The free entropy dimension of hyperfinite von Neumann algebras

Abstract

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