Vol. 2, No. 2, 2009

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
Lower estimates on microstates free entropy dimension

Dimitri Shlyakhtenko

Vol. 2 (2009), No. 2, 119–146
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 ${\delta }_{0}\left({X}_{1},\dots ,{X}_{n}\right)\ge {dim}_{M\overline{\otimes }{M}^{o}}V$, where $M={W}^{\ast }\left({X}_{1},\dots ,{X}_{n}\right)$ and $V=\left\{\left(\partial \left({X}_{1}\right),\dots ,\partial \left({X}_{n}\right)\right):\partial \in \mathsc{C}\right\}$ is the set of values of derivations $A=ℂ\left[{X}_{1},\dots {X}_{n}\right]\to A\otimes A$ with the property that ${\partial }^{\ast }\partial \left(A\right)\subset A$. We show that for $q$ sufficiently small (depending on $n$) and ${X}_{1},\dots ,{X}_{n}$ a $q$-semicircular family, ${\delta }_{0}\left({X}_{1},\dots ,{X}_{n}\right)>1$. 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
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