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The free entropy dimension of hyperfinite von Neumann algebras

Authors:	Kenley Jung

Institution:	Department of Mathematics, University of California, Berkeley, California 94720-3840

Abstract:	Suppose 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 . 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 and $\varphi$ . Consequently, $\delta_0(a_1,\ldots,a_n)$ is independent of the choice of generators for . 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 of $\mathcal R.$ Combined with a result by Voiculescu, this implies that if $\mathcal R$ has a regular diffuse hyperfinite von Neumann subalgebra, then $\delta_0(b_1,\ldots,b_n)=1$ .

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