The free entropy dimension of hyperfinite von Neumann algebras

Abstract

Suppose M is a hyperfinite von Neumann algebra with a tracial state $\phi$ and $\{a_1,...,a_n\}$ is a set of selfadjoint generators for M. We calculate $\delta_0(a_1,...,a_n)$, the modified free entropy dimension of $\{a_1,...,a_n\}$. Moreover we show that $\delta_0(a_1,...,a_n)$ depends only on M and $\phi$. Consequently $\delta_0(a_1,...,a_n)$ is independent of the choice of generators for M. In the course of the argument we show that if $\{b_1,...,b_n\}$ is a set of selfadjoint generators for a von Neumann algebra R with a tracial state and $\{b_1,...,b_n\}$ has finite dimensional approximants, then for any $b\in R$ $\delta_0(b_1,...,b_n)\geq \delta_0(b)$. Combined with a result by Voiculescu this implies that if R has a regular diffuse hyperfinite von Neumann subalgebra, then $\delta_0(b_1,...,b_n)=1$.