# The free entropy dimension of hyperfinite von Neumann algebras

Research paper by **Kenley Jung**

Indexed on: **13 Sep '03**Published on: **13 Sep '03**Published in: **Mathematics - Operator Algebras**

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#### 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 $\delta_0(b_1,...,b_n)=1$.