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

We give an general estimate for the non-microstates free entropy dimension ${\delta }^{\ast }(X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n)$. If $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n$ generate a diffuse von Neumann algebra, we prove that ${\delta }^{\ast }(X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n)\ge 1$. In the case that $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n$ are $q$-semicircular variables as introduced by Bozejko and Speicher and $q^{2}n<1$, we show that $\delta ^{*}(X_{1},…, X_{n})>1$\mathrm{.}Wealsoshowthatfor$|q|<\sqrt{2}-1$, 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 $\mathrm{\Gamma }$.