#### 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
##### 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