δ0(X1,,Xn) > 1. In particular, for small q, q-deformed free group factors have no Cartan subalgebras. An essential tool in our analysis is a free analog of an inequality between Wasserstein distance and Fisher information introduced by Otto and Villani (and also studied in the free case by Biane and Voiculescu).

free stochastic calculus, free probability, von Neumann algebras, $q$-semicircular elements
Mathematical Subject Classification 2000
Primary: 46L54
Received: 10 January 2008
Revised: 10 November 2008
Accepted: 24 March 2009
Published: 1 May 2009
Dimitri Shlyakhtenko
Department of Mathematics
University of California, Los Angeles
Los Angeles, CA 90095
United States