Some estimates for non-microstates free entropy dimension, with applications to $q$-semicircular families

Abstract

We give an general estimate for the non-microstates free entropy dimension ${\delta }^{\ast }(X_1,...,X_n)$. If $X_1,...,X_n$ generate a diffuse von Neumann algebra, we prove that ${\delta }^{\ast }(X_1,...,X_n)\ge 1$. In the case that $X_1,...,X_n$ are $q$-semicircular variables as introduced by Bozejko and Speicher and $q^{2}n<1$, we show that ${\delta }^{\ast }(X_1,...,X_n)>1$. We also show that for $|q|<\sqrt{2}-1$, the von Neumann algebras generated by a finite family of $q$-Gaussian random variables satisfy a condition of Ozawa and are therefore solid: the relative commutant of any diffuse subalgebra must be hyperfinite. In particular, when these algebras are factors, they are prime and do not have property $\Gamma$.

Cited by in corpus (1)

• L^2-Homology for von Neumann Algebras