Lower estimates on microstates free entropy dimension

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 $V=\left\{\left(\partial \left(X_1\right),\dots ,\partial \left(X_n\right)\right):\partial \in \mathcal{C}\right\}$ 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).