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

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

2003-08-10

arxiv.org/abs/math/0308093v1

Mathematics

Operator Algebras

Scientific paper

We give an general estimate for the non-microstates free entropy dimension $\delta ^{*}(X_{1},..., X_{n})$. If $X_{1},..., X_{n}$ generate a diffuse von Neumann algebra, we prove that $\delta ^{*}(X_{1},..., X_{n})\geq 1$. 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 $\delta ^{*}(X_{1},..., X_{n})>1$. 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 $.

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