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 δ∗(X1, . . . , Xn). If X1, . . . , Xn generate a diffuse von Neumann algebra, we prove that δ∗(X1, . . . , Xn) ≥ 1. In the case that X1, . . . , Xn are q-semicircular variables as introduced by Bozejko and Speicher and qn < 1, we show that δ∗(X1, . . . , Xn) > 1. We also show that for |q| < √ 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 Γ.