The free entropy dimension of hyperfinite von Neumann algebras

Suppose M is a hyperfinite von Neumann algebra with a tracial state $\varphi$ and $\{a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_n\}$ is a set of selfadjoint generators for M. We calculate ${\delta }_0(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_n)$, the modified free entropy dimension of $\{a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_n\}$. Moreover we show that ${\delta }_0(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_n)$ depends only on M and $\varphi$. Consequently ${\delta }_0(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_n)$ is independent of the choice of generators for M. In the course of the argument we show that if $\{b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_n\}$ is a set of selfadjoint generators for a von Neumann algebra R with a tracial state and $\{b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_n\}$ has finite dimensional approximants, then for any $b\in R$ ${\delta }_0(b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_n)\ge {\delta }_0(b)$. Combined with a result by Voiculescu this implies that if R has a regular diffuse hyperfinite von Neumann subalgebra, then ${\delta }_0(b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_n)=1$.