Lower estimates on microstates free entropy dimension

TL;DR

This paper establishes lower bounds on the microstates free entropy dimension for certain non-commutative variables using free stochastic differential equations, with implications for the structure of q-deformed free group factors.

Contribution

It introduces a novel method using free stochastic differential equations to estimate free entropy dimension, especially for q-semicircular families, and shows these factors lack Cartan subalgebras for small q.

Findings

Lower bounds on free entropy dimension for specific n-tuples.

q-semicircular families have free entropy dimension greater than 1 for small q.

q-deformed free group factors have no Cartan subalgebras for small q.

Abstract

By proving that certain free stochastic differential equations have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain $n$-tuples $X_{1},...,X_{n}$: we show that 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},...,X_{n}. In particular, we show that \delta_{0}(X_{1},...,X_{n})\geq\dim_{M\bar{\otimes}M^{o}}V where M=W^{*}(X_{1},...,X_{n}) and V=\{(\partial(X_{1}),...,\partial(X_{n})):\partial\in\mathcal{C}\} is the set of values of derivations A=\mathbb{C}[X_{1},... 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},...,X_{n} a q-semicircular family, \delta_{0}(X_{1},...,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).