Lower estimates on microstates free entropy dimension

Shlyakhtenko, Dimitri

doi:10.2140/apde.2009.2.119

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Lower estimates on microstates free entropy dimension

Dimitri Shlyakhtenko

Vol. 2 (2009), No. 2, 119–146

DOI: 10.2140/apde.2009.2.119

Abstract

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 $X_{1}, \dots, X_{n}$ . In particular, we show that $δ_{0} (X_{1}, \dots, X_{n}) \geq {dim}_{M \bar{\otimes} M^{o}} V$ , where $M = W^{*} (X_{1}, \dots, X_{n})$ and $V = {(\partial (X_{1}), \dots, \partial (X_{n})) : \partial \in C}$ is the set of values of derivations $A = ℂ [X_{1}, \dots X_{n}] \to A \otimes A$ with the property that $\partial^{*} \partial (A) \subset A$ . We show that for $q$ sufficiently small (depending on $n$ ) and $X_{1}, \dots, X_{n}$ a $q$ -semicircular family, $δ_{0} (X_{1}, \dots, X_{n}) > 1$ . 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).

Keywords

free stochastic calculus, free probability, von Neumann algebras, $q$-semicircular elements

Mathematical Subject Classification 2000

Primary: 46L54

Milestones

Received: 10 January 2008

Revised: 10 November 2008

Accepted: 24 March 2009

Published: 1 May 2009

Authors

Dimitri Shlyakhtenko
Department of Mathematics University of California, Los Angeles Los Angeles, CA 90095 United States

Vol. 2, No. 2, 2009