The free entropy dimension of hyperfinite von Neumann algebras
Abstract
Description
Suppose M is a hyperfinite von Neumann algebra with a tracial state $ϕ$ and $\{a_1,...,a_n\}$ is a set of selfadjoint generators for M. We calculate $δ_0(a_1,...,a_n)$, the modified free entropy dimension of $\{a_1,...,a_n\}$. Moreover we show that $δ_0(a_1,...,a_n)$ depends only on M and $ϕ$. Consequently $δ_0(a_1,...,a_n)$ 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 $b\in R$ $δ_0(b_1,...,b_n)\geq δ_0(b)$. Combined with a result by Voiculescu this implies that if R has a regular diffuse hyperfinite von Neumann subalgebra, then $δ_0(b_1,...,b_n)=1$.
34 pages, minor corrections
34 pages, minor corrections